WEBVTT 00:00:00.000 --> 00:00:02.759 Welcome to the deep dive. Hey great to be here. 00:00:02.940 --> 00:00:06.040 So today our mission is to decode the hidden 00:00:06.040 --> 00:00:09.199 mathematical engine that basically keeps artificial 00:00:09.199 --> 00:00:11.480 intelligence from going completely off the rails. 00:00:11.539 --> 00:00:13.779 Yeah, it's an absolutely crucial concept. Right 00:00:13.779 --> 00:00:17.339 and we are looking at a single highly comprehensive 00:00:17.339 --> 00:00:19.980 Wikipedia article for this. It's simply titled 00:00:19.980 --> 00:00:23.460 regularization in mathematics. Which sounds intimidating 00:00:23.460 --> 00:00:26.260 I know. Oh totally but we are going to translate 00:00:26.260 --> 00:00:30.109 all those dense formulas into the you know, the 00:00:30.109 --> 00:00:32.770 actual real -world mechanisms that make modern 00:00:32.770 --> 00:00:35.810 AI useful for you. Exactly. Because without this, 00:00:36.090 --> 00:00:38.710 AI is kind of useless. Right. OK. Let's untack 00:00:38.710 --> 00:00:40.950 this. Because to understand why regularization 00:00:40.950 --> 00:00:43.250 matters to you, you really have to understand 00:00:43.250 --> 00:00:46.329 the fundamental flaw in how machines learn. They 00:00:46.329 --> 00:00:49.030 have a major blind spot. Yeah. So imagine you're 00:00:49.030 --> 00:00:51.750 studying for this massive, life -changing exam. 00:00:52.049 --> 00:00:53.890 But instead of actually learning the underlying 00:00:53.890 --> 00:00:56.450 concepts, you just memorize the exact wording 00:00:56.450 --> 00:00:59.039 of every single practice question. And you would 00:00:59.039 --> 00:01:01.359 probably absolutely ace that practice test, right? 00:01:01.380 --> 00:01:03.100 You'd get a perfect score because you know the 00:01:03.100 --> 00:01:06.099 exact answers to those specific questions. Exactly. 00:01:06.359 --> 00:01:09.299 But then you sit down for the actual exam and 00:01:09.299 --> 00:01:11.560 the wording of the questions is slightly different. 00:01:11.579 --> 00:01:13.780 And you completely fail. Right. You didn't learn 00:01:13.780 --> 00:01:15.959 how to solve the overarching problems. You just 00:01:15.959 --> 00:01:19.819 memorized a highly specific, static set of data. 00:01:20.019 --> 00:01:23.819 And that is the ultimate trap for any learning 00:01:23.819 --> 00:01:26.530 system. We all naturally gravitate toward the 00:01:26.530 --> 00:01:28.170 comfort of knowing exactly what is in front of 00:01:28.170 --> 00:01:30.709 us sure But when you step into the world of machine 00:01:30.709 --> 00:01:33.810 learning and statistics that exact same trap 00:01:33.950 --> 00:01:37.489 is the single biggest hurdle to building artificial 00:01:37.489 --> 00:01:39.590 intelligence that actually functions in reality. 00:01:39.790 --> 00:01:41.950 Because the AI just wants to memorize everything. 00:01:42.209 --> 00:01:44.790 Exactly. We're looking at a scenario where algorithms 00:01:44.790 --> 00:01:46.790 naturally want to become massive memorization 00:01:46.790 --> 00:01:49.629 machines. The technical term for this is overfitting. 