WEBVTT 00:00:00.000 --> 00:00:00.500 align:middle line:90% 00:00:00.500 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:03.476 align:middle line:90% [RUSTLING] 00:00:03.476 --> 00:00:05.460 align:middle line:90% [CLICKING] 00:00:05.460 --> 00:00:13.825 align:middle line:90% 00:00:13.825 --> 00:00:14.700 align:middle line:90% PHILLIP ISOLA: Great. 00:00:14.700 --> 00:00:16.360 align:middle line:90% So welcome. 00:00:16.360 --> 00:00:19.680 align:middle line:84% This lecture is going to be on generalization. 00:00:19.680 --> 00:00:22.240 align:middle line:84% So this is the next topic in the series of how 00:00:22.240 --> 00:00:23.580 align:middle line:90% do you approximate functions? 00:00:23.580 --> 00:00:26.040 align:middle line:90% Now do they actually generalize? 00:00:26.040 --> 00:00:30.180 align:middle line:84% And the basic question is, do neural networks generalize? 00:00:30.180 --> 00:00:33.220 align:middle line:84% And well, we'll see, yes, the answer is yes to some degree. 00:00:33.220 --> 00:00:34.780 align:middle line:90% And then the question is why? 00:00:34.780 --> 00:00:36.020 align:middle line:90% Why do they generalize? 00:00:36.020 --> 00:00:39.200 align:middle line:90% 00:00:39.200 --> 00:00:40.940 align:middle line:90% A few logistical reminders. 00:00:40.940 --> 00:00:43.240 align:middle line:90% Problem set one is due today. 00:00:43.240 --> 00:00:46.240 align:middle line:90% Problem set two goes out today. 00:00:46.240 --> 00:00:49.060 align:middle line:84% Problem set two will be posted sometime after this class. 00:00:49.060 --> 00:00:53.640 align:middle line:84% So we always will post problem sets after the lecture. 00:00:53.640 --> 00:00:57.200 align:middle line:84% And you might want to look at the first question of problem 00:00:57.200 --> 00:01:00.570 align:middle line:84% set two if you happen to have time before Thursday's lecture, 00:01:00.570 --> 00:01:03.630 align:middle line:84% because it can help motivate-- you don't have to solve it, 00:01:03.630 --> 00:01:05.610 align:middle line:84% but it can help motivate and put into context 00:01:05.610 --> 00:01:07.130 align:middle line:90% what we'll see on Thursday. 00:01:07.130 --> 00:01:10.770 align:middle line:84% So I'd recommend taking a look at that first problem. 00:01:10.770 --> 00:01:12.570 align:middle line:84% And then you'll kind of know what 00:01:12.570 --> 00:01:15.530 align:middle line:84% to look out for in the lecture to help you solve that problem. 00:01:15.530 --> 00:01:18.710 align:middle line:84% So just a little piece of advice. 00:01:18.710 --> 00:01:19.710 align:middle line:90% And I think that's it. 00:01:19.710 --> 00:01:21.210 align:middle line:90% So we're going to go right in. 00:01:21.210 --> 00:01:24.870 align:middle line:84% So the outline for today will be, what is generalization? 00:01:24.870 --> 00:01:25.970 align:middle line:90% That should be review. 00:01:25.970 --> 00:01:28.530 align:middle line:84% You've all taken machine learning or classes like this 00:01:28.530 --> 00:01:30.390 align:middle line:90% before. 00:01:30.390 --> 00:01:34.410 align:middle line:84% Then, the next question is, do deep nets generalize? 00:01:34.410 --> 00:01:36.650 align:middle line:84% Then, we'll see that there's some classical theories 00:01:36.650 --> 00:01:39.450 align:middle line:84% of generalization, and they don't actually work very well 00:01:39.450 --> 00:01:40.630 align:middle line:90% for deep learning. 00:01:40.630 --> 00:01:43.850 align:middle line:84% So we'll see some failures of classical generalization theory 00:01:43.850 --> 00:01:46.970 align:middle line:84% at explaining phenomena in deep learning. 00:01:46.970 --> 00:01:49.210 align:middle line:84% And then I've already given away the answer-- 00:01:49.210 --> 00:01:52.070 align:middle line:84% that, to some degree, deep nets, will argue, do generalize. 00:01:52.070 --> 00:01:54.370 align:middle line:90% So why do they generalize? 00:01:54.370 --> 00:01:57.830 align:middle line:84% What does this tell us about their inductive biases 00:01:57.830 --> 00:02:01.750 align:middle line:84% and preferences toward solutions that will generalize over 00:02:01.750 --> 00:02:06.590 align:middle line:84% solutions that simply fit the data but don't generalize? 00:02:06.590 --> 00:02:08.830 align:middle line:90% So that's the outline. 00:02:08.830 --> 00:02:12.870 align:middle line:84% And I should also say that the material in this lecture 00:02:12.870 --> 00:02:14.950 align:middle line:90% is not really textbook material. 00:02:14.950 --> 00:02:17.450 align:middle line:84% We gave a few readings, but there are kind of recent papers, 00:02:17.450 --> 00:02:19.870 align:middle line:84% so we just listed them as optional. 00:02:19.870 --> 00:02:22.950 align:middle line:84% This is very much ongoing science. 00:02:22.950 --> 00:02:25.590 align:middle line:84% This is, to me, maybe kind of the biggest, 00:02:25.590 --> 00:02:28.590 align:middle line:84% or one of the biggest open sets of questions 00:02:28.590 --> 00:02:31.390 align:middle line:84% of topics of inquiry in the field of deep learning 00:02:31.390 --> 00:02:34.790 align:middle line:84% and in AI in general, is exactly what are the generalization 00:02:34.790 --> 00:02:36.348 align:middle line:90% properties of deep networks? 00:02:36.348 --> 00:02:38.390 align:middle line:84% They seem to work a lot better than we would have 00:02:38.390 --> 00:02:40.830 align:middle line:90% expected from classical theory. 00:02:40.830 --> 00:02:41.370 align:middle line:90% So why? 00:02:41.370 --> 00:02:42.410 align:middle line:90% We don't actually know. 00:02:42.410 --> 00:02:43.410 align:middle line:90% We don't have all the answers. 00:02:43.410 --> 00:02:45.160 align:middle line:84% This lecture will just pose some questions 00:02:45.160 --> 00:02:47.970 align:middle line:84% and give some initial threads that you could pull on. 00:02:47.970 --> 00:02:49.470 align:middle line:84% But this is going to be a topic that 00:02:49.470 --> 00:02:51.345 align:middle line:84% will change a lot over the next coming years. 00:02:51.345 --> 00:02:54.650 align:middle line:90% I think this will develop a lot. 00:02:54.650 --> 00:02:57.160 align:middle line:84% So let's go over the basics of approximation 00:02:57.160 --> 00:02:58.300 align:middle line:90% versus generalization. 00:02:58.300 --> 00:03:00.280 align:middle line:84% We've so far talked in this course really only 00:03:00.280 --> 00:03:01.920 align:middle line:90% about approximation. 00:03:01.920 --> 00:03:04.900 align:middle line:84% So approximation is, I have some training data. 00:03:04.900 --> 00:03:07.880 align:middle line:84% I want to ask how well can I fit that training data. 00:03:07.880 --> 00:03:11.000 align:middle line:84% And we measure the fit with the empirical risk, which 00:03:11.000 --> 00:03:13.020 align:middle line:90% is the first equation up here. 00:03:13.020 --> 00:03:15.440 align:middle line:84% So we just say, what is the loss L 00:03:15.440 --> 00:03:19.092 align:middle line:84% incurred over all of my training data points? 00:03:19.092 --> 00:03:20.800 align:middle line:84% This is a supervised setting where I have 00:03:20.800 --> 00:03:23.060 align:middle line:90% data points x and outputs y. 00:03:23.060 --> 00:03:26.020 align:middle line:84% I'm trying to predict the correct output y for a given x. 00:03:26.020 --> 00:03:28.120 align:middle line:84% So I just sum up all of my prediction errors 00:03:28.120 --> 00:03:30.240 align:middle line:84% according to my model f and my loss 00:03:30.240 --> 00:03:34.200 align:middle line:84% function L. So this is just, on average, 00:03:34.200 --> 00:03:37.040 align:middle line:90% what is my prediction error. 00:03:37.040 --> 00:03:39.480 align:middle line:84% And we had a few lectures, and in the problems 00:03:39.480 --> 00:03:41.120 align:middle line:84% that you worked out, cases where you 00:03:41.120 --> 00:03:46.600 align:middle line:84% can find that class of functions can minimize that towards 0 00:03:46.600 --> 00:03:50.480 align:middle line:84% with sufficient capacity, with sufficient width or depth. 00:03:50.480 --> 00:03:53.220 align:middle line:84% But that's not actually what we care about in machine learning. 00:03:53.220 --> 00:03:56.530 align:middle line:84% All we actually care about is the test error, 00:03:56.530 --> 00:03:59.090 align:middle line:84% which can be called the population risk. 00:03:59.090 --> 00:04:03.150 align:middle line:84% So the population risk is, on new samples from the world, 00:04:03.150 --> 00:04:05.970 align:middle line:84% new samples from the data generating process p. 00:04:05.970 --> 00:04:08.790 align:middle line:84% So new samples, x, y, new observations, 00:04:08.790 --> 00:04:11.410 align:middle line:90% how well do I do in expectation? 00:04:11.410 --> 00:04:13.570 align:middle line:84% And this is like the population, because it's 00:04:13.570 --> 00:04:17.610 align:middle line:84% over the entire population of observations 00:04:17.610 --> 00:04:21.370 align:middle line:84% I could get from my world's data generating process p. 00:04:21.370 --> 00:04:24.430 align:middle line:84% But you could also have an empirical test risk, 00:04:24.430 --> 00:04:26.090 align:middle line:84% which would just be over a sample, 00:04:26.090 --> 00:04:28.310 align:middle line:84% like my test set is a sample of 10,000 images-- 00:04:28.310 --> 00:04:30.170 align:middle line:90% how well do I do on them? 00:04:30.170 --> 00:04:34.330 align:middle line:84% So we have three different kind of topics 00:04:34.330 --> 00:04:37.630 align:middle line:84% that we can factorize machine learning into. 00:04:37.630 --> 00:04:41.670 align:middle line:84% One is, how well can you approximate the training data? 00:04:41.670 --> 00:04:44.290 align:middle line:84% And that's what we've talked about so far. 00:04:44.290 --> 00:04:48.570 align:middle line:84% In my family of functions I'm considering 00:04:48.570 --> 00:04:52.980 align:middle line:84% parameterized by theta, what is the best theta 00:04:52.980 --> 00:04:56.140 align:middle line:84% star that will achieve the lowest empirical risk? 00:04:56.140 --> 00:04:58.780 align:middle line:84% And we've said that you can get it to go 00:04:58.780 --> 00:05:01.420 align:middle line:90% to 0 under some conditions. 00:05:01.420 --> 00:05:03.480 align:middle line:90% So we can fit our training data. 00:05:03.480 --> 00:05:07.200 align:middle line:84% But then the next question is optimization. 00:05:07.200 --> 00:05:09.012 align:middle line:84% How do we actually find that theta star? 00:05:09.012 --> 00:05:10.720 align:middle line:84% And we've talked a little bit about that. 00:05:10.720 --> 00:05:11.600 align:middle line:90% We saw backprop. 00:05:11.600 --> 00:05:15.500 align:middle line:84% And on Thursday's lecture, we'll talk more about optimization. 00:05:15.500 --> 00:05:19.332 align:middle line:84% But we said that you can often find theta star to some degree. 00:05:19.332 --> 00:05:21.040 align:middle line:84% And this lecture is, well, wait a second. 00:05:21.040 --> 00:05:24.340 align:middle line:84% If you find that really good parameter vector on the training 00:05:24.340 --> 00:05:26.560 align:middle line:84% data, what is the population risk? 00:05:26.560 --> 00:05:29.180 align:middle line:84% What is the risk on the test data? 00:05:29.180 --> 00:05:32.060 align:middle line:84% So what is the gap between the training error and the test 00:05:32.060 --> 00:05:32.560 align:middle line:90% error? 00:05:32.560 --> 00:05:34.000 align:middle line:84% And that's the question of generalization. 00:05:34.000 --> 00:05:35.542 align:middle line:84% And that's all we actually care about 00:05:35.542 --> 00:05:37.140 align:middle line:84% because we want to do well when we 00:05:37.140 --> 00:05:40.100 align:middle line:84% deploy the system in the world and we see new observations. 00:05:40.100 --> 00:05:41.220 align:middle line:90% This should be review. 00:05:41.220 --> 00:05:45.140 align:middle line:90% This is like ML 101. 00:05:45.140 --> 00:05:46.320 align:middle line:90% Yeah, question? 00:05:46.320 --> 00:05:48.100 align:middle line:84% AUDIENCE: If the r in approximations 00:05:48.100 --> 00:05:49.840 align:middle line:84% supposed to have a hat on it, it's 00:05:49.840 --> 00:05:51.900 align:middle line:84% the question about training data? 00:05:51.900 --> 00:05:55.455 align:middle line:84% PHILLIP ISOLA: Yeah, I think you could ask the question-- 00:05:55.455 --> 00:05:57.580 align:middle line:84% so an approximation-- yeah, typically, it would be, 00:05:57.580 --> 00:06:00.740 align:middle line:84% can I minimize the empirical risk on the training data? 00:06:00.740 --> 00:06:02.880 align:middle line:90% Yes. 00:06:02.880 --> 00:06:05.280 align:middle line:90% Oh, I see. 00:06:05.280 --> 00:06:09.160 align:middle line:84% Yeah, we could have denoted that with a hat. 00:06:09.160 --> 00:06:12.400 align:middle line:84% But I suppose you could also ask the question, 00:06:12.400 --> 00:06:15.880 align:middle line:84% is the best possible function in the hypothesis space 00:06:15.880 --> 00:06:20.473 align:middle line:84% going to achieve low risk on the population as well? 00:06:20.473 --> 00:06:23.140 align:middle line:84% So you could ask-- both of those are questions of approximation. 00:06:23.140 --> 00:06:30.620 align:middle line:90% 00:06:30.620 --> 00:06:35.480 align:middle line:84% So let's go over some intuitive ideas about generalization just 00:06:35.480 --> 00:06:40.160 align:middle line:84% to get warmed up and get you thinking about this. 00:06:40.160 --> 00:06:42.933 align:middle line:90% So here's a bad data set. 00:06:42.933 --> 00:06:44.600 align:middle line:84% Suppose we have a data set which is just 00:06:44.600 --> 00:06:47.280 align:middle line:84% a bunch of photos of cats, and we 00:06:47.280 --> 00:06:50.090 align:middle line:84% want to train a cat versus dog classifier. 00:06:50.090 --> 00:06:52.710 align:middle line:84% So clearly, this is not going to work. 00:06:52.710 --> 00:06:53.790 align:middle line:90% You only observe cats. 00:06:53.790 --> 00:06:56.130 align:middle line:84% You're not going to get a cat versus dog classifier. 00:06:56.130 --> 00:06:58.832 align:middle line:90% So what's wrong with this data? 00:06:58.832 --> 00:07:00.290 align:middle line:84% What would make for better training 00:07:00.290 --> 00:07:04.290 align:middle line:84% data for a cat versus dog classifier? 00:07:04.290 --> 00:07:05.390 align:middle line:90% Some obvious ideas. 00:07:05.390 --> 00:07:05.890 align:middle line:90% Yeah? 00:07:05.890 --> 00:07:06.980 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:07:06.980 --> 00:07:08.730 align:middle line:84% PHILLIP ISOLA: You want to have some dogs. 00:07:08.730 --> 00:07:09.950 align:middle line:90% Yeah, what else? 00:07:09.950 --> 00:07:12.250 align:middle line:84% What if you have just one cat, one dog, and then a lot 00:07:12.250 --> 00:07:15.950 align:middle line:90% of copies of that image? 00:07:15.950 --> 00:07:16.830 align:middle line:90% You want diversity. 00:07:16.830 --> 00:07:17.710 align:middle line:90% You want coverage. 00:07:17.710 --> 00:07:19.270 align:middle line:84% You want to show all your different conditions. 00:07:19.270 --> 00:07:21.510 align:middle line:84% So there are certain properties of the data that are critical. 00:07:21.510 --> 00:07:23.260 align:middle line:84% You can't really understand generalization 00:07:23.260 --> 00:07:25.410 align:middle line:84% without understanding the data distribution 00:07:25.410 --> 00:07:26.730 align:middle line:90% that you're training on. 00:07:26.730 --> 00:07:29.050 align:middle line:84% And if your training set doesn't represent 00:07:29.050 --> 00:07:30.770 align:middle line:84% all the interesting aspects of variation 00:07:30.770 --> 00:07:32.110 align:middle line:84% that you'll encounter in the world, 00:07:32.110 --> 00:07:33.830 align:middle line:84% then you're not going to generalize well. 00:07:33.830 --> 00:07:36.830 align:middle line:84% So data is critical to generalization. 00:07:36.830 --> 00:07:40.290 align:middle line:84% And we have to understand data to understand generalization. 00:07:40.290 --> 00:07:44.170 align:middle line:84% The other thing that's critical to understanding generalization 00:07:44.170 --> 00:07:46.640 align:middle line:84% is the model, which will be most of what we focus on. 00:07:46.640 --> 00:07:48.890 align:middle line:84% I just wanted to start by saying that the data is also 00:07:48.890 --> 00:07:52.880 align:middle line:84% a critical part of the recipe for generalization. 00:07:52.880 --> 00:07:56.500 align:middle line:90% So here's a bad model-- 00:07:56.500 --> 00:07:58.060 align:middle line:90% the filing cabinet. 00:07:58.060 --> 00:08:00.300 align:middle line:84% So the filing cabinet model is going 00:08:00.300 --> 00:08:02.500 align:middle line:84% to be this little Python program here, 00:08:02.500 --> 00:08:04.760 align:middle line:84% where we have a dictionary called cabinet. 00:08:04.760 --> 00:08:09.220 align:middle line:84% And if we see a new data point x, we put it into our cabinet 00:08:09.220 --> 00:08:13.060 align:middle line:84% along with the correct prediction y for that x. 00:08:13.060 --> 00:08:14.700 align:middle line:84% And if we come across that x again, 00:08:14.700 --> 00:08:16.755 align:middle line:90% we just hash into that cabinet. 00:08:16.755 --> 00:08:17.880 align:middle line:90% We just pull out that file. 00:08:17.880 --> 00:08:19.460 align:middle line:90% We see what the answer is. 00:08:19.460 --> 00:08:21.520 align:middle line:84% So this is sometimes called memorization. 00:08:21.520 --> 00:08:23.620 align:middle line:84% I'm just memorizing my training data. 00:08:23.620 --> 00:08:26.300 align:middle line:84% But the questions for you are, well, 00:08:26.300 --> 00:08:28.860 align:middle line:84% what will the approximation error be of the filing cabinet 00:08:28.860 --> 00:08:30.280 align:middle line:90% model? 00:08:30.280 --> 00:08:30.780 align:middle line:90% Yeah? 00:08:30.780 --> 00:08:31.420 align:middle line:90% AUDIENCE: 0. 00:08:31.420 --> 00:08:32.462 align:middle line:90% PHILLIP ISOLA: 0-- right. 00:08:32.462 --> 00:08:34.480 align:middle line:84% So the approximation error is zero 00:08:34.480 --> 00:08:38.059 align:middle line:84% because for every data point that I train on, 00:08:38.059 --> 00:08:40.980 align:middle line:84% I can retrieve the exact memorized answer. 00:08:40.980 --> 00:08:42.520 align:middle line:84% What about the generalization error? 00:08:42.520 --> 00:08:45.817 align:middle line:84% What will the generalization error of the filing cabinet be? 00:08:45.817 --> 00:08:47.400 align:middle line:84% Generalization error is the difference 00:08:47.400 --> 00:08:50.020 align:middle line:84% between the empirical risk and the population risk. 00:08:50.020 --> 00:08:52.600 align:middle line:90% 00:08:52.600 --> 00:08:53.880 align:middle line:90% So roughly, what will it be? 00:08:53.880 --> 00:08:55.800 align:middle line:90% Will it be good, will it be bad? 00:08:55.800 --> 00:08:56.520 align:middle line:90% AUDIENCE: Bad. 00:08:56.520 --> 00:08:57.312 align:middle line:90% PHILLIP ISOLA: Bad. 00:08:57.312 --> 00:08:59.560 align:middle line:84% Yeah, it'll generally be bad unless it happens 00:08:59.560 --> 00:09:02.320 align:middle line:84% to be that the true function we're modeling 00:09:02.320 --> 00:09:04.000 align:middle line:84% is 0 almost everywhere, which would 00:09:04.000 --> 00:09:05.680 align:middle line:90% be a very strange function. 00:09:05.680 --> 00:09:06.920 align:middle line:90% Because we said by-- 00:09:06.920 --> 00:09:09.560 align:middle line:84% if I don't have my file in the cabinet, 00:09:09.560 --> 00:09:12.020 align:middle line:90% I will just not know what to do. 00:09:12.020 --> 00:09:15.175 align:middle line:84% But maybe I got lucky, and the true function is 0 everywhere. 00:09:15.175 --> 00:09:16.800 align:middle line:84% So you could have a filing cabinet that 00:09:16.800 --> 00:09:18.160 align:middle line:84% gets low generalization error, but it 00:09:18.160 --> 00:09:20.160 align:middle line:84% would be a very strange function that you're 00:09:20.160 --> 00:09:23.080 align:middle line:90% modeling in that case. 00:09:23.080 --> 00:09:26.060 align:middle line:84% So we'll come back to the filing cabinet model. 00:09:26.060 --> 00:09:29.160 align:middle line:84% And one of the questions is, are deep nets like filing cabinets 00:09:29.160 --> 00:09:31.772 align:middle line:90% or something else? 00:09:31.772 --> 00:09:33.480 align:middle line:84% Do they just memorize and not generalize, 00:09:33.480 --> 00:09:34.780 align:middle line:90% or do they do something else? 00:09:34.780 --> 00:09:37.600 align:middle line:90% 00:09:37.600 --> 00:09:40.920 align:middle line:84% Here's another bad model just, I guess, a little bit for fun. 00:09:40.920 --> 00:09:43.030 align:middle line:90% So this is Paul the Octopus. 