00:01:49.829 --> 00:01:52.569 Overfitting. Okay, so the AI is just memorizing 00:01:52.569 --> 00:01:55.709 the practice test. But the Wikipedia text gives 00:01:55.709 --> 00:01:57.849 us a core definition of how to fight this. It 00:01:57.849 --> 00:02:00.469 does. It says regularization is a process that 00:02:00.469 --> 00:02:02.730 converts the answer to a problem into a simpler 00:02:02.730 --> 00:02:06.349 one. But I gotta say, the idea of forcing a highly 00:02:06.349 --> 00:02:09.949 advanced supercomputing algorithm to intentionally 00:02:09.949 --> 00:02:13.590 be simpler, it just feels counterintuitive. I 00:02:13.590 --> 00:02:16.370 mean, shouldn't we want the most complex answer 00:02:16.370 --> 00:02:18.870 possible? If we connect this to the bigger picture, 00:02:19.210 --> 00:02:22.229 it really comes down to the gap. between what 00:02:22.229 --> 00:02:24.849 an algorithm sees right now and what it will 00:02:24.849 --> 00:02:27.030 eventually encounter out in the real world. Okay. 00:02:27.330 --> 00:02:29.710 In any learning problem, what we really want 00:02:29.710 --> 00:02:33.430 to minimize is the expected error over all possible 00:02:33.430 --> 00:02:36.030 inputs. Meaning, like, how it performs out in 00:02:36.030 --> 00:02:38.849 the wild. Exactly. How it handles data it has 00:02:38.849 --> 00:02:41.409 never, ever seen before. Right. The actual exam, 00:02:41.530 --> 00:02:43.949 not just the practice test. Yeah. But the problem 00:02:43.949 --> 00:02:46.710 is that expected error is practically unmeasurable. 00:02:46.919 --> 00:02:49.080 I mean, we don't have access to all possible 00:02:49.080 --> 00:02:51.439 future inputs. Obviously not. We only have a 00:02:51.439 --> 00:02:54.139 limited set of training data. And that data is, 00:02:54.139 --> 00:02:55.719 you know, it's always measured with some amount 00:02:55.719 --> 00:02:58.560 of noise or errors or random fluctuations. So 00:02:58.560 --> 00:03:01.020 it's messy. Very messy. Yeah. And because the 00:03:01.020 --> 00:03:03.479 model can't measure expected error, it has to 00:03:03.479 --> 00:03:05.479 rely on what is called empirical error. Which 00:03:05.479 --> 00:03:08.819 is what? Exactly. It's error rate on that specific 00:03:08.819 --> 00:03:11.409 limited training set. Gotcha. So without any 00:03:11.409 --> 00:03:14.370 boundaries, a highly complex neural network will 00:03:14.370 --> 00:03:17.710 just bend over backwards to achieve zero empirical 00:03:17.710 --> 00:03:20.610 error. It'll just twist itself into knots to 00:03:20.610 --> 00:03:23.289 perfectly hit every single beta point. Exactly. 00:03:23.710 --> 00:03:26.150 Meaning it ends up memorizing the random noise. 00:03:26.409 --> 00:03:28.530 Oh, so it overthinks it. It overthinks it completely. 00:03:28.629 --> 00:03:30.689 Yeah. And that's where regularization comes in. 00:03:30.810 --> 00:03:33.590 It introduces a mathematical penalty to stop 00:03:33.590 --> 00:03:35.990 that overthinking. OK, so it steps in and says, 00:03:36.409 --> 00:03:40.340 hey, stop it. Pretty much. By forcing the model 00:03:40.340 --> 00:03:43.080 to be simpler, it reduces the generalization 00:03:43.080 --> 00:03:46.120 error, which is the measure of how well the algorithm 00:03:46.120 --> 00:03:49.340 adapts to unseen data. So it essentially tells 00:03:49.340 --> 00:03:51.699 the machine, you know, don't worry about being 00:03:51.699 --> 00:03:53.900 perfect on the training data. Worry about being 00:03:53.900 --> 00:03:56.000 robust for the future. That's a great way to 00:03:56.000 --> 00:03:58.120 put it. Okay, so it's basically giving the algorithm 00:03:58.120 --> 00:04:00.439 some strict boundaries. Now that we understand 00:04:00.439 --> 00:04:02.639 why models need these constraints, how do we 00:04:02.639 --> 00:04:05.000 actually apply them? Well, the source breaks 00:04:05.000 --> 00:04:07.539 this down into two main philosophies. You have 00:04:07.539 --> 00:04:10.719 explicit and implicit regularization. OK, let's 00:04:10.719 --> 00:04:13.680 start with explicit. So explicit regularization 00:04:13.680 --> 00:04:17.600 is whenever you directly add a concrete mathematical 00:04:17.600 --> 00:04:20.560 term to your optimization problem. That's a literal 00:04:20.560 --> 00:04:24.360 rule. Yeah. You are imposing a literal cost on 00:04:24.360 --> 00:04:26.639 the optimization function. Think of it as a penalty 00:04:26.639 --> 00:04:29.459 fee. Oh, like getting a ticket. Exactly. Yeah. 00:04:29.660 --> 00:04:32.399 The algorithm wants to minimize its error. But 00:04:32.399 --> 00:04:35.160 if it gets too complex, you slap it with a mathematical 00:04:35.160 --> 00:04:37.860 fine, which forces the optimal solution to be 00:04:37.860 --> 00:04:40.610 simpler. Makes sense. And implicit. Implicit 00:04:40.610 --> 00:04:43.269 regularization encompasses, well, basically everything 00:04:43.269 --> 00:04:45.829 else. Like what? This includes using a robust 00:04:45.829 --> 00:04:48.930 loss function, or throwing out outlier data points, 00:04:49.089 --> 00:04:52.050 or using ensemble methods like combining multiple 00:04:52.050 --> 00:04:54.610 decision trees into a random forest. Right. It's 00:04:54.610 --> 00:04:56.930 ubiquitous in modern machine learning, especially 00:04:56.930 --> 00:04:58.930 when you're training deep neural networks with 00:04:58.930 --> 00:05:01.430 stochastic gradient descent. I want to push back 00:05:01.430 --> 00:05:03.569 on one of the implicit methods mentioned in the 00:05:03.569 --> 00:05:05.589 text, though, because it talks about something 00:05:05.589 --> 00:05:09.500 called early stopping. Ah, yes. It's early stopping. 00:05:10.220 --> 00:05:12.240 And the definition given is that you simply halt 00:05:12.240 --> 00:05:14.740 the training process when the model's performance 00:05:14.740 --> 00:05:17.120 on a validation data set starts to get worse. 00:05:17.360 --> 00:05:19.500 That's right. Wait, so simply turning the algorithm 00:05:19.500 --> 00:05:22.060 off before it finishes its run is considered 00:05:22.060 --> 00:05:24.560 a mathematical technique. I mean, isn't that 00:05:24.560 --> 00:05:26.519 just giving up early and hoping for the best? 00:05:26.620 --> 00:05:28.639 I know, I know. It definitely sounds like you're 00:05:28.639 --> 00:05:31.139 just pulling the plug. Right. But mathematically, 00:05:31.899 --> 00:05:35.519 early stopping is rigorously viewed as regularization 00:05:35.519 --> 00:05:39.509 in time. regularization in time okay intuitively 00:05:39.509 --> 00:05:43.269 a training procedure like gradient descent it 00:05:43.269 --> 00:05:45.410 starts with a very simple function And then it 00:05:45.410 --> 00:05:47.850 learns more and more complex functions as the 00:05:47.850 --> 00:05:51.230 iterations increase. Oh, I see. So by limiting 00:05:51.230 --> 00:05:53.589 the time parameter, like the number of iterations, 00:05:53.910 --> 00:05:56.730 we restrict the model's complexity. Exactly. 00:05:56.730 --> 00:05:58.709 You're cutting it off. So you're stopping it 00:05:58.709 --> 00:06:01.029 before it has the time to start memorizing the 00:06:01.029 --> 00:06:03.589 useless noise. Yeah. And the text provides a 00:06:03.589 --> 00:06:06.029 fascinating theoretical motivation for this using 00:06:06.029 --> 00:06:08.509 least squares optimization. OK, hit me. So if 00:06:08.509 --> 00:06:10.589 you consider the finite approximation of the 00:06:10.589 --> 00:06:12.889 Newman series for an invertible matrix. Whoa, 00:06:12.889 --> 00:06:15.410 hold on. Newman series? Invertible matrix. Too 00:06:15.410 --> 00:06:17.350 much math. Yeah. Let's translate that for someone 00:06:17.350 --> 00:06:21.410 who doesn't dream in calculus. Like, why does 00:06:21.410 --> 00:06:23.970 stopping early act like a mathematical formula? 