00:09:43.030 --> 00:09:45.130 align:middle line:84% And Paul can decide who's going to win 00:09:45.130 --> 00:09:49.330 align:middle line:84% a soccer match between two different countries. 00:09:49.330 --> 00:09:53.090 align:middle line:84% And Paul was super accurate, and people would make their bets 00:09:53.090 --> 00:09:54.230 align:middle line:90% based on what Paul said. 00:09:54.230 --> 00:09:57.930 align:middle line:84% And what Paul did is just went into the tank of the flag 00:09:57.930 --> 00:09:59.990 align:middle line:84% that it thought would win the match. 00:09:59.990 --> 00:10:03.520 align:middle line:84% And it was like it got 6 out of 6 correct predictions. 00:10:03.520 --> 00:10:05.770 align:middle line:84% So would you trust Paul to make the correct prediction 00:10:05.770 --> 00:10:07.770 align:middle line:90% next year? 00:10:07.770 --> 00:10:11.205 align:middle line:84% Yeah, it's probably like some super intelligent octopus. 00:10:11.205 --> 00:10:12.830 align:middle line:84% No, obviously, you wouldn't trust Paul. 00:10:12.830 --> 00:10:15.970 align:middle line:84% It must have just generalized-- sorry, 00:10:15.970 --> 00:10:17.350 align:middle line:90% it fit the training data well. 00:10:17.350 --> 00:10:18.510 align:middle line:90% It's not a filing cabinet. 00:10:18.510 --> 00:10:21.750 align:middle line:84% It is making some decision, but it happened to get lucky. 00:10:21.750 --> 00:10:23.370 align:middle line:84% It's like some random function that 00:10:23.370 --> 00:10:24.787 align:middle line:84% happened to fit the training data, 00:10:24.787 --> 00:10:26.410 align:middle line:84% but you'd never expect it to actually 00:10:26.410 --> 00:10:30.570 align:middle line:90% have a true model of soccer. 00:10:30.570 --> 00:10:33.610 align:middle line:84% So you can have functions that get lucky 00:10:33.610 --> 00:10:38.610 align:middle line:84% and make interesting predictions out of sample on the test data, 00:10:38.610 --> 00:10:40.190 align:middle line:90% but wrong predictions. 00:10:40.190 --> 00:10:42.940 align:middle line:90% And Paul is an example. 00:10:42.940 --> 00:10:45.860 align:middle line:84% So our big question in this course, in this lecture, 00:10:45.860 --> 00:10:48.980 align:middle line:90% is do deep nets generalize? 00:10:48.980 --> 00:10:52.660 align:middle line:90% So here's a simple warm-up. 00:10:52.660 --> 00:10:56.760 align:middle line:84% Here's the filing cabinet fit to these black data points. 00:10:56.760 --> 00:11:00.060 align:middle line:84% So my training data is the black points. 00:11:00.060 --> 00:11:01.500 align:middle line:90% It's a scalar to scalar mapping. 00:11:01.500 --> 00:11:03.700 align:middle line:90% I'm predicting y given x. 00:11:03.700 --> 00:11:07.160 align:middle line:84% And the red line is the fit of my model. 00:11:07.160 --> 00:11:10.700 align:middle line:84% And if my model is a filing cabinet, what it's saying 00:11:10.700 --> 00:11:15.340 align:middle line:84% is that I've memorized that if I see one of my training data 00:11:15.340 --> 00:11:17.050 align:middle line:84% points, I know exactly what to predict. 00:11:17.050 --> 00:11:19.300 align:middle line:84% That's the kind of red lines, like the delta functions 00:11:19.300 --> 00:11:22.380 align:middle line:84% going down to each of those dots, or up to the dots. 00:11:22.380 --> 00:11:23.960 align:middle line:84% And everywhere else, I'll predict 0. 00:11:23.960 --> 00:11:27.880 align:middle line:84% So anything that is not anything on the x-axis-- 00:11:27.880 --> 00:11:30.820 align:middle line:84% any point I could sample on the x-axis that's not in my training 00:11:30.820 --> 00:11:34.620 align:middle line:84% set that's not where the black dots lie, I will predict 0 for. 00:11:34.620 --> 00:11:38.180 align:middle line:84% So this is a very bad kind of generalization. 00:11:38.180 --> 00:11:40.270 align:middle line:84% So here's the same data, where I've just fit it 00:11:40.270 --> 00:11:43.890 align:middle line:90% with a three layer ReLU MLP. 00:11:43.890 --> 00:11:45.230 align:middle line:90% Not too surprising. 00:11:45.230 --> 00:11:47.510 align:middle line:84% A ReLU MLP can fit data, but notice that it fits it 00:11:47.510 --> 00:11:49.070 align:middle line:90% in a very different way. 00:11:49.070 --> 00:11:51.303 align:middle line:84% So both of these achieve 0 approximation error. 00:11:51.303 --> 00:11:52.970 align:middle line:84% Both of them have fit the training data. 00:11:52.970 --> 00:11:54.428 align:middle line:84% But if that were all that mattered, 00:11:54.428 --> 00:11:56.637 align:middle line:84% then we would have to say these are equally good. 00:11:56.637 --> 00:11:57.970 align:middle line:90% But that's not all that matters. 00:11:57.970 --> 00:12:00.750 align:middle line:84% It also matters how you interpolate and extrapolate 00:12:00.750 --> 00:12:02.030 align:middle line:90% from the training data. 00:12:02.030 --> 00:12:03.790 align:middle line:84% And deep nets do that in a way that's 00:12:03.790 --> 00:12:07.510 align:middle line:84% markedly different than a filing cabinet. 00:12:07.510 --> 00:12:10.850 align:middle line:84% So memorization is just, what do I predict on the training data 00:12:10.850 --> 00:12:11.350 align:middle line:90% points? 00:12:11.350 --> 00:12:13.050 align:middle line:84% And filing cabinets can memorize-- 00:12:13.050 --> 00:12:14.270 align:middle line:90% deep nets can memorize. 00:12:14.270 --> 00:12:16.830 align:middle line:84% And generalization is, what do I do on the points 00:12:16.830 --> 00:12:18.430 align:middle line:84% where I didn't have training data? 00:12:18.430 --> 00:12:21.930 align:middle line:84% And we might expect that this type of interpolation, 00:12:21.930 --> 00:12:24.890 align:middle line:84% this type of generalization is better than that other type, 00:12:24.890 --> 00:12:26.630 align:middle line:84% because most functions in the world 00:12:26.630 --> 00:12:31.702 align:middle line:84% are going to be smooth as opposed to spiky. 00:12:31.702 --> 00:12:34.410 align:middle line:84% You could imagine an adversarial world where that's not the case, 00:12:34.410 --> 00:12:37.710 align:middle line:84% but our world looks more like this on the right. 00:12:37.710 --> 00:12:41.990 align:middle line:90% 00:12:41.990 --> 00:12:43.910 align:middle line:84% So that's a very simple deep net. 00:12:43.910 --> 00:12:45.810 align:middle line:84% But the real feats of generalization 00:12:45.810 --> 00:12:47.570 align:middle line:84% that we get excited about these days 00:12:47.570 --> 00:12:51.250 align:middle line:90% are with fancy, big modern nets. 00:12:51.250 --> 00:12:55.670 align:middle line:84% So Here's a little experiment you can do. 00:12:55.670 --> 00:12:58.818 align:middle line:84% I just ran this for myself this weekend, 00:12:58.818 --> 00:13:00.610 align:middle line:84% but I think it might be a fun thing for you 00:13:00.610 --> 00:13:03.030 align:middle line:84% to play with or maybe take into a final project. 00:13:03.030 --> 00:13:07.970 align:middle line:84% So let's try to ask, are modern deep nets-- 00:13:07.970 --> 00:13:10.148 align:middle line:84% the best model you can think of-- 00:13:10.148 --> 00:13:11.690 align:middle line:84% are they acting like a filing cabinet 00:13:11.690 --> 00:13:14.090 align:middle line:84% or are they doing something more? 00:13:14.090 --> 00:13:17.290 align:middle line:84% It's a little bit hard to say, because if you train on enough 00:13:17.290 --> 00:13:19.930 align:middle line:84% data-- and these are trained on billions or trillions of data 00:13:19.930 --> 00:13:22.290 align:middle line:84% points-- if you trained on enough data and you just 00:13:22.290 --> 00:13:26.170 align:middle line:84% memorize the data, well, your test conditions might look 00:13:26.170 --> 00:13:27.930 align:middle line:90% identical to your-- 00:13:27.930 --> 00:13:29.670 align:middle line:84% if I've trained on infinite things, 00:13:29.670 --> 00:13:32.930 align:middle line:84% then my test cases are just the same as my training data. 00:13:32.930 --> 00:13:36.380 align:middle line:84% So are they close to that-- just memorizing nearly infinite 00:13:36.380 --> 00:13:38.660 align:middle line:90% things, or are they not? 00:13:38.660 --> 00:13:41.580 align:middle line:84% So LLM-- if you don't know, this is Large Language Model-- 00:13:41.580 --> 00:13:46.620 align:middle line:84% ChatGPT, latest greatest deep nets are in that family. 00:13:46.620 --> 00:13:50.800 align:middle line:84% So big model-- so is it acting like a filing cabinet? 00:13:50.800 --> 00:13:54.140 align:middle line:84% Can we actually measure if it's generalizing or not? 00:13:54.140 --> 00:13:56.620 align:middle line:84% So here's the question we can ask. 00:13:56.620 --> 00:13:58.380 align:middle line:84% How big would the filing cabinet have 00:13:58.380 --> 00:14:02.340 align:middle line:84% to be in order to do the feats of prediction 00:14:02.340 --> 00:14:07.300 align:middle line:84% that ChatGPT or some big LLM can do? 00:14:07.300 --> 00:14:09.420 align:middle line:84% So remember our model of the filing cabinet 00:14:09.420 --> 00:14:12.100 align:middle line:84% is just, we return 0 if we have a data point that we've never 00:14:12.100 --> 00:14:14.340 align:middle line:90% seen before. 00:14:14.340 --> 00:14:16.280 align:middle line:84% So here's just a simple counting experiment. 00:14:16.280 --> 00:14:18.655 align:middle line:84% I think you can make more sophisticated versions to this, 00:14:18.655 --> 00:14:20.960 align:middle line:90% but I found it fun. 00:14:20.960 --> 00:14:23.980 align:middle line:90% So we'll sample, a sequence-- 00:14:23.980 --> 00:14:27.140 align:middle line:84% so language models are deep nets that take sequences of words 00:14:27.140 --> 00:14:29.580 align:middle line:84% as input and make some prediction from them. 00:14:29.580 --> 00:14:33.100 align:middle line:84% So we'll sample a sequence of n words, 00:14:33.100 --> 00:14:37.850 align:middle line:84% and we'll be sampling from a vocab of size m. 00:14:37.850 --> 00:14:39.170 align:middle line:90% So we have m nouns-- 00:14:39.170 --> 00:14:42.150 align:middle line:84% 1,000 different nouns, and we make a sentence of length-- 00:14:42.150 --> 00:14:44.450 align:middle line:84% or a set of a sequence of nouns of length 5. 00:14:44.450 --> 00:14:46.190 align:middle line:90% So then n would be 5. 00:14:46.190 --> 00:14:50.790 align:middle line:84% So how many unique input sequences are there? 00:14:50.790 --> 00:14:55.190 align:middle line:84% So just simple combinatorics, m to the n. 00:14:55.190 --> 00:14:58.650 align:middle line:84% So I have m possible words, and I have n of them in a row. 00:14:58.650 --> 00:15:00.550 align:middle line:90% I sample independently each one. 00:15:00.550 --> 00:15:02.910 align:middle line:84% That's the number of such sequences. 00:15:02.910 --> 00:15:06.630 align:middle line:84% So now, we can ask, well, if we take a random such sequence 00:15:06.630 --> 00:15:08.910 align:middle line:84% and we input it into our model, did the model 00:15:08.910 --> 00:15:10.352 align:middle line:90% make a reasonable prediction? 00:15:10.352 --> 00:15:12.810 align:middle line:84% So technically, if this is our model of the filing cabinet, 00:15:12.810 --> 00:15:15.018 align:middle line:84% we're just asking, did it make a non-zero prediction? 00:15:15.018 --> 00:15:18.618 align:middle line:84% But did it output a correct prediction? 00:15:18.618 --> 00:15:20.410 align:middle line:84% So we can take our favorite language model. 00:15:20.410 --> 00:15:24.310 align:middle line:84% We can ask our favorite question about the input sequence 00:15:24.310 --> 00:15:26.690 align:middle line:84% and have the language model try to solve that question 00:15:26.690 --> 00:15:27.565 align:middle line:90% That's how they work. 00:15:27.565 --> 00:15:30.590 align:middle line:84% You can prompt them to answer different questions 00:15:30.590 --> 00:15:32.190 align:middle line:90% about your input sequence. 00:15:32.190 --> 00:15:34.770 align:middle line:84% We'll ask how often is the output-- 00:15:34.770 --> 00:15:38.210 align:middle line:84% something that's non-trivial-- not what a filing cabinet would 00:15:38.210 --> 00:15:39.250 align:middle line:90% do. 00:15:39.250 --> 00:15:42.090 align:middle line:84% And then we can say that the filing cabinet, in order 00:15:42.090 --> 00:15:47.170 align:middle line:84% to get a score of s, equals the number of unique sequences 00:15:47.170 --> 00:15:49.250 align:middle line:84% times the percent of times those sequences mapped 00:15:49.250 --> 00:15:50.510 align:middle line:90% to something non-trivial. 00:15:50.510 --> 00:15:52.370 align:middle line:90% It would have to be that big. 00:15:52.370 --> 00:15:55.290 align:middle line:84% It'd have to be taking that many unique sequences 00:15:55.290 --> 00:15:58.810 align:middle line:84% and achieving this proportion of correct predictions 00:15:58.810 --> 00:16:01.050 align:middle line:84% would have to have that many cabinets. 00:16:01.050 --> 00:16:03.210 align:middle line:84% And if that number is really, really big-- 00:16:03.210 --> 00:16:05.335 align:middle line:84% bigger than we think language models are, or bigger 00:16:05.335 --> 00:16:07.335 align:middle line:84% than the amount of data they've been trained on, 00:16:07.335 --> 00:16:09.050 align:middle line:84% then we know that language models are not 00:16:09.050 --> 00:16:11.250 align:middle line:90% doing pure memorization. 00:16:11.250 --> 00:16:15.830 align:middle line:84% So you can make up your own version of this, 00:16:15.830 --> 00:16:17.950 align:middle line:84% but just to convince myself, I ran it like this. 00:16:17.950 --> 00:16:20.130 align:middle line:90% I took 30 fruit names-- 00:16:20.130 --> 00:16:23.430 align:middle line:90% apple, orange, pear, whatever. 00:16:23.430 --> 00:16:28.610 align:middle line:84% So my vocab is size 30, and my sequence length is size 10. 00:16:28.610 --> 00:16:32.180 align:middle line:84% So I'm going to sample 10 random fruits from that list. 00:16:32.180 --> 00:16:39.380 align:middle line:84% So there's 10 to the 30th possible input sequences. 00:16:39.380 --> 00:16:41.140 align:middle line:84% And then I'm going to have my language 00:16:41.140 --> 00:16:44.580 align:middle line:84% model answer this question about those sequences. 00:16:44.580 --> 00:16:46.620 align:middle line:84% So we'll talk more about language models 00:16:46.620 --> 00:16:47.560 align:middle line:90% and prompting later. 00:16:47.560 --> 00:16:50.460 align:middle line:84% But just you've all done this with things like ChatGPT 00:16:50.460 --> 00:16:52.040 align:middle line:90% or Claude and so forth. 00:16:52.040 --> 00:16:53.880 align:middle line:84% You can say how many citrus fruits-- 00:16:53.880 --> 00:16:57.740 align:middle line:84% this is just my toy problem I'm giving it to try to estimate 00:16:57.740 --> 00:16:58.600 align:middle line:90% these probabilities. 00:16:58.600 --> 00:17:00.480 align:middle line:84% How many citrus fruits are in this list? 00:17:00.480 --> 00:17:02.020 align:middle line:84% And I give it this random sequence 00:17:02.020 --> 00:17:04.940 align:middle line:84% of 10 fruits sampled from that vocab of size 30. 00:17:04.940 --> 00:17:06.819 align:middle line:90% And it gets it correct. 00:17:06.819 --> 00:17:08.240 align:middle line:90% And so I did this 5 times. 00:17:08.240 --> 00:17:10.240 align:middle line:84% I'm not saying this is like the best experiment, 00:17:10.240 --> 00:17:13.400 align:middle line:84% but it got 5 out of 5, so it's correct every single time. 00:17:13.400 --> 00:17:15.619 align:middle line:84% So my estimate for the probability of being correct 00:17:15.619 --> 00:17:17.220 align:middle line:90% is 1. 00:17:17.220 --> 00:17:21.980 align:middle line:84% So now, we can just estimate-- you have p times m to the n 00:17:21.980 --> 00:17:24.700 align:middle line:90% is going to be 100 trillion. 00:17:24.700 --> 00:17:28.710 align:middle line:84% So the filing cabinet would have to have 100 trillion files. 00:17:28.710 --> 00:17:31.950 align:middle line:84% Also the training data-- the way the filing cabinet works 00:17:31.950 --> 00:17:35.292 align:middle line:84% is it only inputs a new data point into the cabinet 00:17:35.292 --> 00:17:37.250 align:middle line:84% if it sees that data point in the training set. 00:17:37.250 --> 00:17:38.917 align:middle line:84% So it means the training set has to have 00:17:38.917 --> 00:17:42.590 align:middle line:84% 100 million examples for that model-- sorry, 100 trillion. 00:17:42.590 --> 00:17:44.310 align:middle line:84% And we think that ChatGPT and things 00:17:44.310 --> 00:17:47.490 align:middle line:84% like this are trained on more like tens of trillions. 00:17:47.490 --> 00:17:49.040 align:middle line:90% So I think I exceeded that. 00:17:49.040 --> 00:17:50.790 align:middle line:84% But you could always make this list longer 00:17:50.790 --> 00:17:52.930 align:middle line:84% and run the same experiment-- get it to a quadrillion, 00:17:52.930 --> 00:17:55.388 align:middle line:84% and really convince yourself that these things are not just 00:17:55.388 --> 00:17:57.510 align:middle line:90% memorizing. 00:17:57.510 --> 00:17:59.010 align:middle line:90% Another just for fun one. 00:17:59.010 --> 00:18:02.485 align:middle line:84% We could do the same thing, but we can use a text-to-image 00:18:02.485 --> 00:18:04.110 align:middle line:84% model, and now have it draw the fruits, 00:18:04.110 --> 00:18:06.910 align:middle line:84% and ask how many unique different-- 00:18:06.910 --> 00:18:09.170 align:middle line:84% how often will it be correct in drawing the fruits? 00:18:09.170 --> 00:18:10.723 align:middle line:90% So I asked that to-- 00:18:10.723 --> 00:18:12.390 align:middle line:84% ChatGPT can do both of these things now. 00:18:12.390 --> 00:18:14.075 align:middle line:84% These are kind of multimodal models. 00:18:14.075 --> 00:18:15.450 align:middle line:84% So you can say draw these fruits. 00:18:15.450 --> 00:18:17.470 align:middle line:84% I sample a random sequence of 10 fruits 00:18:17.470 --> 00:18:20.350 align:middle line:90% and I check if it's correct. 00:18:20.350 --> 00:18:21.870 align:middle line:90% Is this one correct? 00:18:21.870 --> 00:18:23.710 align:middle line:90% AUDIENCE: No. 00:18:23.710 --> 00:18:24.897 align:middle line:90% PHILLIP ISOLA: I said no. 00:18:24.897 --> 00:18:25.730 align:middle line:90% It's almost correct. 00:18:25.730 --> 00:18:26.780 align:middle line:90% I think it got 9 out of 10. 00:18:26.780 --> 00:18:27.980 align:middle line:84% I don't think it has a passion fruit. 00:18:27.980 --> 00:18:30.272 align:middle line:84% But anyway, I don't know what passion fruits look like. 00:18:30.272 --> 00:18:32.700 align:middle line:90% But I called this one incorrect. 00:18:32.700 --> 00:18:34.000 align:middle line:90% And so I estimated-- 00:18:34.000 --> 00:18:35.080 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:18:35.080 --> 00:18:36.160 align:middle line:84% PHILLIP ISOLA: Does have another wrong thing? 00:18:36.160 --> 00:18:36.580 align:middle line:90% OK. 00:18:36.580 --> 00:18:36.840 align:middle line:90% Yeah? 00:18:36.840 --> 00:18:37.560 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:18:37.560 --> 00:18:38.560 align:middle line:90% PHILLIP ISOLA: Cantaloupe or-- 00:18:38.560 --> 00:18:39.185 align:middle line:90% AUDIENCE: Yeah. 00:18:39.185 --> 00:18:40.280 align:middle line:90% AUDIENCE: Coconut as well. 00:18:40.280 --> 00:18:41.580 align:middle line:84% PHILLIP ISOLA: Oh, yeah, coconuts missing. 00:18:41.580 --> 00:18:43.247 align:middle line:84% Maybe that was a passion fruit up there. 00:18:43.247 --> 00:18:45.200 align:middle line:90% But anyway, this one's wrong. 00:18:45.200 --> 00:18:48.280 align:middle line:84% But I ran this eight times, and one out of the eight times 00:18:48.280 --> 00:18:49.540 align:middle line:90% I thought it got it correct. 00:18:49.540 --> 00:18:52.040 align:middle line:84% I really saw each of the fruits represented. 00:18:52.040 --> 00:18:56.520 align:middle line:84% So now, our proportion p is 0.125, 1 over 8. 00:18:56.520 --> 00:18:58.480 align:middle line:84% And that means that you'd have to have a filing 00:18:58.480 --> 00:19:00.560 align:middle line:90% cabinet of size 12 trillion. 00:19:00.560 --> 00:19:04.280 align:middle line:84% So I don't think this has been 12 trillion images, 00:19:04.280 --> 00:19:05.520 align:middle line:90% but it's hard to know. 00:19:05.520 --> 00:19:07.430 align:middle line:90% OpenAI is secretive. 00:19:07.430 --> 00:19:09.180 align:middle line:84% But you can do these types of experiments, 00:19:09.180 --> 00:19:11.760 align:middle line:84% and I think you can pretty quickly convince yourself 00:19:11.760 --> 00:19:14.440 align:middle line:84% that the way that modern deep nets are working 00:19:14.440 --> 00:19:16.100 align:middle line:90% is not like a filing cabinet. 00:19:16.100 --> 00:19:17.900 align:middle line:84% It has some properties in common with the filing cabinet. 00:19:17.900 --> 00:19:19.720 align:middle line:84% We'll come back to that, but that doesn't fully 00:19:19.720 --> 00:19:20.845 align:middle line:90% explain what they're doing. 00:19:20.845 --> 00:19:26.077 align:middle line:90% 00:19:26.077 --> 00:19:28.160 align:middle line:84% Yeah, and so just a suggestion for final project-- 00:19:28.160 --> 00:19:29.400 align:middle line:84% I think it'd be fun to really do that 00:19:29.400 --> 00:19:31.260 align:middle line:84% systematically and rigorously, and try 00:19:31.260 --> 00:19:34.060 align:middle line:84% to actually estimate how many unique concepts these things 00:19:34.060 --> 00:19:34.860 align:middle line:90% have. 00:19:34.860 --> 00:19:36.540 align:middle line:84% I haven't really seen very many papers 00:19:36.540 --> 00:19:41.620 align:middle line:84% that do this type of experiment, but I think it'd be fun. 00:19:41.620 --> 00:19:44.300 align:middle line:84% So I got kind of excited about the feats 00:19:44.300 --> 00:19:48.260 align:middle line:84% of generalization of deep nets in this project 00:19:48.260 --> 00:19:50.220 align:middle line:90% from a few years ago. 00:19:50.220 --> 00:19:53.780 align:middle line:84% And so what we did here is we trained a deep net 00:19:53.780 --> 00:19:58.540 align:middle line:84% to take as input sketches and produces output photos of cats 00:19:58.540 --> 00:20:00.340 align:middle line:90% in this case. 00:20:00.340 --> 00:20:02.980 align:middle line:90% So a neural net just maps. 00:20:02.980 --> 00:20:05.540 align:middle line:84% And we'll talk about these generative models 00:20:05.540 --> 00:20:08.200 align:middle line:84% and image-to-image models a little bit later in the course. 00:20:08.200 --> 00:20:10.500 align:middle line:84% But for now, it's just a mapping from x 00:20:10.500 --> 00:20:14.740 align:middle line:84% to y-- it happens to be sketches to photos. 00:20:14.740 --> 00:20:16.740 align:middle line:84% And so what we did in this project 00:20:16.740 --> 00:20:18.980 align:middle line:84% is we didn't use human hand-drawn sketches 00:20:18.980 --> 00:20:20.980 align:middle line:84% because those were expensive to acquire. 00:20:20.980 --> 00:20:23.770 align:middle line:84% So instead, we just use this thing called HED, 00:20:23.770 --> 00:20:25.170 align:middle line:90% which is like an edge detector. 00:20:25.170 --> 00:20:27.503 align:middle line:84% And it's like a proxy for what a human might have drawn, 00:20:27.503 --> 00:20:30.390 align:middle line:84% but it doesn't quite match what a human would really draw. 00:20:30.390 --> 00:20:33.690 align:middle line:84% So from a kind of naive machine learning perspective, 00:20:33.690 --> 00:20:38.670 align:middle line:84% you would say, if I train this on a ton of examples 00:20:38.670 --> 00:20:41.970 align:middle line:84% of edge maps and cats, and I give it a new edge map, 00:20:41.970 --> 00:20:44.830 align:middle line:84% it should predict a new cat because this is just 00:20:44.830 --> 00:20:47.710 align:middle line:84% generalization to independently and identically sampled 00:20:47.710 --> 00:20:50.330 align:middle line:84% data points from the same population. 00:20:50.330 --> 00:20:53.510 align:middle line:84% So that's like classical generalization theory 00:20:53.510 --> 00:20:55.750 align:middle line:90% in machine learning. 00:20:55.750 --> 00:20:57.922 align:middle line:84% But the really interesting thing is 00:20:57.922 --> 00:21:00.130 align:middle line:84% it also worked when you give it hand-drawn cats which 00:21:00.130 --> 00:21:01.690 align:middle line:90% it had never been trained on. 00:21:01.690 --> 00:21:04.430 align:middle line:84% So these are inputs that are actually out of distribution-- 00:21:04.430 --> 00:21:07.310 align:middle line:84% systematically different than what it was trained on. 00:21:07.310 --> 00:21:12.290 align:middle line:84% So in the prehistory of deep learning, 00:21:12.290 --> 00:21:15.150 align:middle line:84% this was the popular thing of the day. 00:21:15.150 --> 00:21:17.570 align:middle line:84% Now, of course we've seen much more exciting things. 00:21:17.570 --> 00:21:19.790 align:middle line:84% But if I sketch a cat, it doesn't look anything 00:21:19.790 --> 00:21:20.610 align:middle line:90% like an edge map. 00:21:20.610 --> 00:21:25.040 align:middle line:84% And it just works, and it's still cool even today. 00:21:25.040 --> 00:21:27.820 align:middle line:90% So people had a lot of fun. 00:21:27.820 --> 00:21:30.520 align:middle line:90% [LAUGHTER] 00:21:30.520 --> 00:21:32.980 align:middle line:90% And this is generalization. 00:21:32.980 --> 00:21:35.060 align:middle line:84% It's never been trained on these types of things. 00:21:35.060 --> 00:21:37.640 align:middle line:84% It has some kind of creativity almost. 00:21:37.640 --> 00:21:40.060 align:middle line:84% We have to be careful not to anthropomorphize too much, 00:21:40.060 --> 00:21:43.520 align:middle line:84% but there's something here which is a little bit magical 00:21:43.520 --> 00:21:45.800 align:middle line:90% and not fully understood. 00:21:45.800 --> 00:21:48.080 align:middle line:84% But I think we can start to get a sense of what's 00:21:48.080 --> 00:21:50.628 align:middle line:84% going on with an experiment like this. 00:21:50.628 --> 00:21:52.420 align:middle line:84% So I'm just trying to build up intuitions-- 00:21:52.420 --> 00:21:54.340 align:middle line:90% do simple kind of experiments. 00:21:54.340 --> 00:21:57.260 align:middle line:84% We'll get into some more technical theory in a bit. 00:21:57.260 --> 00:21:58.560 align:middle line:90% But here we go. 00:21:58.560 --> 00:22:02.048 align:middle line:90% I just draw two eyes. 00:22:02.048 --> 00:22:03.340 align:middle line:90% Now, what if I draw three eyes? 00:22:03.340 --> 00:22:05.418 align:middle line:90% What's going to happen? 00:22:05.418 --> 00:22:07.460 align:middle line:84% It's never seen a photo of a cat with three eyes. 00:22:07.460 --> 00:22:08.720 align:middle line:90% How would it know what to do? 00:22:08.720 --> 00:22:09.780 align:middle line:90% But it works. 00:22:09.780 --> 00:22:13.700 align:middle line:84% It just puts that eye where that eye was drawn. 00:22:13.700 --> 00:22:14.500 align:middle line:90% A lot of eyes. 00:22:14.500 --> 00:22:18.305 align:middle line:90% 00:22:18.305 --> 00:22:19.520 align:middle line:90% You can add the body. 00:22:19.520 --> 00:22:20.940 align:middle line:90% That makes sense. 00:22:20.940 --> 00:22:23.800 align:middle line:90% So how is it doing this? 00:22:23.800 --> 00:22:25.260 align:middle line:84% How is it generalizing to more eyes 00:22:25.260 --> 00:22:27.460 align:middle line:84% than it ever saw at training time? 00:22:27.460 --> 00:22:31.800 align:middle line:84% And I think there's not a single explanation, 00:22:31.800 --> 00:22:33.620 align:middle line:84% but the best one I have in mind is 00:22:33.620 --> 00:22:36.980 align:middle line:84% it's the inductive biases of convolutional nets. 00:22:36.980 --> 00:22:38.653 align:middle line:90% So this network was a ConvNet. 00:22:38.653 --> 00:22:39.320 align:middle line:90% We've seen that. 00:22:39.320 --> 00:22:41.220 align:middle line:90% What do ConvNets do? 00:22:41.220 --> 00:22:47.420 align:middle line:84% They take an input, they chop it up into patches, 00:22:47.420 --> 00:22:51.220 align:middle line:84% and then they independently and identically process every patch. 00:22:51.220 --> 00:22:54.260 align:middle line:84% So they're factorizing the problem 00:22:54.260 --> 00:22:57.020 align:middle line:84% into a lot of independent decisions. 00:22:57.020 --> 00:23:00.180 align:middle line:84% And when you decompose something into these independent 00:23:00.180 --> 00:23:02.660 align:middle line:84% components, you get what's called compositionality, 00:23:02.660 --> 00:23:08.260 align:middle line:84% where you can take a new composition of things 00:23:08.260 --> 00:23:10.440 align:middle line:84% and factor them into these independent components, 00:23:10.440 --> 00:23:11.420 align:middle line:90% these patches. 00:23:11.420 --> 00:23:14.460 align:middle line:84% And it should it should know what to do, 00:23:14.460 --> 00:23:18.670 align:middle line:84% because it only ever has to imitate the training 00:23:18.670 --> 00:23:19.612 align:middle line:90% data on these patches. 00:23:19.612 --> 00:23:21.070 align:middle line:84% And then the combination of patches 00:23:21.070 --> 00:23:24.190 align:middle line:84% is just the rule of stitch them together. 00:23:24.190 --> 00:23:27.070 align:middle line:84% So for every single patch, it's doing-- 00:23:27.070 --> 00:23:28.830 align:middle line:84% in distribution generalization, it's 00:23:28.830 --> 00:23:31.965 align:middle line:84% seen a lot of things that look like ovals, and they match eyes. 00:23:31.965 --> 00:23:33.590 align:middle line:84% And then it can generalize to more eyes 00:23:33.590 --> 00:23:40.940 align:middle line:84% because I've just factorized the problem in this simple way. 00:23:40.940 --> 00:23:46.110 align:middle line:84% So architectural constraints and biases 00:23:46.110 --> 00:23:48.590 align:middle line:84% and the way that we enforce symmetries and factorization 00:23:48.590 --> 00:23:51.190 align:middle line:84% into our models allows you to do certain kinds 00:23:51.190 --> 00:23:52.710 align:middle line:90% of feats of generalization. 00:23:52.710 --> 00:23:57.230 align:middle line:84% And we saw that with the ability of graph nets 00:23:57.230 --> 00:24:00.190 align:middle line:84% to generalize to new permutations of their inputs, 00:24:00.190 --> 00:24:01.870 align:middle line:84% because they're permutation-invariant or 00:24:01.870 --> 00:24:02.650 align:middle line:90% equivariant. 00:24:02.650 --> 00:24:04.697 align:middle line:84% So that's just built into the architecture. 00:24:04.697 --> 00:24:06.530 align:middle line:84% I don't have to have seen every permutation. 00:24:06.530 --> 00:24:08.010 align:middle line:84% I can just have seen some permutations, 00:24:08.010 --> 00:24:09.630 align:middle line:84% and then it will generalize to other permutations. 00:24:09.630 --> 00:24:11.185 align:middle line:84% That's baked into the architecture. 00:24:11.185 --> 00:24:13.310 align:middle line:84% It's not something you have to learn from the data. 00:24:13.310 --> 00:24:15.440 align:middle line:84% So architecture is one of the main levers 00:24:15.440 --> 00:24:19.080 align:middle line:84% we have for these feats of generalization that 00:24:19.080 --> 00:24:22.800 align:middle line:84% violate just statistical learning theory, 00:24:22.800 --> 00:24:25.660 align:middle line:84% or that go beyond just naive statistical learning theory. 00:24:25.660 --> 00:24:30.320 align:middle line:90% 00:24:30.320 --> 00:24:33.760 align:middle line:84% So let's get a little bit into generalization theory 00:24:33.760 --> 00:24:37.280 align:middle line:84% and start with the classic picture. 00:24:37.280 --> 00:24:42.800 align:middle line:84% So most of generalization theory comes from Occam. 00:24:42.800 --> 00:24:44.580 align:middle line:84% It comes from the Occam's razor idea. 00:24:44.580 --> 00:24:47.020 align:middle line:84% It's all variations on Occam's razor in some sense. 00:24:47.020 --> 00:24:48.760 align:middle line:84% So just to remind you, Occam's razor 00:24:48.760 --> 00:24:51.560 align:middle line:84% says among competing explanations that 00:24:51.560 --> 00:24:55.730 align:middle line:84% fit the data equally well, you should prefer the simplest 00:24:55.730 --> 00:24:57.480 align:middle line:84% And the generalization way of stating that 00:24:57.480 --> 00:24:59.720 align:middle line:84% is that the simplest model that fits the data 00:24:59.720 --> 00:25:01.725 align:middle line:84% will be most likely to be the true model. 00:25:01.725 --> 00:25:04.100 align:middle line:84% Therefore, it will be most likely to generalize the best. 00:25:04.100 --> 00:25:07.080 align:middle line:90% 00:25:07.080 --> 00:25:10.680 align:middle line:84% Just one little note is that I think sometimes people forget 00:25:10.680 --> 00:25:13.940 align:middle line:90% to add the "fits the data." 00:25:13.940 --> 00:25:16.855 align:middle line:84% So it's not true the simplest model is most likely to be true, 00:25:16.855 --> 00:25:18.480 align:middle line:84% or it's most likely to generalize best. 00:25:18.480 --> 00:25:20.438 align:middle line:84% It's the simplest model that fits the data that 00:25:20.438 --> 00:25:21.900 align:middle line:90% actually explains the data. 00:25:21.900 --> 00:25:25.860 align:middle line:84% You need that, otherwise you get these absurd statements like, 00:25:25.860 --> 00:25:27.920 align:middle line:84% oh, well, I should use a linear model. 00:25:27.920 --> 00:25:29.280 align:middle line:90% But no, it doesn't fit the data. 00:25:29.280 --> 00:25:32.247 align:middle line:84% So you have to use a complex enough model that fits the data, 00:25:32.247 --> 00:25:34.080 align:middle line:84% but then you want to prefer the simpler one. 00:25:34.080 --> 00:25:35.740 align:middle line:84% And there's a lot of deep arguments for why 00:25:35.740 --> 00:25:36.740 align:middle line:90% that should be the case. 00:25:36.740 --> 00:25:39.340 align:middle line:84% It's intuitively clear from the idea of Occam's razor 00:25:39.340 --> 00:25:43.420 align:middle line:84% that we learn in grade school or whenever we learn it. 00:25:43.420 --> 00:25:44.940 align:middle line:84% But also there's deep mathematics 00:25:44.940 --> 00:25:46.900 align:middle line:84% that can be used to derive formal versions 00:25:46.900 --> 00:25:48.020 align:middle line:90% of this statement. 00:25:48.020 --> 00:25:50.260 align:middle line:90% Yeah, question? 00:25:50.260 --> 00:25:56.940 align:middle line:84% AUDIENCE: If someone were to compare this [INAUDIBLE] 00:25:56.940 --> 00:25:59.940 align:middle line:90% that all models [INAUDIBLE]. 00:25:59.940 --> 00:26:01.967 align:middle line:84% How would you compare this [INAUDIBLE]? 00:26:01.967 --> 00:26:04.300 align:middle line:84% PHILLIP ISOLA: The question-- that's really interesting. 00:26:04.300 --> 00:26:05.920 align:middle line:84% There's another quote-- "all models are wrong, 00:26:05.920 --> 00:26:07.000 align:middle line:90% but some are useful." 00:26:07.000 --> 00:26:11.020 align:middle line:90% 00:26:11.020 --> 00:26:15.910 align:middle line:84% Yeah, that's intriguing to think about. 00:26:15.910 --> 00:26:21.870 align:middle line:84% I would say that the "all models" is-- 00:26:21.870 --> 00:26:24.110 align:middle line:84% so this is only the set of models 00:26:24.110 --> 00:26:27.470 align:middle line:84% that are in the equivalence class of all equally well 00:26:27.470 --> 00:26:28.950 align:middle line:90% fitting the data. 00:26:28.950 --> 00:26:31.190 align:middle line:84% So we're not really considering models 00:26:31.190 --> 00:26:35.070 align:middle line:84% that are wrong in terms of only approximating the data. 00:26:35.070 --> 00:26:36.630 align:middle line:84% So it's a slightly different setting 00:26:36.630 --> 00:26:40.990 align:middle line:90% that we're looking at there. 00:26:40.990 --> 00:26:42.310 align:middle line:90% But yeah, I don't know. 00:26:42.310 --> 00:26:44.250 align:middle line:84% There's probably more discussion around that, 00:26:44.250 --> 00:26:46.170 align:middle line:84% but I don't have a crisp answer right now. 00:26:46.170 --> 00:26:47.390 align:middle line:90% Yeah, question? 00:26:47.390 --> 00:26:50.350 align:middle line:84% AUDIENCE: Has this statement been proven rigorously or is it 00:26:50.350 --> 00:26:51.437 align:middle line:90% a heuristic? 00:26:51.437 --> 00:26:53.270 align:middle line:84% PHILLIP ISOLA: So I would say in this form-- 00:26:53.270 --> 00:26:54.770 align:middle line:84% so the question-- has this statement 00:26:54.770 --> 00:26:57.190 align:middle line:84% been proven rigorously, or is it a heuristic? 00:26:57.190 --> 00:27:00.230 align:middle line:84% In this form, it's a heuristic because how do we 00:27:00.230 --> 00:27:01.970 align:middle line:90% measure simplest? 00:27:01.970 --> 00:27:03.650 align:middle line:84% How do we measure generalization? 00:27:03.650 --> 00:27:05.150 align:middle line:84% We need to make these mathematically 00:27:05.150 --> 00:27:06.690 align:middle line:90% precise to make it rigorous. 00:27:06.690 --> 00:27:12.040 align:middle line:84% But there are rigorous versions of the Occam's razor argument. 00:27:12.040 --> 00:27:15.420 align:middle line:84% My favorite one is this one from various people, 00:27:15.420 --> 00:27:23.320 align:middle line:84% including Solomonoff, which is that the shortest program that 00:27:23.320 --> 00:27:25.240 align:middle line:84% perfectly fits or generates the data-- 00:27:25.240 --> 00:27:27.960 align:middle line:90% that just outputs the data set-- 00:27:27.960 --> 00:27:30.720 align:middle line:84% is the one that will generalize best and make the best 00:27:30.720 --> 00:27:31.510 align:middle line:90% predictions. 00:27:31.510 --> 00:27:33.760 align:middle line:84% There's an equivalent version of this statement, which 00:27:33.760 --> 00:27:37.240 align:middle line:84% is that the most compressed representation of your data 00:27:37.240 --> 00:27:38.797 align:middle line:90% is the best model of your data. 00:27:38.797 --> 00:27:40.880 align:middle line:84% There's some equivalency between these two things. 00:27:40.880 --> 00:27:44.080 align:middle line:84% So I'm not going to get into all the technical details of that. 00:27:44.080 --> 00:27:47.220 align:middle line:84% But this is the place to start reading 00:27:47.220 --> 00:27:50.840 align:middle line:84% if you want mathematically precise arguments for Occam's 00:27:50.840 --> 00:27:52.400 align:middle line:90% razor. 00:27:52.400 --> 00:27:55.600 align:middle line:84% And the problem, though, is that finding the shortest 00:27:55.600 --> 00:27:57.697 align:middle line:84% program that fits the data requires 00:27:57.697 --> 00:27:59.780 align:middle line:84% searching over the space of all possible programs. 00:27:59.780 --> 00:28:03.920 align:middle line:84% Imagine searching over the space of all possible Python programs. 00:28:03.920 --> 00:28:05.322 align:middle line:90% You just can't do that. 00:28:05.322 --> 00:28:07.280 align:middle line:84% And deep learning doesn't search over the space 00:28:07.280 --> 00:28:08.390 align:middle line:90% of all possible programs. 00:28:08.390 --> 00:28:10.765 align:middle line:84% It just searches over the space of all weights and biases 00:28:10.765 --> 00:28:12.890 align:middle line:84% of a particular model architecture. 00:28:12.890 --> 00:28:15.090 align:middle line:84% So this is completely intractable, 00:28:15.090 --> 00:28:17.570 align:middle line:84% but it comes up in algorithmic information theory 00:28:17.570 --> 00:28:19.130 align:middle line:90% and areas like that. 00:28:19.130 --> 00:28:21.010 align:middle line:84% So it's good to keep in mind, this 00:28:21.010 --> 00:28:23.970 align:middle line:84% is kind of like the mathematical basis for generalization theory. 00:28:23.970 --> 00:28:28.370 align:middle line:84% But actually finding the optimal program or the shortest program 00:28:28.370 --> 00:28:30.910 align:middle line:84% is intractable, so we're not going to quite do that. 00:28:30.910 --> 00:28:33.790 align:middle line:90% But this is what we want. 00:28:33.790 --> 00:28:36.610 align:middle line:84% We want a way of measuring simplicity or complexity 00:28:36.610 --> 00:28:43.810 align:middle line:84% of a hypothesis space or of a solution to a problem. 00:28:43.810 --> 00:28:47.547 align:middle line:84% And this is kind of the best notion of simplicity, 00:28:47.547 --> 00:28:48.630 align:middle line:90% but we can't quite use it. 00:28:48.630 --> 00:28:51.130 align:middle line:90% So what else can we use? 00:28:51.130 --> 00:28:55.370 align:middle line:84% So let's do a quick review of overfitting-- 00:28:55.370 --> 00:28:58.410 align:middle line:84% the idea of overfitting and the bias variance trade-off. 00:28:58.410 --> 00:29:00.947 align:middle line:84% So remember that the test error is 00:29:00.947 --> 00:29:03.530 align:middle line:84% equal to the train error plus the difference between the train 00:29:03.530 --> 00:29:06.870 align:middle line:90% error and the test error. 00:29:06.870 --> 00:29:09.162 align:middle line:84% And we call the training error the bias 00:29:09.162 --> 00:29:10.870 align:middle line:84% and the difference between the test error 00:29:10.870 --> 00:29:14.830 align:middle line:84% and the train error the variance. 