00:06:24.269 --> 00:06:26.769 Fair enough. Let's break that down. A Newman 00:06:26.769 --> 00:06:29.589 series is essentially an infinite sum, a bit 00:06:29.589 --> 00:06:32.829 like a Taylor series, but for matrices. OK. In 00:06:32.829 --> 00:06:35.329 machine learning, finding the absolute perfect 00:06:35.329 --> 00:06:38.209 set of weights often requires calculating the 00:06:38.209 --> 00:06:41.350 inverse of a massive matrix of data. Which is 00:06:41.350 --> 00:06:43.550 hard to do. Right. And calculating that exact 00:06:43.550 --> 00:06:46.389 inverse perfectly is a recipe for memorizing 00:06:46.389 --> 00:06:49.470 noise. So the Newman series lets us approximate 00:06:49.470 --> 00:06:51.949 that inverse by adding up terms one by one. Got 00:06:51.949 --> 00:06:54.610 it. But each turn you add makes the model more 00:06:54.610 --> 00:06:57.810 complex. So by limiting the iterations, which 00:06:57.810 --> 00:07:00.870 is represented by a time parameter, tu are essentially 00:07:00.870 --> 00:07:04.399 cutting off that infinite sum early. Oh. So you 00:07:04.399 --> 00:07:06.379 get a good enough approximation of the solution 00:07:06.379 --> 00:07:09.000 before the model starts adding in the highly 00:07:09.000 --> 00:07:11.660 complex terms that represent all that noisy data. 00:07:12.199 --> 00:07:13.639 Exactly. It's like taking the test away from 00:07:13.639 --> 00:07:15.660 the student right when they've grasped the broad 00:07:15.660 --> 00:07:17.920 strokes before they start overthinking and second 00:07:17.920 --> 00:07:20.060 -guessing every single answer. That's exactly 00:07:20.060 --> 00:07:22.779 what it is. And the text uses a really great 00:07:22.779 --> 00:07:25.019 visual example to illustrate this underlying 00:07:25.019 --> 00:07:27.720 philosophy. Occam's razor, right? Yes, Occam's 00:07:27.720 --> 00:07:31.699 razor. It describes a graph with a green function 00:07:31.699 --> 00:07:34.899 and a blue function. OK, picture a graph. Right. 00:07:35.360 --> 00:07:38.339 And both functions perfectly hit all the given 00:07:38.339 --> 00:07:41.699 data points, meaning they both incur zero loss 00:07:41.699 --> 00:07:44.399 on the training data. So they both get 100 % 00:07:44.399 --> 00:07:46.980 on the practice test. Yep. But the blue line 00:07:46.980 --> 00:07:49.600 is incredibly squiggly. It hits every point, 00:07:49.860 --> 00:07:52.100 but it fluctuates wildly up and down in between 00:07:52.100 --> 00:07:54.819 them. It's all over the place. Exactly. While 00:07:54.819 --> 00:07:58.610 the green line is a smooth, simple curve. So 00:07:58.610 --> 00:08:01.329 regularization mathematically induces the model 00:08:01.329 --> 00:08:04.269 to prefer the simpler green function. Yes, honoring 00:08:04.269 --> 00:08:07.069 Occam's razor. Because the smooth curve is just 00:08:07.069 --> 00:08:09.589 far more likely to generalize better to an unknown 00:08:09.589 --> 00:08:11.910 distribution of data. Because simpler is better. 00:08:12.069 --> 00:08:14.230 Okay, but early stopping is an implicit method, 00:08:14.389 --> 00:08:16.670 right? It relies on manipulating time. Correct. 