00:29:14.830 --> 00:29:17.350 align:middle line:84% And the kind of classical machine learning 101-- 00:29:17.350 --> 00:29:18.750 align:middle line:90% picture of how things work-- 00:29:18.750 --> 00:29:23.130 align:middle line:84% is that as I increase the kind of capacity, 00:29:23.130 --> 00:29:24.950 align:middle line:84% the number of functions under consideration 00:29:24.950 --> 00:29:26.967 align:middle line:84% in my hypothesis space that I'm searching over, 00:29:26.967 --> 00:29:28.550 align:middle line:84% as I make bigger and bigger and bigger 00:29:28.550 --> 00:29:33.350 align:middle line:84% deep nets with more and more and more parameters, then eventually 00:29:33.350 --> 00:29:34.950 align:middle line:84% I'll get universal approximation. 00:29:34.950 --> 00:29:37.530 align:middle line:84% If I get width goes to infinity, I can fit any function. 00:29:37.530 --> 00:29:40.730 align:middle line:84% So I can minimize my loss on the training error, 00:29:40.730 --> 00:29:42.870 align:middle line:90% so the training risk goes down. 00:29:42.870 --> 00:29:46.230 align:middle line:84% But I'll start overfitting, and the test risk will eventually 00:29:46.230 --> 00:29:49.670 align:middle line:84% go up, because I'm finding fits to the data which 00:29:49.670 --> 00:29:52.750 align:middle line:84% are kind of fitting to properties of the data that 00:29:52.750 --> 00:29:54.410 align:middle line:84% don't actually generalize, like noise, 00:29:54.410 --> 00:29:57.710 align:middle line:84% or the fact that I got this particular batch of samples 00:29:57.710 --> 00:30:00.070 align:middle line:84% as opposed to a different batch of samples. 00:30:00.070 --> 00:30:05.800 align:middle line:84% So this is the overfitting scenario that you have likely 00:30:05.800 --> 00:30:07.880 align:middle line:90% seen before. 00:30:07.880 --> 00:30:10.680 align:middle line:84% And one of the ways of measuring model capacity-- modeling 00:30:10.680 --> 00:30:15.320 align:middle line:84% the x-axis or measuring the complexity of a hypothesis space 00:30:15.320 --> 00:30:16.660 align:middle line:90% is via number of parameters. 00:30:16.660 --> 00:30:17.760 align:middle line:84% And this is one of the classical ways 00:30:17.760 --> 00:30:19.093 align:middle line:90% that you might have encountered. 00:30:19.093 --> 00:30:21.748 align:middle line:84% If I want to find my best machine learning model, 00:30:21.748 --> 00:30:23.540 align:middle line:84% I need to control the number of parameters. 00:30:23.540 --> 00:30:26.040 align:middle line:84% If I have too many parameters, it will overfit. 00:30:26.040 --> 00:30:28.440 align:middle line:84% So let's look at what happens if I'm 00:30:28.440 --> 00:30:33.520 align:middle line:84% doing some kind of polynomial regression onto these red points 00:30:33.520 --> 00:30:34.160 align:middle line:90% here. 00:30:34.160 --> 00:30:39.280 align:middle line:84% So the orange line is my fit to the data. 00:30:39.280 --> 00:30:41.720 align:middle line:84% And if I use a degree 1 polynomial-- 00:30:41.720 --> 00:30:43.220 align:middle line:84% I think it's not quite polynomials. 00:30:43.220 --> 00:30:45.400 align:middle line:84% It's a particular family of polynomials. 00:30:45.400 --> 00:30:49.640 align:middle line:84% I'm finding the coefficients of a weighted sum 00:30:49.640 --> 00:30:52.240 align:middle line:84% of a particular family of simple functions, 00:30:52.240 --> 00:30:54.620 align:middle line:84% like Legendre polynomials, I think, in this case. 00:30:54.620 --> 00:30:56.760 align:middle line:90% But it doesn't matter too much. 00:30:56.760 --> 00:30:59.080 align:middle line:84% So I'm going to say with d equals 00:30:59.080 --> 00:31:01.660 align:middle line:84% 1 is the degree of that family of polynomials. 00:31:01.660 --> 00:31:04.330 align:middle line:84% It's just d equals 1 means linear, 00:31:04.330 --> 00:31:06.490 align:middle line:84% and a linear fit can't do a great job. 00:31:06.490 --> 00:31:08.290 align:middle line:84% The truth function is the blue line, 00:31:08.290 --> 00:31:12.490 align:middle line:84% and the training data is the red points. 00:31:12.490 --> 00:31:16.930 align:middle line:84% What happens as I increase the wiggliness of the polynomials 00:31:16.930 --> 00:31:17.710 align:middle line:90% in my family? 00:31:17.710 --> 00:31:19.610 align:middle line:84% I include more and more polynomials 00:31:19.610 --> 00:31:21.190 align:middle line:84% that are higher and higher frequency 00:31:21.190 --> 00:31:23.850 align:middle line:84% and the kind of wiggles that they can express. 00:31:23.850 --> 00:31:27.950 align:middle line:84% So if I go up to a quadratic or cubic model in this case-- well, 00:31:27.950 --> 00:31:30.250 align:middle line:84% the ground truth function happened to be cubic 00:31:30.250 --> 00:31:33.770 align:middle line:84% or happened to have order 3 Legendre polynomial, 00:31:33.770 --> 00:31:36.150 align:middle line:90% and I get a really good fit. 00:31:36.150 --> 00:31:40.530 align:middle line:84% And then overfitting is, if I go up to-- 00:31:40.530 --> 00:31:42.350 align:middle line:84% have too many degrees of freedom, 00:31:42.350 --> 00:31:46.910 align:middle line:84% I can actually not only recover the true data, 00:31:46.910 --> 00:31:48.510 align:middle line:84% but I'm also fitting to the noise. 00:31:48.510 --> 00:31:51.690 align:middle line:84% This deviation from the blue line is just noise. 00:31:51.690 --> 00:31:56.010 align:middle line:84% And the model is wildly deviating 00:31:56.010 --> 00:31:59.450 align:middle line:84% in order to capture that aspect that doesn't actually 00:31:59.450 --> 00:31:59.950 align:middle line:90% generalize. 00:31:59.950 --> 00:32:01.575 align:middle line:84% It doesn't actually tell them something 00:32:01.575 --> 00:32:03.930 align:middle line:84% about the true underlying function. 00:32:03.930 --> 00:32:06.050 align:middle line:84% So this is our classic picture of overfitting, 00:32:06.050 --> 00:32:09.910 align:middle line:84% but here's the surprise if you haven't seen it before. 00:32:09.910 --> 00:32:14.590 align:middle line:84% What happens if I go not to d equals 20, but d equals 1,000? 00:32:14.590 --> 00:32:17.000 align:middle line:84% I have a really, really expressive class 00:32:17.000 --> 00:32:18.750 align:middle line:84% of polynomials that can fit really, really 00:32:18.750 --> 00:32:20.470 align:middle line:90% crazy wiggly functions. 00:32:20.470 --> 00:32:23.790 align:middle line:84% Is it going to go more crazy and overfit even worse in the sense 00:32:23.790 --> 00:32:27.990 align:middle line:84% that these deviations will just go to infinity, 00:32:27.990 --> 00:32:29.630 align:middle line:84% or will it do something different? 00:32:29.630 --> 00:32:31.378 align:middle line:90% What do you think? 00:32:31.378 --> 00:32:33.170 align:middle line:84% Some of you probably have seen this before, 00:32:33.170 --> 00:32:34.050 align:middle line:90% but let's go up here. 00:32:34.050 --> 00:32:34.550 align:middle line:90% Yeah? 00:32:34.550 --> 00:32:37.470 align:middle line:84% AUDIENCE: I mean, you should have a large manifold of fits 00:32:37.470 --> 00:32:38.918 align:middle line:90% that all fit perfectly, right? 00:32:38.918 --> 00:32:39.710 align:middle line:90% PHILLIP ISOLA: Yes. 00:32:39.710 --> 00:32:41.590 align:middle line:84% AUDIENCE: So some of them could be good, some of them 00:32:41.590 --> 00:32:42.210 align:middle line:90% could be bad. 00:32:42.210 --> 00:32:43.690 align:middle line:84% PHILLIP ISOLA: Yeah, so the answer to-- 00:32:43.690 --> 00:32:45.398 align:middle line:84% and that's a quite sophisticated answer-- 00:32:45.398 --> 00:32:48.230 align:middle line:84% is you should have a large set of functions 00:32:48.230 --> 00:32:50.350 align:middle line:90% that all perfectly fit the data. 00:32:50.350 --> 00:32:53.110 align:middle line:84% And so some of them should be good and some of them 00:32:53.110 --> 00:32:53.715 align:middle line:90% should be bad. 00:32:53.715 --> 00:32:55.090 align:middle line:84% But which one will you arrive at? 00:32:55.090 --> 00:32:56.923 align:middle line:84% Well, it's going to depend on other factors. 00:32:56.923 --> 00:32:58.430 align:middle line:84% It's going to depend on how you're 00:32:58.430 --> 00:33:01.020 align:middle line:84% searching over that space, or what optimizer are you using? 00:33:01.020 --> 00:33:03.298 align:middle line:84% How are you minimizing your training error? 00:33:03.298 --> 00:33:04.840 align:middle line:84% It's going to depend on other factors 00:33:04.840 --> 00:33:11.480 align:middle line:84% beyond just randomly selecting one of the solutions that 00:33:11.480 --> 00:33:14.140 align:middle line:90% fits the data. 00:33:14.140 --> 00:33:16.680 align:middle line:84% So here's what happens for this regression 00:33:16.680 --> 00:33:19.040 align:middle line:90% problem with these polynomials. 00:33:19.040 --> 00:33:23.880 align:middle line:84% So when d equals 1,000, you actually are able to fit this 00:33:23.880 --> 00:33:27.560 align:middle line:84% really spiky function, but it adheres much more closely 00:33:27.560 --> 00:33:31.760 align:middle line:90% to the true smooth solution. 00:33:31.760 --> 00:33:33.960 align:middle line:84% And this phenomena is called least-- 00:33:33.960 --> 00:33:35.400 align:middle line:90% is called double descent. 00:33:35.400 --> 00:33:38.220 align:middle line:84% So it's the idea that overfitting is actually-- 00:33:38.220 --> 00:33:39.720 align:middle line:84% the classical picture of overfitting 00:33:39.720 --> 00:33:41.560 align:middle line:90% is a little bit inaccurate. 00:33:41.560 --> 00:33:45.680 align:middle line:84% And what actually happens is you start overfitting really badly 00:33:45.680 --> 00:33:47.445 align:middle line:90% as you add more parameters. 00:33:47.445 --> 00:33:49.320 align:middle line:84% But eventually, if you add enough parameters, 00:33:49.320 --> 00:33:52.440 align:middle line:84% for certain types of models, including deep nets 00:33:52.440 --> 00:33:55.680 align:middle line:84% and also including these polynomial regression problems, 00:33:55.680 --> 00:33:59.130 align:middle line:84% you end up getting this kind of behavior 00:33:59.130 --> 00:34:01.490 align:middle line:84% where you start actually getting better generalization 00:34:01.490 --> 00:34:04.250 align:middle line:90% error with more parameters. 00:34:04.250 --> 00:34:07.410 align:middle line:84% And the basic hypothesis here is that-- 00:34:07.410 --> 00:34:09.630 align:middle line:84% and this is provable for certain simple systems, 00:34:09.630 --> 00:34:12.010 align:middle line:84% but it's not really provable for deep nets. 00:34:12.010 --> 00:34:13.764 align:middle line:84% But the basic hypothesis for deep nets 00:34:13.764 --> 00:34:15.889 align:middle line:84% and the general statement is that the learned model 00:34:15.889 --> 00:34:18.730 align:middle line:90% has two components. 00:34:18.730 --> 00:34:22.889 align:middle line:84% With enough capacity, it will find a simple fit to the data. 00:34:22.889 --> 00:34:26.889 align:middle line:84% And then it will also use little spikes of capacity 00:34:26.889 --> 00:34:29.770 align:middle line:90% to fit the outliers. 00:34:29.770 --> 00:34:32.370 align:middle line:84% So there's some kind of simple component, which 00:34:32.370 --> 00:34:34.929 align:middle line:84% is predictive and generalizes well to new samples 00:34:34.929 --> 00:34:37.449 align:middle line:84% from the population, and a spiking component 00:34:37.449 --> 00:34:40.489 align:middle line:84% that overfits noise and things that don't generalize 00:34:40.489 --> 00:34:41.429 align:middle line:90% in the training data. 00:34:41.429 --> 00:34:42.590 align:middle line:90% So it's like memorization. 00:34:42.590 --> 00:34:44.449 align:middle line:90% Spikiness is like memorization. 00:34:44.449 --> 00:34:47.820 align:middle line:90% I'm just memorizing those-- 00:34:47.820 --> 00:34:50.270 align:middle line:84% when my input is this, I memorized what it was. 00:34:50.270 --> 00:34:52.310 align:middle line:84% So it's memorization plus generalization. 00:34:52.310 --> 00:34:54.739 align:middle line:84% It's a really interesting combination 00:34:54.739 --> 00:35:00.040 align:middle line:84% that, until a few years ago, people didn't widely appreciate. 00:35:00.040 --> 00:35:02.460 align:middle line:84% There's a nice visualization of this with an MLP 00:35:02.460 --> 00:35:04.202 align:middle line:90% that you can find at this link. 00:35:04.202 --> 00:35:05.660 align:middle line:84% It looks basically the same as what 00:35:05.660 --> 00:35:08.920 align:middle line:84% I'm showing you, but not quite as spiky. 00:35:08.920 --> 00:35:12.980 align:middle line:84% So deep nets always are a little bit Messier 00:35:12.980 --> 00:35:15.700 align:middle line:84% than more classical models, so it's easier 00:35:15.700 --> 00:35:18.260 align:middle line:84% to see with this polynomial regression model. 00:35:18.260 --> 00:35:19.200 align:middle line:90% Yeah, question? 00:35:19.200 --> 00:35:20.860 align:middle line:84% AUDIENCE: So from my understanding, 00:35:20.860 --> 00:35:24.160 align:middle line:84% a polynomial regression is a closed form solution. 00:35:24.160 --> 00:35:26.220 align:middle line:84% So I guess you could see that this is always 00:35:26.220 --> 00:35:27.518 align:middle line:90% true in this case. 00:35:27.518 --> 00:35:29.060 align:middle line:84% But for a neural network where you're 00:35:29.060 --> 00:35:31.820 align:middle line:84% doing like a stochastic optimization process, 00:35:31.820 --> 00:35:34.180 align:middle line:84% I'm wondering how you can be sure you'll always 00:35:34.180 --> 00:35:38.940 align:middle line:84% arrive at this function and not, as you said, a function that 00:35:38.940 --> 00:35:40.113 align:middle line:90% is wildly inaccurate? 00:35:40.113 --> 00:35:41.780 align:middle line:84% PHILLIP ISOLA: Yeah, so the question is, 00:35:41.780 --> 00:35:44.080 align:middle line:84% for polynomial regression, there's a closed form solution, 00:35:44.080 --> 00:35:46.413 align:middle line:84% and you can really prove that this will be what happens. 00:35:46.413 --> 00:35:49.700 align:middle line:84% And that solution will have this property if that's the case. 00:35:49.700 --> 00:35:52.860 align:middle line:84% For a neural net, the solution you arrive at 00:35:52.860 --> 00:35:54.663 align:middle line:84% will depend on the initialization, 00:35:54.663 --> 00:35:55.580 align:middle line:90% which might be random. 00:35:55.580 --> 00:35:57.520 align:middle line:84% It will depend on the stochastic mini batches, 00:35:57.520 --> 00:35:59.040 align:middle line:90% which are going to be random. 00:35:59.040 --> 00:36:01.360 align:middle line:84% There's a lot of randomness, and it's much harder 00:36:01.360 --> 00:36:04.560 align:middle line:84% to make clean statements about what solution you'll arrive at. 00:36:04.560 --> 00:36:06.820 align:middle line:84% So instead, we can make empirical statements. 00:36:06.820 --> 00:36:09.600 align:middle line:84% We can say that if we run this neural network over and over 00:36:09.600 --> 00:36:12.660 align:middle line:84% again with the Adam optimizer, or the SGD optimizer, 00:36:12.660 --> 00:36:13.920 align:middle line:90% here's what we observe. 00:36:13.920 --> 00:36:15.738 align:middle line:84% And I'll show you that on the next slide. 00:36:15.738 --> 00:36:17.280 align:middle line:84% Or we can make theoretical statements 00:36:17.280 --> 00:36:20.240 align:middle line:84% like, the distribution of solutions 00:36:20.240 --> 00:36:22.240 align:middle line:84% that will be arrived at under certain conditions 00:36:22.240 --> 00:36:26.240 align:middle line:84% will be smooth and spiky with high probability, stuff 00:36:26.240 --> 00:36:26.740 align:middle line:90% like this. 00:36:26.740 --> 00:36:28.400 align:middle line:90% Yep, one more question? 00:36:28.400 --> 00:36:30.400 align:middle line:84% AUDIENCE: When would you know that you're 00:36:30.400 --> 00:36:34.280 align:middle line:84% using the right computational resources rather 00:36:34.280 --> 00:36:37.100 align:middle line:90% than optimizing to a new model? 00:36:37.100 --> 00:36:42.260 align:middle line:84% So if it takes 1,000 to make an optimal solution, 00:36:42.260 --> 00:36:45.440 align:middle line:84% but it takes a lot of computational effort, 00:36:45.440 --> 00:36:50.120 align:middle line:84% should you stop at some point, or turn to other model that will 00:36:50.120 --> 00:36:52.130 align:middle line:90% potentially fit better? 00:36:52.130 --> 00:36:54.970 align:middle line:84% PHILLIP ISOLA: Yeah, so when should you 00:36:54.970 --> 00:36:59.847 align:middle line:84% decide where to stop in an optimization process 00:36:59.847 --> 00:37:01.430 align:middle line:84% is the question, or at least stopping. 00:37:01.430 --> 00:37:04.130 align:middle line:90% Yeah, so that's a hard question. 00:37:04.130 --> 00:37:07.650 align:middle line:84% And so the number of steps of gradient descent that you do 00:37:07.650 --> 00:37:10.790 align:middle line:84% is kind of a hyperparameter, and it can affect the solution. 00:37:10.790 --> 00:37:13.230 align:middle line:84% And if you do too many, then you might be overfitting. 00:37:13.230 --> 00:37:14.850 align:middle line:84% And that's kind of a classical idea. 00:37:14.850 --> 00:37:18.510 align:middle line:84% But what we'll see is that in this neural net era, 00:37:18.510 --> 00:37:20.190 align:middle line:84% it tends to be just train forever. 00:37:20.190 --> 00:37:21.690 align:middle line:84% And the longer you train, the better 00:37:21.690 --> 00:37:24.530 align:middle line:84% you do because there's a double descent phenomenon, which 00:37:24.530 --> 00:37:27.410 align:middle line:84% I'll show in the next slide in terms of computation. 00:37:27.410 --> 00:37:29.130 align:middle line:84% The longer you train, the better you'll 00:37:29.130 --> 00:37:33.190 align:middle line:84% do in this particular family of models is the general principle. 00:37:33.190 --> 00:37:34.810 align:middle line:84% But yeah, of course, there's more 00:37:34.810 --> 00:37:38.083 align:middle line:90% that can be said about that. 00:37:38.083 --> 00:37:40.250 align:middle line:84% So this was a really cool paper from a few years ago 00:37:40.250 --> 00:37:43.240 align:middle line:90% that got people excited. 00:37:43.240 --> 00:37:45.490 align:middle line:84% So the double descent phenomenon is just a consequence 00:37:45.490 --> 00:37:46.910 align:middle line:90% of what I already showed you. 00:37:46.910 --> 00:37:50.100 align:middle line:84% So we have regular classical overfitting. 00:37:50.100 --> 00:37:53.405 align:middle line:84% As I increase my capacity of my hypothesis space, which we're 00:37:53.405 --> 00:37:55.780 align:middle line:84% right now naively calculating as the number of parameters 00:37:55.780 --> 00:37:59.360 align:middle line:84% in the hypothesis space, then my training error goes down, 00:37:59.360 --> 00:38:01.080 align:middle line:84% but my test error eventually goes up. 00:38:01.080 --> 00:38:02.260 align:middle line:90% I overfit. 00:38:02.260 --> 00:38:05.180 align:middle line:84% But then after overfitting, eventually, 00:38:05.180 --> 00:38:08.220 align:middle line:84% I learned the simple function and the spikes, 00:38:08.220 --> 00:38:13.860 align:middle line:84% and so I actually start to generalize better again. 00:38:13.860 --> 00:38:17.060 align:middle line:84% And there's this thing called interpolation threshold, which 00:38:17.060 --> 00:38:18.820 align:middle line:84% is the point at which I can perfectly 00:38:18.820 --> 00:38:20.820 align:middle line:90% memorize all the training data. 00:38:20.820 --> 00:38:22.175 align:middle line:90% And I think one intuitive-- 00:38:22.175 --> 00:38:24.800 align:middle line:84% I'm not sure if this is going to turn out to be precisely true, 00:38:24.800 --> 00:38:26.722 align:middle line:84% but my intuitive picture of what's going on 00:38:26.722 --> 00:38:28.680 align:middle line:84% is that you're trying to fit the training data, 00:38:28.680 --> 00:38:30.038 align:middle line:90% so you have to memorize it. 00:38:30.038 --> 00:38:32.580 align:middle line:84% And you have to learn a really crazy function to memorize it. 00:38:32.580 --> 00:38:36.380 align:middle line:84% But then once you add more capacity, where now there 00:38:36.380 --> 00:38:39.500 align:middle line:84% are a lot of solutions that equally well memorize 00:38:39.500 --> 00:38:42.800 align:middle line:84% the data equally well, interpolate the data, 00:38:42.800 --> 00:38:46.020 align:middle line:84% then you can select within all of those set of solutions that 00:38:46.020 --> 00:38:48.960 align:middle line:84% fit the data the one that is the smoothest 00:38:48.960 --> 00:38:51.240 align:middle line:90% or has other nice properties. 00:38:51.240 --> 00:38:54.040 align:middle line:84% And the pressures that select within the set of things that 00:38:54.040 --> 00:38:55.800 align:middle line:84% fit the data the one that is the smoothest 00:38:55.800 --> 00:38:57.217 align:middle line:84% are sometimes called regularizers, 00:38:57.217 --> 00:38:59.380 align:middle line:84% or implicit regularizers, or inductive biases. 00:38:59.380 --> 00:39:01.140 align:middle line:84% And we'll talk more about those in a bit. 00:39:01.140 --> 00:39:03.060 align:middle line:84% It's like, for a while, you're just fitting the data. 00:39:03.060 --> 00:39:04.300 align:middle line:90% You have to find a crazy fit. 00:39:04.300 --> 00:39:06.040 align:middle line:84% And then once you have more capacity, 00:39:06.040 --> 00:39:08.880 align:middle line:84% you can search over that kind of degeneracy 00:39:08.880 --> 00:39:12.400 align:middle line:84% to find the one that is the best according to other criteria, 00:39:12.400 --> 00:39:13.380 align:middle line:90% like smoothness. 00:39:13.380 --> 00:39:15.748 align:middle line:90% 00:39:15.748 --> 00:39:17.040 align:middle line:90% Yeah, we'll go with a question. 00:39:17.040 --> 00:39:23.240 align:middle line:84% AUDIENCE: For [INAUDIBLE] descent, [INAUDIBLE] 00:39:23.240 --> 00:39:25.600 align:middle line:84% you should stop at the first place spot, 00:39:25.600 --> 00:39:30.160 align:middle line:84% or you should train like [INAUDIBLE] training 00:39:30.160 --> 00:39:32.640 align:middle line:84% until the second coverage [INAUDIBLE]. 00:39:32.640 --> 00:39:34.240 align:middle line:84% PHILLIP ISOLA: Yeah, so the question 00:39:34.240 --> 00:39:38.120 align:middle line:84% is, should you stop at the first minima of the double descent 00:39:38.120 --> 00:39:40.980 align:middle line:84% curve, where you have low risk on the population, 00:39:40.980 --> 00:39:43.560 align:middle line:84% or should you stop at-- or should you just train forever? 00:39:43.560 --> 00:39:47.210 align:middle line:84% And generally, I would say that it 00:39:47.210 --> 00:39:53.890 align:middle line:84% will depend on how much capacity you have available to you, 00:39:53.890 --> 00:39:56.610 align:middle line:84% because this is costly to move out in the x-axis. 00:39:56.610 --> 00:39:58.910 align:middle line:84% Capacity can be measured in parameters, 00:39:58.910 --> 00:40:00.590 align:middle line:84% but it could also be measured in flops. 00:40:00.590 --> 00:40:03.330 align:middle line:84% So there's kind of double descent also in compute time. 00:40:03.330 --> 00:40:08.390 align:middle line:84% And so it will depend on the cost-benefit analysis there. 00:40:08.390 --> 00:40:10.870 align:middle line:84% But generally, in deep learning, we're out in this regime. 00:40:10.870 --> 00:40:12.610 align:middle line:84% We're past the first peak, and we're 00:40:12.610 --> 00:40:14.810 align:middle line:84% in the part where you just train longer, and longer, and longer, 00:40:14.810 --> 00:40:16.150 align:middle line:90% and it smoothly will get better. 00:40:16.150 --> 00:40:19.070 align:middle line:84% That seems to be the empirical property we have. 00:40:19.070 --> 00:40:21.050 align:middle line:84% One more, and then I'll have to move on. 00:40:21.050 --> 00:40:25.050 align:middle line:84% AUDIENCE: What's the correlation between doing 00:40:25.050 --> 00:40:29.090 align:middle line:84% polynomial regression and training for a higher order 00:40:29.090 --> 00:40:29.650 align:middle line:90% polynomial? 00:40:29.650 --> 00:40:32.190 align:middle line:84% Is it similar to a Taylor series as that sum 00:40:32.190 --> 00:40:34.690 align:middle line:84% approaches infinity that you could approximate any arbitrary 00:40:34.690 --> 00:40:35.667 align:middle line:90% function? 00:40:35.667 --> 00:40:38.250 align:middle line:84% What's the relationship between that and a polynomial function 00:40:38.250 --> 00:40:39.227 align:middle line:90% in that [INAUDIBLE]? 00:40:39.227 --> 00:40:40.810 align:middle line:84% PHILLIP ISOLA: What's the relationship 00:40:40.810 --> 00:40:45.340 align:middle line:84% between polynomial regression and deep nets-- 00:40:45.340 --> 00:40:50.820 align:middle line:84% so roughly, you can think of the last layer of an MLP 00:40:50.820 --> 00:40:54.140 align:middle line:84% as a linear combination of some basis functions, 00:40:54.140 --> 00:40:55.320 align:middle line:90% of some features. 00:40:55.320 --> 00:40:58.240 align:middle line:84% So every neuron on the previous layer is a feature of the data. 00:40:58.240 --> 00:41:02.122 align:middle line:84% And those features could be like polynomial functions of a scalar 00:41:02.122 --> 00:41:04.080 align:middle line:84% input, and that would be polynomial regression. 00:41:04.080 --> 00:41:06.700 align:middle line:90% Or they could be Fourier. 00:41:06.700 --> 00:41:09.380 align:middle line:84% They could be sines and cosines with different frequencies. 00:41:09.380 --> 00:41:11.560 align:middle line:84% And in deep learning, they'll be learned functions, 00:41:11.560 --> 00:41:14.060 align:middle line:84% but they have the capacity to learn those types of functions 00:41:14.060 --> 00:41:16.580 align:middle line:84% if those are the right ones to learn. 00:41:16.580 --> 00:41:21.020 align:middle line:84% So here's an empirical result of double descent. 00:41:21.020 --> 00:41:23.240 align:middle line:84% And this is training an MNIST classifier. 00:41:23.240 --> 00:41:24.840 align:middle line:84% And you can see this really happens. 00:41:24.840 --> 00:41:28.560 align:middle line:84% So the train error just always goes down as I add more weights. 00:41:28.560 --> 00:41:30.300 align:middle line:84% So I increase the width of my model. 00:41:30.300 --> 00:41:34.220 align:middle line:84% I get universal approximation, but test error spikes 00:41:34.220 --> 00:41:36.940 align:middle line:90% and then it goes back down. 00:41:36.940 --> 00:41:39.900 align:middle line:84% This stuff is a little finicky to actually observe in practice. 00:41:39.900 --> 00:41:42.510 align:middle line:84% So I actually tried to make my own example of this, 00:41:42.510 --> 00:41:43.860 align:middle line:90% and I couldn't get it to work. 00:41:43.860 --> 00:41:45.610 align:middle line:84% But if you read the double descent papers, 00:41:45.610 --> 00:41:48.670 align:middle line:84% you'll realize that it only really shows up 00:41:48.670 --> 00:41:49.730 align:middle line:90% in certain regimes. 00:41:49.730 --> 00:41:52.397 align:middle line:84% And you have to make sure not to have momentum in this and that, 00:41:52.397 --> 00:41:53.490 align:middle line:90% and weight decay. 00:41:53.490 --> 00:41:55.710 align:middle line:84% There's a lot of aspects of the optimization process 00:41:55.710 --> 00:41:58.550 align:middle line:90% that can make this disappear. 00:41:58.550 --> 00:42:00.790 align:middle line:84% But it is a real phenomena that does 00:42:00.790 --> 00:42:03.470 align:middle line:90% exist in the right conditions. 00:42:03.470 --> 00:42:05.788 align:middle line:84% So don't worry if you don't see it in your graphs. 00:42:05.788 --> 00:42:07.330 align:middle line:84% You might just be out in this regime, 00:42:07.330 --> 00:42:08.830 align:middle line:84% or you might have other effects that 00:42:08.830 --> 00:42:10.210 align:middle line:90% are dampening out that spike. 00:42:10.210 --> 00:42:11.290 align:middle line:90% Yeah, question over here? 00:42:11.290 --> 00:42:13.682 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:42:13.682 --> 00:42:16.662 align:middle line:90% 00:42:16.662 --> 00:42:18.370 align:middle line:84% PHILLIP ISOLA: Yeah, I'll have to think-- 00:42:18.370 --> 00:42:19.830 align:middle line:84% I'd have to think a bit more about the interaction 00:42:19.830 --> 00:42:20.890 align:middle line:90% of momentum with this. 00:42:20.890 --> 00:42:24.390 align:middle line:84% But I would say that, generally, all the tricks that we 00:42:24.390 --> 00:42:27.270 align:middle line:84% use in practice are just getting rid of that spike 00:42:27.270 --> 00:42:28.970 align:middle line:84% and pushing us more out to the right. 00:42:28.970 --> 00:42:31.550 align:middle line:84% So it's not like we're losing that benefit. 00:42:31.550 --> 00:42:33.030 align:middle line:84% We're reducing the negative effect 00:42:33.030 --> 00:42:35.610 align:middle line:84% of having this spike in the middle. 00:42:35.610 --> 00:42:39.650 align:middle line:90% 00:42:39.650 --> 00:42:42.290 align:middle line:84% So one of the interesting things that 00:42:42.290 --> 00:42:44.410 align:middle line:84% could help explain this phenomenon 00:42:44.410 --> 00:42:48.623 align:middle line:84% is that the more features that you have, 00:42:48.623 --> 00:42:50.290 align:middle line:84% that you're regressing on top of-- these 00:42:50.290 --> 00:42:54.715 align:middle line:84% could be features, like the hidden units of an MLP, 00:42:54.715 --> 00:42:57.090 align:middle line:84% or in this case, these are what are called random Fourier 00:42:57.090 --> 00:43:00.690 align:middle line:84% features-- like, random sines and cosines, 00:43:00.690 --> 00:43:04.370 align:middle line:84% random periodic functions to the data. 00:43:04.370 --> 00:43:08.290 align:middle line:84% As I have more and more features, well-- 00:43:08.290 --> 00:43:09.990 align:middle line:90% at first, I underfit the data. 00:43:09.990 --> 00:43:12.930 align:middle line:84% And then I start using all the features. 00:43:12.930 --> 00:43:16.450 align:middle line:84% And my weights are assigned to those features spike 00:43:16.450 --> 00:43:20.110 align:middle line:84% because I have to make a linear combination that fits the data. 00:43:20.110 --> 00:43:23.170 align:middle line:90% I have to be using all of them. 00:43:23.170 --> 00:43:26.170 align:middle line:84% But then what happens is, as I add more and more features, 00:43:26.170 --> 00:43:28.958 align:middle line:84% the norm of the weight vector, or the parameter 00:43:28.958 --> 00:43:31.250 align:middle line:84% norm-- the norm of that vector, the size of that vector 00:43:31.250 --> 00:43:32.850 align:middle line:90% actually decays. 00:43:32.850 --> 00:43:35.530 align:middle line:84% So I have more features, but the weights 00:43:35.530 --> 00:43:37.850 align:middle line:84% I'm applying to those features are 00:43:37.850 --> 00:43:39.460 align:middle line:84% getting smaller and smaller in terms 00:43:39.460 --> 00:43:42.002 align:middle line:84% of their overall magnitude-- the norm of the entire parameter 00:43:42.002 --> 00:43:42.660 align:middle line:90% vector. 00:43:42.660 --> 00:43:46.940 align:middle line:84% So it's like, more features means lower norm parameter 00:43:46.940 --> 00:43:49.300 align:middle line:84% solutions, so in some sense kind of smoother 00:43:49.300 --> 00:43:51.020 align:middle line:90% in a particular sense. 00:43:51.020 --> 00:43:53.912 align:middle line:84% And so this is just saying that counting number of parameters 00:43:53.912 --> 00:43:55.120 align:middle line:90% is not the right thing to do. 00:43:55.120 --> 00:43:57.980 align:middle line:84% It might be that the parameter norm or some other property 00:43:57.980 --> 00:44:03.180 align:middle line:84% of the parameters is what is the more meaningful measure 00:44:03.180 --> 00:44:06.200 align:middle line:84% of the complexity of the solution, or how wiggly it is, 00:44:06.200 --> 00:44:08.420 align:middle line:84% or how much it deviates from a nice smooth fit 00:44:08.420 --> 00:44:11.100 align:middle line:90% to the underlying trend. 00:44:11.100 --> 00:44:11.920 align:middle line:90% Yeah, question? 00:44:11.920 --> 00:44:12.980 align:middle line:84% AUDIENCE: Did that line up-- could that 00:44:12.980 --> 00:44:15.165 align:middle line:84% be lined up with the interpolation threshold? 00:44:15.165 --> 00:44:17.540 align:middle line:84% PHILLIP ISOLA: So the question is, does that peak line up 00:44:17.540 --> 00:44:19.300 align:middle line:84% with the interpolation threshold where you perfectly 00:44:19.300 --> 00:44:19.842 align:middle line:90% fit the data? 00:44:19.842 --> 00:44:22.260 align:middle line:84% Yes, it's exactly at the interpolation threshold. 00:44:22.260 --> 00:44:24.880 align:middle line:84% And you can show these things for linear models, 00:44:24.880 --> 00:44:26.160 align:middle line:90% and it's all provable. 00:44:26.160 --> 00:44:28.205 align:middle line:84% But for deep nets, it's just empirical. 00:44:28.205 --> 00:44:30.580 align:middle line:84% Let me keep going, and we can come back to more questions 00:44:30.580 --> 00:44:31.340 align:middle line:90% later. 00:44:31.340 --> 00:44:34.140 align:middle line:84% So how should we measure complexity? 00:44:34.140 --> 00:44:37.810 align:middle line:84% So number of neurons, number of weights, number of parameters? 00:44:37.810 --> 00:44:41.070 align:middle line:90% No, that just doesn't work. 00:44:41.070 --> 00:44:44.910 align:middle line:84% So double descent shows that, as I increase 00:44:44.910 --> 00:44:47.610 align:middle line:84% the number of neurons, the number of parameters, 00:44:47.610 --> 00:44:51.230 align:middle line:90% I don't necessarily overfit. 00:44:51.230 --> 00:44:52.830 align:middle line:84% So number of parameters is not really 00:44:52.830 --> 00:44:55.230 align:middle line:84% the right notion of complexity according 00:44:55.230 --> 00:44:57.470 align:middle line:84% to the classical theory that having 00:44:57.470 --> 00:45:01.710 align:middle line:84% a more expressive, higher capacity hypothesis 00:45:01.710 --> 00:45:03.850 align:middle line:90% space will cause overfitting. 00:45:03.850 --> 00:45:06.000 align:middle line:90% So number of parameters, no. 00:45:06.000 --> 00:45:08.250 align:middle line:84% Parameter norm-- maybe that seems a little bit better, 00:45:08.250 --> 00:45:12.550 align:middle line:90% but it has its own flaws. 00:45:12.550 --> 00:45:18.350 align:middle line:84% Just to emphasize this with maybe a really obvious example-- 00:45:18.350 --> 00:45:23.710 align:middle line:84% does the number of parameters a deep net has-- 00:45:23.710 --> 00:45:25.650 align:middle line:84% why would we actually expect that to matter? 00:45:25.650 --> 00:45:28.990 align:middle line:84% So consider these two deep nets, f and g. 00:45:28.990 --> 00:45:33.870 align:middle line:84% And f has a lot of parameters, a lot of edges. 00:45:33.870 --> 00:45:36.953 align:middle line:84% And g has only a few parameters, a few edges. 00:45:36.953 --> 00:45:38.870 align:middle line:84% And we're going to consider a new function, h, 00:45:38.870 --> 00:45:41.970 align:middle line:84% which is just a combination of a tiny, tiny bit of f-- 00:45:41.970 --> 00:45:44.250 align:middle line:84% 10 to the negative 100 worth of f. 00:45:44.250 --> 00:45:49.250 align:middle line:84% And then almost entirely we're going to use g. 00:45:49.250 --> 00:45:53.730 align:middle line:84% And now, we're going to ask how many parameters would it 00:45:53.730 --> 00:45:59.370 align:middle line:84% take to fit h to some epsilon degree of approximation? 00:45:59.370 --> 00:46:05.325 align:middle line:84% And the point here is that if I just fit h with g, 00:46:05.325 --> 00:46:06.950 align:middle line:84% I would only be using a few parameters, 00:46:06.950 --> 00:46:10.530 align:middle line:84% but get within epsilon is 10 to the negative 100 00:46:10.530 --> 00:46:12.450 align:middle line:90% of the correct solution. 00:46:12.450 --> 00:46:16.670 align:middle line:84% So it's kind of the count of parameters is not what matters. 00:46:16.670 --> 00:46:18.810 align:middle line:84% Something about the size has to also 00:46:18.810 --> 00:46:20.770 align:middle line:84% be what matters in terms of how well I'm 00:46:20.770 --> 00:46:24.610 align:middle line:84% able to approximate different types of functions. 00:46:24.610 --> 00:46:27.250 align:middle line:90% 00:46:27.250 --> 00:46:29.330 align:middle line:84% So parameter norm, size of the parameters 00:46:29.330 --> 00:46:31.330 align:middle line:90% matters more than count. 00:46:31.330 --> 00:46:34.220 align:middle line:84% But parameter norm is also not everything. 00:46:34.220 --> 00:46:38.080 align:middle line:90% 00:46:38.080 --> 00:46:42.540 align:middle line:84% So let's try another idea from classical theory, which is-- 00:46:42.540 --> 00:46:44.020 align:middle line:84% OK, it's not number of parameters 00:46:44.020 --> 00:46:46.980 align:middle line:84% that is the proper measure of the complexity 00:46:46.980 --> 00:46:51.220 align:middle line:84% of a hypothesis space or a neural architecture. 00:46:51.220 --> 00:46:54.380 align:middle line:84% How about the number of unique functions 00:46:54.380 --> 00:46:57.860 align:middle line:90% in the hypothesis space? 00:46:57.860 --> 00:47:01.520 align:middle line:84% It's another measure of capacity. 00:47:01.520 --> 00:47:08.768 align:middle line:84% And this is related to a theory that Vapnik and Chervonenkis-- 00:47:08.768 --> 00:47:11.060 align:middle line:84% I don't know if I got the pronunciation right-- anyway, 00:47:11.060 --> 00:47:11.960 align:middle line:90% VC theory. 00:47:11.960 --> 00:47:14.500 align:middle line:90% 00:47:14.500 --> 00:47:16.365 align:middle line:84% So remember, generalization error 00:47:16.365 --> 00:47:18.740 align:middle line:84% is the difference between the training error and the test 00:47:18.740 --> 00:47:22.300 align:middle line:84% error, or the training error and the population error. 00:47:22.300 --> 00:47:24.500 align:middle line:84% And the basic argument of this theory 00:47:24.500 --> 00:47:28.060 align:middle line:84% says that if the number of training data 00:47:28.060 --> 00:47:32.190 align:middle line:84% points I fit to is much larger than the number of functions 00:47:32.190 --> 00:47:34.550 align:middle line:84% under consideration in my hypothesis space 00:47:34.550 --> 00:47:38.630 align:middle line:84% I'm searching over, then you can guarantee to some degree 00:47:38.630 --> 00:47:43.310 align:middle line:84% that the population error matches the training error. 00:47:43.310 --> 00:47:46.450 align:middle line:84% So something about the size of the training data and the size-- 00:47:46.450 --> 00:47:49.750 align:middle line:84% the number of distinct functions that I'm searching over-- 00:47:49.750 --> 00:47:51.910 align:middle line:84% something about the ratio between these things. 00:47:51.910 --> 00:47:55.012 align:middle line:90% So the intuition is like this. 00:47:55.012 --> 00:47:57.470 align:middle line:84% So we could define something called a false positive, which 00:47:57.470 --> 00:47:59.430 align:middle line:84% would be a function that fits the training data 00:47:59.430 --> 00:48:02.870 align:middle line:90% but has high population risk-- 00:48:02.870 --> 00:48:04.330 align:middle line:90% a function that just gets lucky. 00:48:04.330 --> 00:48:07.230 align:middle line:90% It fits but doesn't generalize. 00:48:07.230 --> 00:48:09.690 align:middle line:84% So what is the chance of getting a false positive 00:48:09.690 --> 00:48:13.710 align:middle line:84% if I'm searching over all of the different candidate 00:48:13.710 --> 00:48:16.750 align:middle line:90% functions in my function class? 00:48:16.750 --> 00:48:18.830 align:middle line:84% So the chance of getting a false positive 00:48:18.830 --> 00:48:23.610 align:middle line:84% will be high if we just have more candidate functions. 00:48:23.610 --> 00:48:27.130 align:middle line:84% Like, if I just have more candidate functions, 00:48:27.130 --> 00:48:29.840 align:middle line:84% there's more random ones that will get lucky-- 00:48:29.840 --> 00:48:32.160 align:middle line:90% so that intuition. 00:48:32.160 --> 00:48:37.460 align:middle line:84% And then it should be lower if I have more data points, 00:48:37.460 --> 00:48:40.297 align:middle line:84% because every additional data point rules out 00:48:40.297 --> 00:48:41.880 align:middle line:84% certain candidate functions from being 00:48:41.880 --> 00:48:42.980 align:middle line:90% able to fit the data right. 00:48:42.980 --> 00:48:44.688 align:middle line:84% Every new data point is a new constraint. 00:48:44.688 --> 00:48:46.800 align:middle line:84% And it says there are fewer functions 00:48:46.800 --> 00:48:49.240 align:middle line:84% in my hypothesis space, in my function 00:48:49.240 --> 00:48:51.400 align:middle line:90% class that can fit the data. 00:48:51.400 --> 00:48:53.760 align:middle line:84% So therefore, if the function class is small 00:48:53.760 --> 00:48:56.320 align:middle line:84% and the training data is big, the chance of a false positive, 00:48:56.320 --> 00:48:58.298 align:middle line:90% of just getting lucky, is low. 00:48:58.298 --> 00:48:59.840 align:middle line:84% And therefore, I have a higher chance 00:48:59.840 --> 00:49:02.140 align:middle line:84% that I really found a solution that generalizes. 00:49:02.140 --> 00:49:03.820 align:middle line:84% That's kind of the rough argument. 00:49:03.820 --> 00:49:05.900 align:middle line:84% And you can make this mathematically precise. 