00:08:16.949 --> 00:08:19.230 What if we want to actively force a model to 00:08:19.230 --> 00:08:21.389 be simple through those explicit mathematical 00:08:21.389 --> 00:08:23.990 penalties you mentioned earlier? Well, then we 00:08:23.990 --> 00:08:26.069 have to look at how we measure the complexity 00:08:26.069 --> 00:08:29.889 of the model's weights. The text introduces two 00:08:29.889 --> 00:08:32.909 primary methods for this, L1 and L2 regularization. 00:08:32.990 --> 00:08:36.190 L1 and L2. Yeah. L1, which is also called LASSO, 00:08:36.690 --> 00:08:39.230 adds a penalty to the cost function based on 00:08:39.230 --> 00:08:41.529 the absolute value of the model's coefficients. 00:08:42.029 --> 00:08:43.970 So it looks at the sheer size of the weights. 00:08:44.309 --> 00:08:46.230 Right. And it penalizes the model for having 00:08:46.230 --> 00:08:49.360 too many of them. Mathematically, this penalty 00:08:49.360 --> 00:08:51.779 induces what we call sparsity. OK, here's where 00:08:51.779 --> 00:08:54.500 it gets really interesting. Sparsity means it 00:08:54.500 --> 00:08:56.679 forces a lot of the weights to become exactly 00:08:56.679 --> 00:08:59.399 zero? Yes, it zeroes them out. To wrap our heads 00:08:59.399 --> 00:09:02.179 around this, imagine an AI model is packing for 00:09:02.179 --> 00:09:05.559 a trip. L1 regularization is the ruthless minimalist 00:09:05.559 --> 00:09:08.279 who forces you to throw away 90 % of your clothes, 00:09:08.620 --> 00:09:11.200 bringing only the absolute essentials. I love 00:09:11.200 --> 00:09:13.379 that. It basically decides certain features of 00:09:13.379 --> 00:09:15.620 the data are completely irrelevant and tosses 00:09:15.620 --> 00:09:18.350 them out. A perfect analogy. Now... Contrast 00:09:18.350 --> 00:09:21.549 that ruthless minimalist with L2 regularization, 00:09:21.649 --> 00:09:24.029 which is also known as ridge regression. Here, 00:09:24.090 --> 00:09:26.710 the penalty is based on the square of the coefficients. 00:09:27.309 --> 00:09:30.610 Oh, so L2 is the meticulous organizer. It doesn't 00:09:30.610 --> 00:09:32.610 necessarily throw things away, but it makes sure 00:09:32.610 --> 00:09:35.570 everything you do bring is packed evenly in small, 00:09:35.830 --> 00:09:37.970 manageable bags so the suitcase doesn't burst. 00:09:38.210 --> 00:09:40.210 Right, because it's squaring the weights. So 00:09:40.210 --> 00:09:42.690 it punishes really large weights much more severely 00:09:42.690 --> 00:09:45.029 than small ones, right? Exactly. Squaring a large 00:09:45.029 --> 00:09:47.860 number makes it exponentially larger. which results 00:09:47.860 --> 00:09:51.360 in a massive mathematical penalty. So L2 encourages 00:09:51.360 --> 00:09:54.460 the model to have smaller, more evenly distributed 00:09:54.460 --> 00:09:56.740 weights. That makes total sense. And the text 00:09:56.740 --> 00:10:00.379 notes that L2 is heavily tied to Tikhonov regularization, 00:10:00.799 --> 00:10:03.240 named after Andrei Nikolaevich Tikhonov. Okay. 00:10:03.450 --> 00:10:06.309 And unlike some really complex neural network 00:10:06.309 --> 00:10:09.210 math, taking off regularization for a least squares 00:10:09.210 --> 00:10:11.889 problem can actually be solved analytically. 00:10:12.090 --> 00:10:14.490 Meaning there is a single exact mathematical 00:10:14.490 --> 00:10:17.149 answer you can calculate? Yes. You can write 00:10:17.149 --> 00:10:19.590 it in matrix form and find the exact optimal 00:10:19.590 --> 00:10:22.070 weights just by setting the gradient of the loss 00:10:22.070 --> 00:10:24.090 function to zero. But there's a catch, isn't 00:10:24.090 --> 00:10:26.149 there? There is a catch. During training, this 00:10:26.149 --> 00:10:29.429 algorithm takes O of D cubed plus N times D squared 00:10:29.429 --> 00:10:31.919 time. Okay, let me jump in again. O of D cubed. 