00:49:05.900 --> 00:49:08.960 align:middle line:90% 00:49:08.960 --> 00:49:11.520 align:middle line:84% I have one more picture to make this argument just 00:49:11.520 --> 00:49:13.420 align:middle line:90% in case that wasn't clear. 00:49:13.420 --> 00:49:16.980 align:middle line:84% But let's imagine that I have this very simple setting. 00:49:16.980 --> 00:49:20.320 align:middle line:84% I have a hypothesis space, which is a discrete set of candidate 00:49:20.320 --> 00:49:21.920 align:middle line:90% functions, these circles. 00:49:21.920 --> 00:49:24.320 align:middle line:84% So every circle is a candidate function. 00:49:24.320 --> 00:49:26.840 align:middle line:84% The purple circles are the functions 00:49:26.840 --> 00:49:29.020 align:middle line:90% that fit my training data. 00:49:29.020 --> 00:49:31.292 align:middle line:84% And the true function is in green. 00:49:31.292 --> 00:49:33.000 align:middle line:84% And we're going to make some assumptions. 00:49:33.000 --> 00:49:34.260 align:middle line:84% We're going to assume that there's 00:49:34.260 --> 00:49:35.510 align:middle line:90% no noise in the training data. 00:49:35.510 --> 00:49:39.720 align:middle line:84% We're going to assume that the true function is in this set. 00:49:39.720 --> 00:49:42.980 align:middle line:84% So it's not like I have a chance of finding it. 00:49:42.980 --> 00:49:47.500 align:middle line:84% And we'll assume that my way of selecting from this set 00:49:47.500 --> 00:49:50.680 align:middle line:84% is just I pick at random one of the purple points. 00:49:50.680 --> 00:49:52.347 align:middle line:84% And the true function is also one of-- 00:49:52.347 --> 00:49:54.680 align:middle line:84% the true function also perfectly fits the training data. 00:49:54.680 --> 00:49:57.900 align:middle line:84% So my optimization algorithm is just pick at random one 00:49:57.900 --> 00:50:01.020 align:middle line:84% of the points that fits the training data. 00:50:01.020 --> 00:50:05.780 align:middle line:84% So what happens as I increase the training data? 00:50:05.780 --> 00:50:08.500 align:middle line:84% As I increase the training data, some of those purple points 00:50:08.500 --> 00:50:10.880 align:middle line:84% turn off, because if I add a new data point, 00:50:10.880 --> 00:50:14.140 align:middle line:84% some of the functions that fit the data previously no longer 00:50:14.140 --> 00:50:16.560 align:middle line:84% will fit the new data because there's more constraints. 00:50:16.560 --> 00:50:19.300 align:middle line:84% Every new data point adds new constraints 00:50:19.300 --> 00:50:22.160 align:middle line:90% that you might violate. 00:50:22.160 --> 00:50:23.940 align:middle line:84% So as I increase data, I strictly 00:50:23.940 --> 00:50:27.390 align:middle line:84% will decrease, or at least not increase, 00:50:27.390 --> 00:50:29.543 align:middle line:84% the number of these purple points. 00:50:29.543 --> 00:50:31.210 align:middle line:84% So what happened is I add training data. 00:50:31.210 --> 00:50:35.790 align:middle line:84% Right now, I have a 1 out of 6 chance of if I randomly 00:50:35.790 --> 00:50:39.250 align:middle line:84% sample one of the purple or green points, 1 out of 6 times 00:50:39.250 --> 00:50:40.950 align:middle line:90% I will get the true function. 00:50:40.950 --> 00:50:44.790 align:middle line:84% But if I add more data now, it's 1 out of 4 times. 00:50:44.790 --> 00:50:46.790 align:middle line:84% So adding data will increase my chance 00:50:46.790 --> 00:50:50.973 align:middle line:84% of getting the true function under this procedure. 00:50:50.973 --> 00:50:52.890 align:middle line:84% If I reduce the capacity, what does that mean? 00:50:52.890 --> 00:50:54.670 align:middle line:84% That just means removing circles. 00:50:54.670 --> 00:50:58.210 align:middle line:84% So the capacity is the number of functions in my function class. 00:50:58.210 --> 00:51:00.750 align:middle line:84% If I just remove circles at random-- 00:51:00.750 --> 00:51:03.250 align:middle line:84% it's not necessarily how you would normally reduce capacity, 00:51:03.250 --> 00:51:06.950 align:middle line:84% but this is one model of reducing capacity-- 00:51:06.950 --> 00:51:09.790 align:middle line:84% while still making sure that the true function is 00:51:09.790 --> 00:51:13.790 align:middle line:84% within my hypothesis space, then I 00:51:13.790 --> 00:51:16.950 align:middle line:84% will remove some of the purple spurious fits, 00:51:16.950 --> 00:51:20.190 align:middle line:84% and I'll increase the probability of, 00:51:20.190 --> 00:51:22.810 align:middle line:84% if I just select a random function, getting the right one. 00:51:22.810 --> 00:51:25.720 align:middle line:84% So there's these two things that are happening. 00:51:25.720 --> 00:51:27.920 align:middle line:84% Control capacity-- add more data. 00:51:27.920 --> 00:51:30.543 align:middle line:84% They'll both end up increasing my probability 00:51:30.543 --> 00:51:32.460 align:middle line:84% of getting the true function that generalizes. 00:51:32.460 --> 00:51:35.900 align:middle line:90% 00:51:35.900 --> 00:51:41.140 align:middle line:90% So a few more technical details. 00:51:41.140 --> 00:51:44.440 align:middle line:84% How do you count the number of functions in a function class? 00:51:44.440 --> 00:51:46.440 align:middle line:84% I said we could have a discrete set of functions 00:51:46.440 --> 00:51:47.920 align:middle line:90% under consideration. 00:51:47.920 --> 00:51:50.460 align:middle line:84% But with a neural net, it's a continuous thing. 00:51:50.460 --> 00:51:53.480 align:middle line:84% So it turns out that for the VC theory, 00:51:53.480 --> 00:51:55.300 align:middle line:90% it suffices to do the following. 00:51:55.300 --> 00:52:04.200 align:middle line:84% You simply count up how many discrete outputs your function 00:52:04.200 --> 00:52:06.440 align:middle line:90% can apply to the training data. 00:52:06.440 --> 00:52:10.360 align:middle line:84% So the simple version of this is I'm 00:52:10.360 --> 00:52:14.420 align:middle line:84% going to assume that my function is outputting-- 00:52:14.420 --> 00:52:16.260 align:middle line:90% it's a binary output. 00:52:16.260 --> 00:52:18.280 align:middle line:84% So it's going to output negative 1 or 1. 00:52:18.280 --> 00:52:22.420 align:middle line:84% So it's like binary classifications of my data. 00:52:22.420 --> 00:52:24.380 align:middle line:90% These are called dichotomies. 00:52:24.380 --> 00:52:27.120 align:middle line:84% And let's say I'm doing an image classification system. 00:52:27.120 --> 00:52:28.740 align:middle line:90% So I have some number of images. 00:52:28.740 --> 00:52:32.380 align:middle line:84% And one unique dichotomy is plus 1 to the first image, 00:52:32.380 --> 00:52:33.900 align:middle line:84% plus 1 to the next image, negative 1 00:52:33.900 --> 00:52:36.700 align:middle line:84% to the thousandth image, and so forth. 00:52:36.700 --> 00:52:40.420 align:middle line:84% And I can count up how many unique dichotomies 00:52:40.420 --> 00:52:43.660 align:middle line:84% I can apply to my data, and that will be called 00:52:43.660 --> 00:52:49.060 align:middle line:90% the VC dimension of that model. 00:52:49.060 --> 00:52:49.840 align:middle line:90% Just one sec. 00:52:49.840 --> 00:52:54.660 align:middle line:84% So VC dimension is the number of dichotomies over my n training 00:52:54.660 --> 00:52:55.240 align:middle line:90% points. 00:52:55.240 --> 00:52:58.140 align:middle line:90% 00:52:58.140 --> 00:53:02.020 align:middle line:84% And you can prove, under some assumptions, this generalization 00:53:02.020 --> 00:53:05.340 align:middle line:84% bound that the population risk will 00:53:05.340 --> 00:53:11.140 align:middle line:84% be bounded by something related to this square root of d over n. 00:53:11.140 --> 00:53:14.980 align:middle line:84% So if my model class can represent more dichotomies, 00:53:14.980 --> 00:53:19.980 align:middle line:84% can classify the training data in more different ways, 00:53:19.980 --> 00:53:22.230 align:middle line:84% then my generalization bound will increase, 00:53:22.230 --> 00:53:24.270 align:middle line:84% and I will not be able to say how well I'll 00:53:24.270 --> 00:53:25.250 align:middle line:90% do on the test data. 00:53:25.250 --> 00:53:27.990 align:middle line:84% It's like, more functions under consideration, so decreasing. 00:53:27.990 --> 00:53:30.990 align:middle line:84% But as I add more data, n, then that 00:53:30.990 --> 00:53:34.110 align:middle line:84% will rule out more and more of the spurious fits. 00:53:34.110 --> 00:53:35.830 align:middle line:84% And so we have this ratio, and we 00:53:35.830 --> 00:53:39.870 align:middle line:84% say that our population error will be bounded by this ratio. 00:53:39.870 --> 00:53:43.270 align:middle line:84% So the cartoon argument is the same, 00:53:43.270 --> 00:53:45.230 align:middle line:84% but you can prove under some conditions 00:53:45.230 --> 00:53:49.390 align:middle line:84% that this is the exact bound that you get. 00:53:49.390 --> 00:53:54.470 align:middle line:84% Let me finish this argument, and then we'll go a few questions. 00:53:54.470 --> 00:53:59.630 align:middle line:90% So here's the interesting thing. 00:53:59.630 --> 00:54:05.870 align:middle line:84% Neural nets can basically fit any dichotomy over regular data 00:54:05.870 --> 00:54:06.450 align:middle line:90% sets. 00:54:06.450 --> 00:54:08.310 align:middle line:84% So given a data set of cats versus dogs 00:54:08.310 --> 00:54:11.670 align:middle line:84% I'm trying to classify, I can train the neural net, 00:54:11.670 --> 00:54:13.610 align:middle line:84% and it will correctly classify cats and dogs. 00:54:13.610 --> 00:54:15.330 align:middle line:84% But I can do the following thing. 00:54:15.330 --> 00:54:18.210 align:middle line:84% I can randomly shuffle the labels. 00:54:18.210 --> 00:54:20.680 align:middle line:84% So I take my data set where the cats are labeled cat 00:54:20.680 --> 00:54:24.240 align:middle line:84% and dogs are labeled dog, and now I'll just 00:54:24.240 --> 00:54:28.280 align:middle line:90% randomly permute the labels-- 00:54:28.280 --> 00:54:30.220 align:middle line:84% assign each image a random label-- 00:54:30.220 --> 00:54:32.480 align:middle line:84% just choose different random ordering. 00:54:32.480 --> 00:54:36.060 align:middle line:84% So now, some dogs are labeled cats, some cats are labeled dog, 00:54:36.060 --> 00:54:37.300 align:middle line:90% some cats are labeled cats. 00:54:37.300 --> 00:54:39.360 align:middle line:90% It's completely random labels. 00:54:39.360 --> 00:54:42.760 align:middle line:84% And it turns out that, in practice, the neural 00:54:42.760 --> 00:54:45.560 align:middle line:84% nets we use can fit any random labeling. 00:54:45.560 --> 00:54:48.480 align:middle line:84% So in other words, they can achieve any dichotomy. 00:54:48.480 --> 00:54:51.640 align:middle line:84% So in other words, the VC dimension 00:54:51.640 --> 00:54:56.560 align:middle line:84% is the total possible numbers of dichotomies of n points. 00:54:56.560 --> 00:55:02.680 align:middle line:84% So how many different ways can I binarily classify n points? 00:55:02.680 --> 00:55:07.000 align:middle line:90% 2 to the n possible ways. 00:55:07.000 --> 00:55:10.150 align:middle line:84% So every single point I can assign either a plus 1 or minus 00:55:10.150 --> 00:55:12.850 align:middle line:84% 1, and then I can do that for every other point-- for n 00:55:12.850 --> 00:55:14.160 align:middle line:90% points. 00:55:14.160 --> 00:55:18.515 align:middle line:84% And so the VC dimension d is equal to 2 to the n. 00:55:18.515 --> 00:55:20.890 align:middle line:84% And the generalization bound is exponential in the number 00:55:20.890 --> 00:55:22.515 align:middle line:84% of data training points-- is 2 to the n 00:55:22.515 --> 00:55:24.690 align:middle line:90% over square root of that. 00:55:24.690 --> 00:55:27.630 align:middle line:84% And it turns out that this is just extremely loose. 00:55:27.630 --> 00:55:29.270 align:middle line:84% And in practice, it's just vacuous. 00:55:29.270 --> 00:55:34.070 align:middle line:84% It says that my population error will be greater than 100% 00:55:34.070 --> 00:55:38.370 align:middle line:84% if this is a classifier, or my population error 00:55:38.370 --> 00:55:44.470 align:middle line:84% is less than 10 billion percent or something like this. 00:55:44.470 --> 00:55:46.850 align:middle line:84% So as n becomes large, this scales exponentially, 00:55:46.850 --> 00:55:48.910 align:middle line:90% and it becomes a vacuous bound. 00:55:48.910 --> 00:55:54.170 align:middle line:84% It tells us nothing about how well the system will generalize. 00:55:54.170 --> 00:56:01.730 align:middle line:84% So the VC theory doesn't explain deep learning. 00:56:01.730 --> 00:56:04.930 align:middle line:84% It has the right, I think, paradigm 00:56:04.930 --> 00:56:07.690 align:middle line:84% of thinking about the size of the hypothesis space, the number 00:56:07.690 --> 00:56:10.890 align:middle line:84% of constraints, how many functions could fit the data 00:56:10.890 --> 00:56:12.450 align:middle line:90% without generalizing well. 00:56:12.450 --> 00:56:14.557 align:middle line:84% But the assumptions it makes are violated 00:56:14.557 --> 00:56:16.390 align:middle line:84% and don't take into account the full picture 00:56:16.390 --> 00:56:20.230 align:middle line:84% of all of the factors that are affecting deep learning. 00:56:20.230 --> 00:56:22.690 align:middle line:84% One of the main ones is that, in deep learning, 00:56:22.690 --> 00:56:25.290 align:middle line:84% we're not just picking a random hypothesis that fits the data. 00:56:25.290 --> 00:56:26.870 align:middle line:84% We're using gradient descent, and that's 00:56:26.870 --> 00:56:29.230 align:middle line:84% going to pick certain hypotheses and prefer certain hypotheses 00:56:29.230 --> 00:56:29.910 align:middle line:90% over others. 00:56:29.910 --> 00:56:32.490 align:middle line:84% Among all hypotheses that fit the data, 00:56:32.490 --> 00:56:34.410 align:middle line:84% it might prefer simple ones over complex ones, 00:56:34.410 --> 00:56:35.577 align:middle line:90% which we'll see in a minute. 00:56:35.577 --> 00:56:38.690 align:middle line:90% 00:56:38.690 --> 00:56:40.550 align:middle line:84% And this is one of the optional readings. 00:56:40.550 --> 00:56:44.270 align:middle line:84% There's a nice paper from a few years ago called "Understanding 00:56:44.270 --> 00:56:46.750 align:middle line:84% Deep Learning Requires Rethinking Generalization." 00:56:46.750 --> 00:56:48.430 align:middle line:90% It ran that experiment. 00:56:48.430 --> 00:56:51.810 align:middle line:84% It took MNIST, and CIFAR, and some of the classic data sets, 00:56:51.810 --> 00:56:53.850 align:middle line:84% and it randomly shuffled the labels. 00:56:53.850 --> 00:56:56.670 align:middle line:84% And it showed that you can get 100% training accuracy 00:56:56.670 --> 00:56:59.150 align:middle line:84% on any random shuffling of the labels. 00:56:59.150 --> 00:57:03.510 align:middle line:84% But if you use the true labels, then your test accuracy 00:57:03.510 --> 00:57:04.210 align:middle line:90% will be high. 00:57:04.210 --> 00:57:07.790 align:middle line:84% But if you just randomly shuffle the labels, 00:57:07.790 --> 00:57:10.330 align:middle line:84% now, there's no ability to generalize at all. 00:57:10.330 --> 00:57:12.310 align:middle line:90% Your test accuracy will be low. 00:57:12.310 --> 00:57:17.160 align:middle line:84% And VC theory would say that the generalization bound 00:57:17.160 --> 00:57:18.300 align:middle line:90% is completely vacuous. 00:57:18.300 --> 00:57:20.058 align:middle line:84% It doesn't tell us anything about how well 00:57:20.058 --> 00:57:21.600 align:middle line:84% one or the other of these things will 00:57:21.600 --> 00:57:22.660 align:middle line:90% generalize to the test data. 00:57:22.660 --> 00:57:24.480 align:middle line:84% So there must be something more about what 00:57:24.480 --> 00:57:27.260 align:middle line:84% neural nets are doing that's not explained by VC theory. 00:57:27.260 --> 00:57:30.260 align:middle line:90% 00:57:30.260 --> 00:57:34.000 align:middle line:84% So neural nets can fit random labels, 00:57:34.000 --> 00:57:38.320 align:middle line:84% yet when they're trained on real labels, they generalize. 00:57:38.320 --> 00:57:41.520 align:middle line:84% This is the violation of classical theory. 00:57:41.520 --> 00:57:44.392 align:middle line:90% Any questions there? 00:57:44.392 --> 00:57:46.600 align:middle line:84% I think we had a few, so we can go back to the slides 00:57:46.600 --> 00:57:49.560 align:middle line:84% where you had them if you still have them. 00:57:49.560 --> 00:57:50.200 align:middle line:90% Yeah? 00:57:50.200 --> 00:57:55.000 align:middle line:84% AUDIENCE: So a dichotomy is just [INAUDIBLE]. 00:57:55.000 --> 00:57:59.200 align:middle line:84% PHILLIP ISOLA: Yeah, dichotomy is a binary labeling of all 00:57:59.200 --> 00:58:02.880 align:middle line:84% of the training data points, so a vector 00:58:02.880 --> 00:58:07.960 align:middle line:84% of assignments of the training points to 0 or 1-- 00:58:07.960 --> 00:58:08.980 align:middle line:90% negative 1 or 1. 00:58:08.980 --> 00:58:15.830 align:middle line:90% 00:58:15.830 --> 00:58:19.410 align:middle line:84% So how should we measure model complexity 00:58:19.410 --> 00:58:20.887 align:middle line:90% in the deep learning era? 00:58:20.887 --> 00:58:22.970 align:middle line:84% Should it just be the number of distinct functions 00:58:22.970 --> 00:58:24.530 align:middle line:90% the model can represent? 00:58:24.530 --> 00:58:27.570 align:middle line:84% No, because deep nets can represent 00:58:27.570 --> 00:58:30.410 align:middle line:84% a huge number of functions, and so many functions 00:58:30.410 --> 00:58:32.730 align:middle line:90% that the classical bounds on-- 00:58:32.730 --> 00:58:35.810 align:middle line:84% that tell us how capacity relates to generalization 00:58:35.810 --> 00:58:36.590 align:middle line:90% don't hold. 00:58:36.590 --> 00:58:39.290 align:middle line:84% They would say that this is way too high capacity a model. 00:58:39.290 --> 00:58:40.595 align:middle line:90% There must be some other thing. 00:58:40.595 --> 00:58:41.970 align:middle line:84% There must be something else that 00:58:41.970 --> 00:58:45.590 align:middle line:84% is helping us to find solutions that generalize 00:58:45.590 --> 00:58:47.470 align:middle line:84% rather than solutions that don't generalize. 00:58:47.470 --> 00:58:51.090 align:middle line:90% 00:58:51.090 --> 00:58:53.790 align:middle line:84% And unfortunately, I have to say that for deep learning, 00:58:53.790 --> 00:58:55.543 align:middle line:90% we really don't know what it is. 00:58:55.543 --> 00:58:56.710 align:middle line:90% It's still an open question. 00:58:56.710 --> 00:58:57.710 align:middle line:90% There's a lot of papers. 00:58:57.710 --> 00:58:59.450 align:middle line:84% There's a lot of new ideas on this. 00:58:59.450 --> 00:59:01.410 align:middle line:84% I'm not saying that we don't know anything, 00:59:01.410 --> 00:59:05.430 align:middle line:84% but a lot of the classical tools aren't working in this era. 00:59:05.430 --> 00:59:10.490 align:middle line:84% And it's a great thing for us to try to make progress on. 00:59:10.490 --> 00:59:15.270 align:middle line:84% So I'm going to use the final section to talk about some 00:59:15.270 --> 00:59:17.590 align:middle line:84% of the inductive biases and pressures that 00:59:17.590 --> 00:59:21.990 align:middle line:84% do lead deep nets to find solutions that generalize, 00:59:21.990 --> 00:59:26.270 align:middle line:84% that give us some inkling of where the theory might go 00:59:26.270 --> 00:59:27.530 align:middle line:90% and some practical tips. 00:59:27.530 --> 00:59:30.370 align:middle line:84% But the recap so far is that, empirically, 00:59:30.370 --> 00:59:32.430 align:middle line:90% deep nets generalize. 00:59:32.430 --> 00:59:34.850 align:middle line:84% They can make reasonable predictions, 00:59:34.850 --> 00:59:37.270 align:middle line:84% often accurate predictions on inputs that 00:59:37.270 --> 00:59:38.730 align:middle line:90% are not in their training data. 00:59:38.730 --> 00:59:42.230 align:middle line:84% And you can even show this for large language models 00:59:42.230 --> 00:59:44.610 align:middle line:84% and things where they've been trained on so much data. 00:59:44.610 --> 00:59:46.630 align:middle line:84% But you can just make a counting argument 00:59:46.630 --> 00:59:50.250 align:middle line:84% that they still can't possibly be just memorizing. 00:59:50.250 --> 00:59:54.190 align:middle line:84% They have to be doing some generalization. 00:59:54.190 --> 00:59:56.990 align:middle line:84% And generalization, just from first principles, 00:59:56.990 --> 00:59:58.930 align:middle line:84% requires some kind of inductive bias. 00:59:58.930 --> 01:00:02.430 align:middle line:84% You have to have something that rules out the filing cabinet. 