00:10:32.399 --> 00:10:34.259 You're talking about big O notation, right? The 00:10:34.259 --> 00:10:37.019 computational cost. Yeah, big O notation. Explain 00:10:37.019 --> 00:10:39.519 why squaring those coefficients suddenly makes 00:10:39.519 --> 00:10:42.419 the computer work so much harder. Well, big O 00:10:42.419 --> 00:10:44.799 notation describes how the runtime of an algorithm 00:10:44.799 --> 00:10:48.539 scales as the data grows. The D there stands 00:10:48.539 --> 00:10:51.600 for the number of dimensions or features in your 00:10:51.600 --> 00:10:54.419 data. Right. Because finding that exact mathematical 00:10:54.419 --> 00:10:57.000 answer requires a process called matrix inversion. 00:10:57.600 --> 00:11:00.870 The math just... gets incredibly heavy. How heavy? 00:11:01.169 --> 00:11:04.169 Well, O of D cubed means if you double the number 00:11:04.169 --> 00:11:07.330 of features your AI is analyzing, the time it 00:11:07.330 --> 00:11:10.230 takes to compute, the answer doesn't double. 00:11:10.350 --> 00:11:13.269 It multiplies by eight. Wow. By eight. Yeah. 00:11:13.690 --> 00:11:16.750 So if you have a massive data set, that L2 analytical 00:11:16.750 --> 00:11:18.990 solution becomes a massive computational burden. 00:11:19.210 --> 00:11:21.169 Which is why we need algorithms that approximate 00:11:21.169 --> 00:11:23.600 the answer. Exactly. But, you know, abstract 00:11:23.600 --> 00:11:26.080 map aside, what's amazing is how these penalties 00:11:26.080 --> 00:11:28.519 translate to real life. What's fascinating here 00:11:28.519 --> 00:11:30.960 is the real -world value of that L1 sparsity 00:11:30.960 --> 00:11:33.019 we just talked about. The minimalist packer. 00:11:33.259 --> 00:11:36.100 Right. The text highlights this brilliant application 00:11:36.100 --> 00:11:39.700 in computational biology. Okay. Imagine you are 00:11:39.700 --> 00:11:43.080 developing a predictive test for a disease. You 00:11:43.080 --> 00:11:45.820 might have 10 ,000 biological markers you could 00:11:45.820 --> 00:11:48.919 theoretically test for. But requiring a patient 00:11:48.919 --> 00:11:52.320 to undergo 10 ,000 individual lab tests is insanely 00:11:52.320 --> 00:11:54.639 expensive. And totally impractical. Yeah. But 00:11:54.639 --> 00:11:58.220 by enforcing a sparsity constraint using L1 regularization, 00:11:58.679 --> 00:12:01.899 the algorithm looks at all 10 ,000 markers, realizes 00:12:01.899 --> 00:12:05.100 most of them are redundant, and ruthlessly forces 00:12:05.100 --> 00:12:07.340 their weights to zero. It just throws them out. 00:12:07.399 --> 00:12:10.240 It throws out the noise and identifies the tiny 00:12:10.240 --> 00:12:12.419 handful of critical markers, maybe just five 00:12:12.419 --> 00:12:15.039 or 10, that actually matter. So you get a simple, 00:12:15.159 --> 00:12:17.940 cost -effective medical test that maximizes predictive 00:12:17.940 --> 00:12:20.799 power. Exactly. It literally finds the signal 00:12:20.799 --> 00:12:24.120 in the noise. That is amazing. But the text mentions 00:12:24.120 --> 00:12:26.720 a bit of a catch with L1, too, doesn't it? A 00:12:26.720 --> 00:12:28.879 technical hurdle regarding how we actually define 00:12:28.879 --> 00:12:31.220 sparsity. Yeah, it drops a great technical nugget. 