01:00:02.430 --> 01:00:05.190 align:middle line:84% Only fitting the training data cannot fully explain 01:00:05.190 --> 01:00:08.000 align:middle line:84% generalization, because you can fit the training data by just 01:00:08.000 --> 01:00:11.360 align:middle line:84% memorizing the training points, but not having-- 01:00:11.360 --> 01:00:13.880 align:middle line:84% saying anything about the samples in between 01:00:13.880 --> 01:00:18.350 align:middle line:84% the out-of-distribution samples in between the training points-- 01:00:18.350 --> 01:00:22.260 align:middle line:84% or sorry, the out-of-sample samples in the test set. 01:00:22.260 --> 01:00:25.320 align:middle line:90% 01:00:25.320 --> 01:00:28.600 align:middle line:84% So the inductive biases can't really just 01:00:28.600 --> 01:00:31.780 align:middle line:84% be about classical notions of model capacity, 01:00:31.780 --> 01:00:34.220 align:middle line:84% like number of parameters or VC dimension, 01:00:34.220 --> 01:00:36.480 align:middle line:84% which is number of functions that can be represented 01:00:36.480 --> 01:00:38.200 align:middle line:90% by that hypothesis space. 01:00:38.200 --> 01:00:41.160 align:middle line:84% So we need to understand what are the inductive biases, 01:00:41.160 --> 01:00:43.600 align:middle line:84% and how do they relate to meaningful controls 01:00:43.600 --> 01:00:45.780 align:middle line:84% on the expressivity and the capacity of the model? 01:00:45.780 --> 01:00:48.680 align:middle line:84% How do they prefer simple or smooth solutions 01:00:48.680 --> 01:00:51.720 align:middle line:84% over other solutions, and how do we measure that? 01:00:51.720 --> 01:00:55.040 align:middle line:84% And this is very much open and ongoing research. 01:00:55.040 --> 01:00:58.868 align:middle line:84% So deep nets must have some nice inductive biases. 01:00:58.868 --> 01:01:01.160 align:middle line:84% They must control the complexity in ways we don't fully 01:01:01.160 --> 01:01:03.280 align:middle line:90% know how to characterize. 01:01:03.280 --> 01:01:04.840 align:middle line:84% So what are these inductive biases 01:01:04.840 --> 01:01:07.730 align:middle line:84% that deep nets have that allow them to generalize? 01:01:07.730 --> 01:01:10.570 align:middle line:90% 01:01:10.570 --> 01:01:13.215 align:middle line:84% So why do deep nets learn functions that generalize? 01:01:13.215 --> 01:01:15.090 align:middle line:84% And the basic question is, other than fitting 01:01:15.090 --> 01:01:16.290 align:middle line:84% the training data-- we understand that. 01:01:16.290 --> 01:01:17.165 align:middle line:90% That's approximation. 01:01:17.165 --> 01:01:19.540 align:middle line:84% We understand that fitting the training data is critical. 01:01:19.540 --> 01:01:21.470 align:middle line:84% That's part of the recipe for generalization, 01:01:21.470 --> 01:01:22.930 align:middle line:84% but there has to be something else. 01:01:22.930 --> 01:01:24.230 align:middle line:84% Other than fitting the training data, 01:01:24.230 --> 01:01:26.170 align:middle line:84% what are the other pressures that affect the solution 01:01:26.170 --> 01:01:27.230 align:middle line:90% that deep nets arrive at? 01:01:27.230 --> 01:01:29.050 align:middle line:84% There must be some other pressures 01:01:29.050 --> 01:01:31.970 align:middle line:84% that affect which of the two solutions that both perfectly 01:01:31.970 --> 01:01:34.942 align:middle line:84% fit the training data does the deep net choose? 01:01:34.942 --> 01:01:36.650 align:middle line:84% And why does it choose the one that works 01:01:36.650 --> 01:01:39.370 align:middle line:84% and not the filing cabinet one, which potentially it 01:01:39.370 --> 01:01:41.430 align:middle line:84% would have the capacity to do in some cases? 01:01:41.430 --> 01:01:44.417 align:middle line:90% 01:01:44.417 --> 01:01:46.750 align:middle line:84% So if we fit the training data, what's left to consider? 01:01:46.750 --> 01:01:50.182 align:middle line:84% So this purple space is meant to be-- it's the same space as I 01:01:50.182 --> 01:01:51.390 align:middle line:90% showed in purple dots before. 01:01:51.390 --> 01:01:53.765 align:middle line:84% But now, I'm going to give it this name, "version space." 01:01:53.765 --> 01:01:57.690 align:middle line:84% So the version space is the set of all hypotheses 01:01:57.690 --> 01:02:01.770 align:middle line:84% in the hypothesis space that fit the training data. 01:02:01.770 --> 01:02:04.110 align:middle line:84% So in neural nets, if I have an MLP, 01:02:04.110 --> 01:02:08.450 align:middle line:84% the version space is the set of all settings of the parameters, 01:02:08.450 --> 01:02:11.350 align:middle line:84% the weights and the biases, that perfectly interpolate, 01:02:11.350 --> 01:02:15.270 align:middle line:84% that perfectly fit my training data that achieves zero training 01:02:15.270 --> 01:02:16.150 align:middle line:90% error. 01:02:16.150 --> 01:02:20.450 align:middle line:84% So some blob within the set of functions I'm searching over, 01:02:20.450 --> 01:02:23.650 align:middle line:84% which is the set of weights and biases of the MLP, 01:02:23.650 --> 01:02:24.650 align:middle line:90% if that is my model. 01:02:24.650 --> 01:02:28.150 align:middle line:90% 01:02:28.150 --> 01:02:31.807 align:middle line:84% So basically, the point is that some points in this set 01:02:31.807 --> 01:02:33.390 align:middle line:84% are bad and don't generalize, and some 01:02:33.390 --> 01:02:34.515 align:middle line:90% are good and do generalize. 01:02:34.515 --> 01:02:37.590 align:middle line:84% Why do deep nets find the ones that are good and do generalize 01:02:37.590 --> 01:02:40.690 align:middle line:90% to some degree? 01:02:40.690 --> 01:02:42.990 align:middle line:84% So these are all kind of ideas that are out there 01:02:42.990 --> 01:02:47.710 align:middle line:84% but are not yet completely standard or proven. 01:02:47.710 --> 01:02:50.850 align:middle line:84% But I think that they're fun to think about. 01:02:50.850 --> 01:02:54.110 align:middle line:84% So the first one is the idea that there's 01:02:54.110 --> 01:02:56.870 align:middle line:84% a simplicity bias in what's called the parameter function 01:02:56.870 --> 01:02:58.950 align:middle line:90% map. 01:02:58.950 --> 01:03:01.410 align:middle line:84% The parameter function map is-- remember 01:03:01.410 --> 01:03:03.660 align:middle line:84% that we're trying to find functions that fit the data, 01:03:03.660 --> 01:03:07.400 align:middle line:84% but we're trying to find them by searching over parametrization 01:03:07.400 --> 01:03:08.238 align:middle line:90% of those functions. 01:03:08.238 --> 01:03:10.280 align:middle line:84% So we have access to the parameters-- the weights 01:03:10.280 --> 01:03:11.020 align:middle line:90% and biases. 01:03:11.020 --> 01:03:13.520 align:middle line:84% But every setting of the weights and biases defines actually 01:03:13.520 --> 01:03:16.080 align:middle line:90% some mathematical mapping f. 01:03:16.080 --> 01:03:20.400 align:middle line:84% And there might be some parameter vectors-- 01:03:20.400 --> 01:03:23.840 align:middle line:84% two parameter vectors could define the same f. 01:03:23.840 --> 01:03:30.042 align:middle line:84% Like, if I have a network which just adds a bias of one 01:03:30.042 --> 01:03:32.000 align:middle line:84% on one layer and then subtracts the bias of one 01:03:32.000 --> 01:03:34.800 align:middle line:84% on the next layer, and otherwise it is an identity, 01:03:34.800 --> 01:03:39.000 align:middle line:84% well, that is equivalent to just a network that adds a bias of 0 01:03:39.000 --> 01:03:41.420 align:middle line:90% and then adds a bias of 0. 01:03:41.420 --> 01:03:43.020 align:middle line:90% So plus 1 minus 1 equals 0. 01:03:43.020 --> 01:03:46.420 align:middle line:84% There are a multiplicity-- this is a many-to-one mapping. 01:03:46.420 --> 01:03:49.003 align:middle line:84% There's a multiplicity of parameter vectors 01:03:49.003 --> 01:03:50.420 align:middle line:84% that all map to the same function. 01:03:50.420 --> 01:03:52.742 align:middle line:84% So the parameters and the function space 01:03:52.742 --> 01:03:53.700 align:middle line:90% are not the same thing. 01:03:53.700 --> 01:03:57.200 align:middle line:84% And we can look at the parameter function map. 01:03:57.200 --> 01:04:00.130 align:middle line:84% And this is what was observed by these authors a few years ago 01:04:00.130 --> 01:04:00.790 align:middle line:90% empirically. 01:04:00.790 --> 01:04:04.170 align:middle line:84% They didn't prove this, but they observed it empirically. 01:04:04.170 --> 01:04:08.410 align:middle line:84% If you just sample a random set of parameters 01:04:08.410 --> 01:04:12.450 align:middle line:84% in an MLP or some other neural net models, 01:04:12.450 --> 01:04:16.450 align:middle line:84% most of them will map to functions f 01:04:16.450 --> 01:04:19.655 align:middle line:90% that are simple according to-- 01:04:19.655 --> 01:04:21.030 align:middle line:84% this is not number of parameters. 01:04:21.030 --> 01:04:22.890 align:middle line:90% This is not number of distinct-- 01:04:22.890 --> 01:04:23.990 align:middle line:90% this is not VC dimension. 01:04:23.990 --> 01:04:25.782 align:middle line:84% This is another one, Lempel-Ziv complexity. 01:04:25.782 --> 01:04:28.890 align:middle line:84% It's like saying Lempel-Ziv is one of these compression 01:04:28.890 --> 01:04:29.390 align:middle line:90% algorithms. 01:04:29.390 --> 01:04:30.932 align:middle line:84% And I think this is saying like, if I 01:04:30.932 --> 01:04:32.310 align:middle line:84% run that function f on some data, 01:04:32.310 --> 01:04:34.410 align:middle line:84% how well can I compress it with this algorithm? 01:04:34.410 --> 01:04:37.020 align:middle line:84% So another crazy complexity measure 01:04:37.020 --> 01:04:38.770 align:middle line:84% that's hard to understand exactly how that 01:04:38.770 --> 01:04:42.690 align:middle line:84% would relate to generalization, but they tried that one. 01:04:42.690 --> 01:04:44.810 align:middle line:90% And so they're saying that-- 01:04:44.810 --> 01:04:47.360 align:middle line:84% this is like a probability distribution, I believe. 01:04:47.360 --> 01:04:49.110 align:middle line:84% It's a little hard to interpret this plot, 01:04:49.110 --> 01:04:54.170 align:middle line:84% but there is high probability of finding-- 01:04:54.170 --> 01:04:55.670 align:middle line:84% if I take a random parameter vector, 01:04:55.670 --> 01:04:57.420 align:middle line:84% a random setting of the weights and biases 01:04:57.420 --> 01:05:00.420 align:middle line:84% in a network, most random settings will be down here. 01:05:00.420 --> 01:05:02.260 align:middle line:84% It'll be higher probability, and only a few 01:05:02.260 --> 01:05:04.260 align:middle line:84% will be functions-- will define functions 01:05:04.260 --> 01:05:07.500 align:middle line:84% that are really uncompressable and complex 01:05:07.500 --> 01:05:09.700 align:middle line:90% in that particular sense. 01:05:09.700 --> 01:05:13.060 align:middle line:84% So here's a way to look at-- an intuitive 01:05:13.060 --> 01:05:14.460 align:middle line:90% way of looking at this. 01:05:14.460 --> 01:05:17.300 align:middle line:84% So if we randomly pick a point in the parameter 01:05:17.300 --> 01:05:20.140 align:middle line:84% space of the neural network, this 01:05:20.140 --> 01:05:22.800 align:middle line:84% will tend to map towards simple functions. 01:05:22.800 --> 01:05:25.100 align:middle line:84% So imagine that we're organizing this hypothesis space 01:05:25.100 --> 01:05:26.960 align:middle line:90% by the simple functions. 01:05:26.960 --> 01:05:29.780 align:middle line:84% The smooth compressible functions are down at the left, 01:05:29.780 --> 01:05:32.060 align:middle line:84% and more complex, wiggly, incompressible functions 01:05:32.060 --> 01:05:34.020 align:middle line:90% are up at the right. 01:05:34.020 --> 01:05:37.188 align:middle line:84% So most random samples will be simple. 01:05:37.188 --> 01:05:38.480 align:middle line:90% And then what does learning do? 01:05:38.480 --> 01:05:40.462 align:middle line:84% Learning just moves our parameters 01:05:40.462 --> 01:05:42.420 align:middle line:84% and, therefore, moves the functions they map to 01:05:42.420 --> 01:05:44.440 align:middle line:90% toward the version space. 01:05:44.440 --> 01:05:46.600 align:middle line:84% It moves you toward the space that fits the data. 01:05:46.600 --> 01:05:48.500 align:middle line:84% And so we'll end up in this little section 01:05:48.500 --> 01:05:51.420 align:middle line:84% where we're at the corner of the version space 01:05:51.420 --> 01:05:53.460 align:middle line:84% where we have simple functions just by chance, 01:05:53.460 --> 01:05:58.680 align:middle line:84% because most random initializations of my parameters 01:05:58.680 --> 01:06:00.200 align:middle line:90% map to simple functions. 01:06:00.200 --> 01:06:03.000 align:middle line:84% And most of the volume, as I walk through parameter space, 01:06:03.000 --> 01:06:04.700 align:middle line:84% will also map to simple functions. 01:06:04.700 --> 01:06:08.320 align:middle line:84% And so just by chance, we will find a solution 01:06:08.320 --> 01:06:09.860 align:middle line:90% that is simple in that sense. 01:06:09.860 --> 01:06:14.193 align:middle line:90% 01:06:14.193 --> 01:06:20.700 align:middle line:84% So I was involved in another paper that looked at that, 01:06:20.700 --> 01:06:24.300 align:middle line:84% but looked particularly at a measure of simplicity, 01:06:24.300 --> 01:06:26.432 align:middle line:90% which is the rank of the kernel. 01:06:26.432 --> 01:06:28.140 align:middle line:84% So I'm going to describe what that means. 01:06:28.140 --> 01:06:30.880 align:middle line:84% But we found that there's also this bias in the parameter 01:06:30.880 --> 01:06:34.120 align:middle line:84% function map toward solutions that are 01:06:34.120 --> 01:06:37.960 align:middle line:90% low rank in a certain sense. 01:06:37.960 --> 01:06:45.440 align:middle line:84% So here's a MLP, W, ReLU, W. And here's a deeper MLP. 01:06:45.440 --> 01:06:46.783 align:middle line:90% And one of the classic stories-- 01:06:46.783 --> 01:06:49.200 align:middle line:84% And we'll come back to this in the representation learning 01:06:49.200 --> 01:06:49.700 align:middle line:90% lectures. 01:06:49.700 --> 01:06:53.400 align:middle line:84% But one of the classical stories is that shallow nets-- 01:06:53.400 --> 01:06:56.730 align:middle line:84% let's say I'm trying to classify between 10 different classes 01:06:56.730 --> 01:06:58.290 align:middle line:90% indicated by the colors. 01:06:58.290 --> 01:07:00.712 align:middle line:84% I'm trying to take my data and map them 01:07:00.712 --> 01:07:02.670 align:middle line:84% to a space where they're all linearly separable 01:07:02.670 --> 01:07:03.630 align:middle line:90% so I can classify them. 01:07:03.630 --> 01:07:07.410 align:middle line:84% So the classic story is that deeper nets with more parameters 01:07:07.410 --> 01:07:09.050 align:middle line:84% have more capacity to organize the data 01:07:09.050 --> 01:07:10.550 align:middle line:90% and separate all those classes. 01:07:10.550 --> 01:07:13.150 align:middle line:90% 01:07:13.150 --> 01:07:14.770 align:middle line:90% So that's the picture here. 01:07:14.770 --> 01:07:17.810 align:middle line:84% We get better linear separation of all these classes 01:07:17.810 --> 01:07:20.290 align:middle line:90% at the top of the network. 01:07:20.290 --> 01:07:23.490 align:middle line:84% Now, I'm going to characterize that distribution 01:07:23.490 --> 01:07:26.970 align:middle line:84% at the top of the network with a kernel matrix, which 01:07:26.970 --> 01:07:32.130 align:middle line:84% is going to be the similarity between every data 01:07:32.130 --> 01:07:33.750 align:middle line:84% point and every other data point. 01:07:33.750 --> 01:07:37.170 align:middle line:84% So this creates a matrix, a bivariate function. 01:07:37.170 --> 01:07:41.570 align:middle line:84% And it's called the kernel of my embeddings 01:07:41.570 --> 01:07:44.090 align:middle line:84% that the neural network arrives at. 01:07:44.090 --> 01:07:48.450 align:middle line:90% So every row and column is-- 01:07:48.450 --> 01:07:50.130 align:middle line:84% every row is a data point and the column 01:07:50.130 --> 01:07:51.230 align:middle line:90% is another data point. 01:07:51.230 --> 01:07:54.220 align:middle line:84% And the blue value is the similarity 01:07:54.220 --> 01:07:58.300 align:middle line:84% in the representational vectors at the output of that network. 01:07:58.300 --> 01:08:00.720 align:middle line:84% So we'll see these kinds of kernels a few times. 01:08:00.720 --> 01:08:03.300 align:middle line:84% This is a kernel that characterizes the representation 01:08:03.300 --> 01:08:06.340 align:middle line:84% at the output of the network over the input data. 01:08:06.340 --> 01:08:08.420 align:middle line:84% And if it's kind of block-structured 01:08:08.420 --> 01:08:12.860 align:middle line:84% like it is here, that means that it's found clusters. 01:08:12.860 --> 01:08:18.300 align:middle line:84% So that's like all the red points have grouped together 01:08:18.300 --> 01:08:20.368 align:middle line:84% into one block, and all the purple points 01:08:20.368 --> 01:08:21.160 align:middle line:90% into another block. 01:08:21.160 --> 01:08:22.080 align:middle line:90% And that's good. 01:08:22.080 --> 01:08:23.740 align:middle line:84% So that's kind of what we're looking 01:08:23.740 --> 01:08:25.819 align:middle line:84% for if we want a good organization of my data 01:08:25.819 --> 01:08:27.526 align:middle line:90% that's easy to classify. 01:08:27.526 --> 01:08:28.359 align:middle line:90% It's very clustered. 01:08:28.359 --> 01:08:32.380 align:middle line:84% It will look block-structured like this. 01:08:32.380 --> 01:08:35.700 align:middle line:84% So the common understanding is that deeper nets 01:08:35.700 --> 01:08:37.220 align:middle line:84% have greater capacity to organize 01:08:37.220 --> 01:08:39.380 align:middle line:84% the data into this clustered structure that's 01:08:39.380 --> 01:08:43.308 align:middle line:84% easy to then make a prediction on top of to classify. 01:08:43.308 --> 01:08:45.100 align:middle line:84% But the interesting thing is if you get rid 01:08:45.100 --> 01:08:47.939 align:middle line:84% of the nonlinearities, you actually get the same effect. 01:08:47.939 --> 01:08:52.600 align:middle line:84% So just a linear network-- just a stack of linear layers also 01:08:52.600 --> 01:08:56.439 align:middle line:84% will result in a deeper linear net-- 01:08:56.439 --> 01:09:00.960 align:middle line:90% results in more blocky kernels. 01:09:00.960 --> 01:09:04.620 align:middle line:84% So we're going to characterize now the kernel with its rank. 01:09:04.620 --> 01:09:07.960 align:middle line:84% So the rank of that matrix is how many 01:09:07.960 --> 01:09:11.200 align:middle line:84% linearly independent rows and columns there are. 01:09:11.200 --> 01:09:15.040 align:middle line:84% So blockier matrices have lower rank. 01:09:15.040 --> 01:09:17.359 align:middle line:84% Lower rank is like, I have clustered 01:09:17.359 --> 01:09:20.319 align:middle line:90% the data into a simpler format. 01:09:20.319 --> 01:09:23.680 align:middle line:84% And that's useful for classification. 01:09:23.680 --> 01:09:26.560 align:middle line:84% So you get the same phenomenon with deep linear nets. 01:09:26.560 --> 01:09:29.760 align:middle line:84% And deep linear nets are equivalent to one 01:09:29.760 --> 01:09:31.060 align:middle line:90% linear transformation. 01:09:31.060 --> 01:09:32.920 align:middle line:84% So you can't explain this phenomena 01:09:32.920 --> 01:09:38.479 align:middle line:84% as the deeper nets have more capacity to organize the data. 01:09:38.479 --> 01:09:39.535 align:middle line:90% So what's going on? 01:09:39.535 --> 01:09:41.160 align:middle line:84% The argument that we make in this paper 01:09:41.160 --> 01:09:46.120 align:middle line:84% is that products of matrices tend to be low rank. 01:09:46.120 --> 01:09:50.130 align:middle line:84% So again, it's like a parameter function map. 01:09:50.130 --> 01:09:53.290 align:middle line:84% If I have a random set of weight matrices, 01:09:53.290 --> 01:09:58.810 align:middle line:84% if I have more of them, just it will be higher probability 01:09:58.810 --> 01:10:04.170 align:middle line:84% that I will output vectors whose kernel is lower rank. 01:10:04.170 --> 01:10:07.010 align:middle line:84% So this is a property of random matrix theory 01:10:07.010 --> 01:10:09.490 align:middle line:84% that products of random Gaussian matrices 01:10:09.490 --> 01:10:12.570 align:middle line:84% will result in a low rank matrix. 01:10:12.570 --> 01:10:17.690 align:middle line:84% Now, deep nets are not products of random matrices, 01:10:17.690 --> 01:10:21.090 align:middle line:84% but the full argument is that if I'm looking at the weight 01:10:21.090 --> 01:10:23.690 align:middle line:84% space of a neural network and I'm sampling a random point, 01:10:23.690 --> 01:10:25.650 align:middle line:84% if I have a shallow network, then 01:10:25.650 --> 01:10:29.370 align:middle line:84% most random points will be full rank. 01:10:29.370 --> 01:10:31.650 align:middle line:84% If I have a deeper network, whether it's 01:10:31.650 --> 01:10:34.730 align:middle line:84% a linear net or a nonlinear net, I 01:10:34.730 --> 01:10:39.250 align:middle line:84% will tend to map to lower rank solutions just by chance. 01:10:39.250 --> 01:10:41.845 align:middle line:84% Now, optimization will pick out which of the parameter 01:10:41.845 --> 01:10:43.970 align:middle line:84% vectors actually is in the version space-- actually 01:10:43.970 --> 01:10:45.130 align:middle line:90% fits the data. 01:10:45.130 --> 01:10:47.600 align:middle line:84% But because most of the volume of parameter space is 01:10:47.600 --> 01:10:49.860 align:middle line:90% mapping to low rank solutions-- 01:10:49.860 --> 01:10:55.030 align:middle line:84% low rank representations of the data, when the network is deep, 01:10:55.030 --> 01:10:56.280 align:middle line:90% there'll be this kind of bias. 01:10:56.280 --> 01:10:58.420 align:middle line:84% There'll be the bias that if I'm just 01:10:58.420 --> 01:11:00.120 align:middle line:84% searching over some random location, 01:11:00.120 --> 01:11:02.820 align:middle line:84% I'll tend to be more likely to find low rank solutions 01:11:02.820 --> 01:11:04.220 align:middle line:90% than high rank solutions. 01:11:04.220 --> 01:11:05.980 align:middle line:84% And empirically, this is what happens. 01:11:05.980 --> 01:11:07.420 align:middle line:84% So a lot of these things are only 01:11:07.420 --> 01:11:11.440 align:middle line:84% kind of provable and analyzable in linear models. 01:11:11.440 --> 01:11:15.420 align:middle line:84% But then we can show empirically in real deep nets. 01:11:15.420 --> 01:11:18.395 align:middle line:84% And what I'm showing here is the rank-- 01:11:18.395 --> 01:11:19.520 align:middle line:90% it's called effective rank. 01:11:19.520 --> 01:11:21.437 align:middle line:84% It's just kind of like the rank of the matrix, 01:11:21.437 --> 01:11:24.260 align:middle line:84% but a slightly different version of it. 01:11:24.260 --> 01:11:29.178 align:middle line:84% So higher rank means you have these representations 01:11:29.178 --> 01:11:30.720 align:middle line:84% of your data that are less clustered, 01:11:30.720 --> 01:11:34.540 align:middle line:84% and lower rank means this more clustered-like structure. 01:11:34.540 --> 01:11:37.738 align:middle line:84% And on the y-axis is the probability. 01:11:37.738 --> 01:11:39.280 align:middle line:84% These are like probability densities. 01:11:39.280 --> 01:11:41.020 align:middle line:90% So this is empirical PDFs-- 01:11:41.020 --> 01:11:45.060 align:middle line:84% Probability Densities-- of if I randomly 01:11:45.060 --> 01:11:48.767 align:middle line:84% sample parameters of a neural network of different depths. 01:11:48.767 --> 01:11:50.350 align:middle line:84% So if I randomly sample the parameters 01:11:50.350 --> 01:11:56.150 align:middle line:84% of a depth-16 network, the functions that it maps to 01:11:56.150 --> 01:12:01.190 align:middle line:84% will be these functions that map to low rank embeddings. 01:12:01.190 --> 01:12:03.430 align:middle line:84% I know there's a lot of moving pieces here, 01:12:03.430 --> 01:12:05.830 align:middle line:84% but basically deeper nets will be 01:12:05.830 --> 01:12:10.510 align:middle line:84% biased toward functions that are simpler in the sense of having 01:12:10.510 --> 01:12:12.130 align:middle line:90% lower rank structure. 01:12:12.130 --> 01:12:16.230 align:middle line:90% 01:12:16.230 --> 01:12:18.310 align:middle line:84% So one more way of looking at it-- so this 01:12:18.310 --> 01:12:26.310 align:middle line:84% is the rank which is in color and in height. 01:12:26.310 --> 01:12:31.150 align:middle line:84% If I'm navigating two different dimensions of my hypothesis 01:12:31.150 --> 01:12:33.810 align:middle line:84% space-- so my hypothesis space is parameterized by-- 01:12:33.810 --> 01:12:36.730 align:middle line:84% my parameter space has a million parameters, weights and biases, 01:12:36.730 --> 01:12:38.510 align:middle line:84% but I'm going to look at just two of them. 01:12:38.510 --> 01:12:42.830 align:middle line:84% And I'm going to chart out as I walk around that space what 01:12:42.830 --> 01:12:44.050 align:middle line:90% is the effective rank. 01:12:44.050 --> 01:12:47.530 align:middle line:84% And the point here is that as I add depth-- 01:12:47.530 --> 01:12:51.010 align:middle line:84% so with a single layer model, I will 01:12:51.010 --> 01:12:56.050 align:middle line:84% have this kind of narrow ridge of low rank functions 01:12:56.050 --> 01:12:58.390 align:middle line:84% as I navigate my parameter space. 01:12:58.390 --> 01:13:01.110 align:middle line:84% But a two-layer model, I'll pull that ridge apart. 01:13:01.110 --> 01:13:04.170 align:middle line:84% And so more volume of the parameter space 01:13:04.170 --> 01:13:06.570 align:middle line:90% will map to low rank functions. 01:13:06.570 --> 01:13:10.050 align:middle line:84% And if I combine this with optimization, I'll just-- 01:13:10.050 --> 01:13:12.830 align:middle line:84% more of the solutions will lie in the low rank space as well. 01:13:12.830 --> 01:13:20.325 align:middle line:90% 01:13:20.325 --> 01:13:22.450 align:middle line:84% Yeah, one set of ideas is that deep nets are biased 01:13:22.450 --> 01:13:24.750 align:middle line:84% toward simplicity according to Lempel-Ziv 01:13:24.750 --> 01:13:27.730 align:middle line:84% or according to the rank of the functions that they learn just 01:13:27.730 --> 01:13:30.650 align:middle line:84% because most parameterizations-- most settings of the parameters 01:13:30.650 --> 01:13:33.050 align:middle line:90% map to these simple functions. 01:13:33.050 --> 01:13:36.290 align:middle line:84% Another idea is that actually the optimization dynamics really 01:13:36.290 --> 01:13:40.210 align:middle line:84% play a big role here, and that I'm not just 01:13:40.210 --> 01:13:43.240 align:middle line:84% picking a random solution in my version space. 01:13:43.240 --> 01:13:46.060 align:middle line:84% I'm actually using an optimizer, which will prefer some solutions 01:13:46.060 --> 01:13:48.540 align:middle line:90% over other solutions. 01:13:48.540 --> 01:13:52.820 align:middle line:84% And sometimes, I'll have explicit regularizers 01:13:52.820 --> 01:13:55.860 align:middle line:84% that prefer simple solutions in my optimizer 01:13:55.860 --> 01:13:57.608 align:middle line:90% or in my objective function. 01:13:57.608 --> 01:13:59.400 align:middle line:84% And so one of those is called weight decay. 01:13:59.400 --> 01:14:02.400 align:middle line:84% It's just saying that, as I optimize, 01:14:02.400 --> 01:14:07.380 align:middle line:84% I'll tend to shrink the weights a little bit 01:14:07.380 --> 01:14:09.160 align:middle line:90% on each update step. 01:14:09.160 --> 01:14:11.540 align:middle line:84% And this tends to make weights that aren't actually 01:14:11.540 --> 01:14:13.560 align:middle line:90% being used go to 0. 01:14:13.560 --> 01:14:17.628 align:middle line:84% And so I'll end up learning a set of weights 01:14:17.628 --> 01:14:19.420 align:middle line:84% of my neural network, which have lower norm 01:14:19.420 --> 01:14:20.800 align:middle line:90% and are simple in that sense. 01:14:20.800 --> 01:14:23.340 align:middle line:90% 01:14:23.340 --> 01:14:25.740 align:middle line:84% Often, we initialize our weights and biases 01:14:25.740 --> 01:14:27.420 align:middle line:90% of our networks near 0. 01:14:27.420 --> 01:14:30.980 align:middle line:84% And this acts like a bias toward finding solutions that have, 01:14:30.980 --> 01:14:37.780 align:middle line:84% again, low norm solutions-- low norm parameter vectors. 01:14:37.780 --> 01:14:39.540 align:middle line:84% And then there's a lot of interesting work 01:14:39.540 --> 01:14:41.790 align:middle line:84% on the implicit biases of gradient descent 01:14:41.790 --> 01:14:43.690 align:middle line:90% and stochastic gradient descent. 01:14:43.690 --> 01:14:47.710 align:middle line:84% And so one idea here is that stochastic gradient descent-- 01:14:47.710 --> 01:14:49.670 align:middle line:84% well, gradient descent with a fixed step size 01:14:49.670 --> 01:14:50.850 align:middle line:90% kind of descend down. 01:14:50.850 --> 01:14:54.830 align:middle line:84% But let's say that I have a really, really sharp well 01:14:54.830 --> 01:14:57.550 align:middle line:84% in my objective function, in my energy landscape. 01:14:57.550 --> 01:15:00.130 align:middle line:84% Well, gradient descent can't go into a really sharp well 01:15:00.130 --> 01:15:01.900 align:middle line:90% if I'm using fixed step sizes. 01:15:01.900 --> 01:15:03.150 align:middle line:90% It'll just jump right over it. 01:15:03.150 --> 01:15:05.258 align:middle line:90% That well will be too small. 01:15:05.258 --> 01:15:07.550 align:middle line:84% So what gradient descent with a fixed step size will do 01:15:07.550 --> 01:15:11.230 align:middle line:84% is it will tend to find these, what are called, flat minima. 01:15:11.230 --> 01:15:14.510 align:middle line:84% So kind of basins that are low loss, but 01:15:14.510 --> 01:15:19.590 align:middle line:84% are low loss over kind of a large region of the parameter 01:15:19.590 --> 01:15:21.550 align:middle line:90% space that I'm searching over. 01:15:21.550 --> 01:15:25.190 align:middle line:84% And these flat minima can be argued to generalize better 01:15:25.190 --> 01:15:26.530 align:middle line:90% than these narrow minima. 01:15:26.530 --> 01:15:29.968 align:middle line:84% The narrow minima might be more like just a weird solution that 01:15:29.968 --> 01:15:31.510 align:middle line:84% got lucky, but the flat minima is one 01:15:31.510 --> 01:15:33.610 align:middle line:90% that will generalize better. 01:15:33.610 --> 01:15:36.110 align:middle line:84% So I'm not going to go into all the arguments for why that's 01:15:36.110 --> 01:15:39.610 align:middle line:84% the case, but you can see these papers if you're interested. 01:15:39.610 --> 01:15:43.030 align:middle line:84% But the optimizer is a critical part of the recipe too, 01:15:43.030 --> 01:15:46.743 align:middle line:84% and it can affect the generalization properties. 01:15:46.743 --> 01:15:48.410 align:middle line:84% But I think all of what I've said so far 01:15:48.410 --> 01:15:50.210 align:middle line:84% is not actually the most important. 01:15:50.210 --> 01:15:53.370 align:middle line:84% I think the most important is the following two slides-- is 01:15:53.370 --> 01:15:55.450 align:middle line:90% the architectural symmetries. 01:15:55.450 --> 01:15:57.523 align:middle line:84% And we saw these in ConvNets and graph nets, 01:15:57.523 --> 01:15:59.690 align:middle line:84% and we'll see these in transformers and other models 01:15:59.690 --> 01:16:00.190 align:middle line:90% too. 01:16:00.190 --> 01:16:02.770 align:middle line:90% 01:16:02.770 --> 01:16:06.970 align:middle line:84% So our architectures are built to have certain invariances. 01:16:06.970 --> 01:16:10.330 align:middle line:84% Like, if I'm pooling over channels 01:16:10.330 --> 01:16:13.090 align:middle line:84% with a max pooling operation, I'll be invariant. 01:16:13.090 --> 01:16:15.830 align:middle line:84% And my filters are like edge detectors. 01:16:15.830 --> 01:16:19.890 align:middle line:84% I'll be invariant to the orientation of that bird's beak. 01:16:19.890 --> 01:16:22.810 align:middle line:84% I'll fire regardless of what the orientation of that bird's beak 01:16:22.810 --> 01:16:23.310 align:middle line:90% is. 01:16:23.310 --> 01:16:26.570 align:middle line:84% So max pooling achieves this invariance over the dimension 01:16:26.570 --> 01:16:27.710 align:middle line:90% we're maxing over. 01:16:27.710 --> 01:16:31.370 align:middle line:90% 01:16:31.370 --> 01:16:33.570 align:middle line:84% Equivariances-- so graph nets and ConvNets 01:16:33.570 --> 01:16:34.750 align:middle line:90% have these equivariances. 01:16:34.750 --> 01:16:37.960 align:middle line:84% If I permute the labeling of the nodes, 01:16:37.960 --> 01:16:41.340 align:middle line:84% then I will permute the predictions. 01:16:41.340 --> 01:16:45.620 align:middle line:84% Translation equivariance-- if I have a filter, 01:16:45.620 --> 01:16:49.340 align:middle line:84% and I slide it across the image, then 01:16:49.340 --> 01:16:51.360 align:middle line:84% if I shift the image by some amount, 01:16:51.360 --> 01:16:55.300 align:middle line:84% I'll shift my predictions by that same amount. 01:16:55.300 --> 01:16:58.860 align:middle line:84% So these are symmetries that are kind of built 01:16:58.860 --> 01:17:02.620 align:middle line:84% into our architectures that cause them to exhibit 01:17:02.620 --> 01:17:07.100 align:middle line:84% out-of-sample generalization, because I know 01:17:07.100 --> 01:17:10.660 align:middle line:84% if I just shift my image, it will respond 01:17:10.660 --> 01:17:13.360 align:middle line:84% in the way it was trained to respond on the unshifted image. 01:17:13.360 --> 01:17:16.397 align:middle line:84% That's what the symmetry gives you. 01:17:16.397 --> 01:17:17.980 align:middle line:84% And then this kind of compositionality 01:17:17.980 --> 01:17:19.730 align:middle line:84% I was mentioning before-- that the ConvNet 01:17:19.730 --> 01:17:22.680 align:middle line:84% and a lot of architectures, they factor this input signal. 01:17:22.680 --> 01:17:25.037 align:middle line:84% They maybe chop it up into little components. 01:17:25.037 --> 01:17:27.120 align:middle line:84% They learn something about every little component, 01:17:27.120 --> 01:17:29.340 align:middle line:84% but then the conjunction of components 01:17:29.340 --> 01:17:30.993 align:middle line:84% is just given by something that's 01:17:30.993 --> 01:17:32.160 align:middle line:90% defined by the architecture. 01:17:32.160 --> 01:17:33.140 align:middle line:90% It's not learned. 01:17:33.140 --> 01:17:35.990 align:middle line:84% The conjunction of the outputs of 01:17:35.990 --> 01:17:39.750 align:middle line:84% these little patch-wise filters, patch-wise predictors 01:17:39.750 --> 01:17:40.610 align:middle line:90% is not learned. 01:17:40.610 --> 01:17:44.890 align:middle line:84% It's just you stitch them all back together. 01:17:44.890 --> 01:17:48.110 align:middle line:84% So I think this is where really the real power 01:17:48.110 --> 01:17:49.750 align:middle line:84% and the real explanation of why things 01:17:49.750 --> 01:17:52.440 align:middle line:84% like ChatGPT, why they actually do 01:17:52.440 --> 01:17:53.690 align:middle line:90% these feats of generalization. 01:17:53.690 --> 01:17:56.350 align:middle line:84% It's not so much in my-- well, I don't know. 01:17:56.350 --> 01:17:59.273 align:middle line:84% But I think it's a lot about it factorizes the world. 01:17:59.273 --> 01:18:00.690 align:middle line:84% It carves the world at its joints. 01:18:00.690 --> 01:18:03.610 align:middle line:84% It factors its world into parcels, 01:18:03.610 --> 01:18:05.670 align:middle line:84% which are words, and sentences, and paragraphs. 01:18:05.670 --> 01:18:07.630 align:middle line:84% And then compositions of these things 01:18:07.630 --> 01:18:12.690 align:middle line:84% will be arrived at not from learning, 01:18:12.690 --> 01:18:18.150 align:middle line:84% but from just how the architecture is built. 01:18:18.150 --> 01:18:20.575 align:middle line:84% And then a lot of neural networks 01:18:20.575 --> 01:18:22.450 align:middle line:84% have domain knowledge that's baked into them. 01:18:22.450 --> 01:18:26.910 align:middle line:84% For example, Sarah gave the example of the NeRF model, which 01:18:26.910 --> 01:18:29.550 align:middle line:84% is a neural architecture, but it uses equations 01:18:29.550 --> 01:18:32.230 align:middle line:84% of perspective projection, and it uses 01:18:32.230 --> 01:18:33.958 align:middle line:90% equations of light transport. 01:18:33.958 --> 01:18:35.500 align:middle line:84% And these are built into the process, 01:18:35.500 --> 01:18:38.080 align:middle line:84% and so it will generalize to new viewpoints 01:18:38.080 --> 01:18:41.130 align:middle line:90% because it's using structures. 01:18:41.130 --> 01:18:42.380 align:middle line:90% It's not just fitting to data. 01:18:42.380 --> 01:18:44.760 align:middle line:84% It's data plus structure, data plus constraints. 01:18:44.760 --> 01:18:49.160 align:middle line:84% And this is a drug discovery, drug prediction problem, where 01:18:49.160 --> 01:18:50.300 align:middle line:90% they've baked in structure. 01:18:50.300 --> 01:18:52.333 align:middle line:84% They know about how drugs are supposed 01:18:52.333 --> 01:18:53.500 align:middle line:90% to interact with each other. 01:18:53.500 --> 01:18:56.200 align:middle line:84% And the conjunction of fitting to data 01:18:56.200 --> 01:18:57.800 align:middle line:84% under structural constraints is what 01:18:57.800 --> 01:19:00.700 align:middle line:90% can allow you to generalize. 01:19:00.700 --> 01:19:05.105 align:middle line:84% So I want to end with one final point, which is, well, maybe we 01:19:05.105 --> 01:19:06.480 align:middle line:84% can actually go back, ultimately, 01:19:06.480 --> 01:19:07.560 align:middle line:90% to this theory answer. 01:19:07.560 --> 01:19:09.560 align:middle line:84% The theory answer is that the shortest 01:19:09.560 --> 01:19:12.880 align:middle line:84% program that fits the data will be the one that generalizes. 01:19:12.880 --> 01:19:14.762 align:middle line:84% I said it's intractable, but what if we say, 01:19:14.762 --> 01:19:16.220 align:middle line:84% we don't have to find the shortest? 01:19:16.220 --> 01:19:18.445 align:middle line:84% Finding the shortest is intractable. 01:19:18.445 --> 01:19:20.820 align:middle line:84% Maybe we don't have to optimize for the shortest program, 01:19:20.820 --> 01:19:22.920 align:middle line:84% but we can just say a short enough program that 01:19:22.920 --> 01:19:25.520 align:middle line:90% fits the data will generalize. 01:19:25.520 --> 01:19:28.840 align:middle line:84% And so this is something that I've just kind come back to 01:19:28.840 --> 01:19:31.300 align:middle line:90% over the years. 01:19:31.300 --> 01:19:33.360 align:middle line:84% This is a conversation I had with Ilya Sutskever, 01:19:33.360 --> 01:19:37.620 align:middle line:84% who's well-known as one of the founders of OpenAI. 01:19:37.620 --> 01:19:39.480 align:middle line:84% And so he had this thing he said, 01:19:39.480 --> 01:19:41.560 align:middle line:84% which is deep-- this is not a direct quote. 01:19:41.560 --> 01:19:42.920 align:middle line:90% This is just the rough memory. 01:19:42.920 --> 01:19:45.620 align:middle line:84% "Deep nets generalize because they find small circuits that 01:19:45.620 --> 01:19:47.210 align:middle line:90% fit the data." 01:19:47.210 --> 01:19:48.960 align:middle line:84% And so at the time, I didn't quite get it. 01:19:48.960 --> 01:19:50.240 align:middle line:90% I was like, small circuits? 01:19:50.240 --> 01:19:50.880 align:middle line:90% What are you talking about? 01:19:50.880 --> 01:19:51.672 align:middle line:90% Deep nets are huge. 01:19:51.672 --> 01:19:54.100 align:middle line:84% Don't we need all of these inductive biases, 01:19:54.100 --> 01:19:56.545 align:middle line:84% and regularizers, and simplicity biases? 01:19:56.545 --> 01:19:58.920 align:middle line:84% Like, that seems critical to understanding deep learning. 01:19:58.920 --> 01:20:00.128 align:middle line:90% And we need to bake those in. 01:20:00.128 --> 01:20:02.860 align:middle line:84% We need to put those in our models. 01:20:02.860 --> 01:20:04.560 align:middle line:84% And he said, "no, no, no, it's OK. 01:20:04.560 --> 01:20:05.560 align:middle line:90% Deep nets are finite. 01:20:05.560 --> 01:20:07.260 align:middle line:90% That's enough. 01:20:07.260 --> 01:20:08.160 align:middle line:90% Finite is small. 01:20:08.160 --> 01:20:11.100 align:middle line:84% Anything finite will look small if you have enough data." 01:20:11.100 --> 01:20:12.860 align:middle line:84% And I think that was a deep insight 01:20:12.860 --> 01:20:15.080 align:middle line:84% and maybe led to some of the success that he's had. 01:20:15.080 --> 01:20:16.660 align:middle line:90% So I'll end there. 01:20:16.660 --> 01:20:18.490 align:middle line:90% Thank you. 01:20:18.490 --> 01:20:30.000 align:middle line:90%