00:12:31.480 --> 00:12:33.620 The most mathematically sensible way to enforce 00:12:33.620 --> 00:12:36.580 sparsity isn't actually L1. It's something called 00:12:36.580 --> 00:12:39.639 the L0 norm. The L0 norm doesn't look at the 00:12:39.639 --> 00:12:41.299 size of the weights at all. It simply counts 00:12:41.299 --> 00:12:44.279 the number of non -zero elements. Oh, so it is 00:12:44.279 --> 00:12:47.120 the purest form of counting your luggage. Exactly. 00:12:47.340 --> 00:12:49.740 Yeah. But solving an L0 regularized learning 00:12:49.740 --> 00:12:51.860 problem has been mathematically demonstrated 00:12:51.860 --> 00:12:55.879 to be NP -hard. NP -hard. Basically meaning it's 00:12:55.879 --> 00:12:58.080 practically impossible for a computer to solve 00:12:58.080 --> 00:13:00.740 in a reasonable amount of time if the data set 00:13:00.740 --> 00:13:04.029 is large. Exactly. To solve L0... The computer 00:13:04.029 --> 00:13:06.529 would have to test every single possible combination 00:13:06.529 --> 00:13:09.110 of keeping or throwing away features, which for 00:13:09.110 --> 00:13:12.509 10 ,000 biological markers is an astronomical 00:13:12.509 --> 00:13:15.350 number of combinations. It is computationally 00:13:15.350 --> 00:13:18.529 intractable. So L1 is used as what mathematicians 00:13:18.529 --> 00:13:21.529 call a convex relaxation. Convex relaxation? 00:13:21.649 --> 00:13:23.330 Okay, I'm gonna need you to untack that one, 00:13:23.350 --> 00:13:25.169 too. Sure. So imagine the optimization landscape, 00:13:25.309 --> 00:13:27.889 like the map the AI uses to find the lowest error, 00:13:28.389 --> 00:13:30.690 is incredibly bumpy and filled with jagged valleys. 00:13:30.850 --> 00:13:33.409 Okay, a bumpy map. That's the L0 landscape. Yeah. 00:13:33.490 --> 00:13:35.769 A computer gets stuck there very easily, but 00:13:35.769 --> 00:13:38.230 a convex relaxation smooths out all those bumps, 00:13:38.309 --> 00:13:40.710 turning the landscape into a single, smooth, 00:13:40.950 --> 00:13:44.230 giant bowl. Oh. L1 creates that smooth bowl, 00:13:44.590 --> 00:13:46.860 making it incredibly - easy for the algorithm 00:13:46.860 --> 00:13:49.500 to slide down to the bottom and find a solution 00:13:49.500 --> 00:13:52.899 that approximates the impossible L0 norm. That 00:13:52.899 --> 00:13:55.820 is a brilliant workaround. It really is. But 00:13:55.820 --> 00:13:58.039 L1 isn't flawless either, right? Because the 00:13:58.039 --> 00:13:59.899 text points out that if you have input features 00:13:59.899 --> 00:14:03.100 that are highly correlated, like two biological 00:14:03.100 --> 00:14:05.200 markers that always show up together, L1 might 00:14:05.200 --> 00:14:07.840 just randomly pick one and throw the other away. 00:14:08.019 --> 00:14:10.679 Yeah, which can make the model unstable. Right. 00:14:11.039 --> 00:14:13.730 It can. which is why we use something called 00:14:13.730 --> 00:14:17.090 elastic net regularization. Elastic net. It literally 00:14:17.090 --> 00:14:20.370 combines both the L1 and L2 penalties into a 00:14:20.370 --> 00:14:23.110 single equation. Oh, best of both worlds. Exactly. 00:14:23.309 --> 00:14:26.409 It gives you the ruthless sparsity of L1, throwing 00:14:26.409 --> 00:14:28.750 out the useless features. But the meticulous 00:14:28.750 --> 00:14:31.850 L2 component ensures that highly correlated features 00:14:31.850 --> 00:14:34.690 get assigned equal weights. Creating a grouping 00:14:34.690 --> 00:14:37.070 effect so you don't lose valuable context. Nailed 00:14:37.070 --> 00:14:40.299 it. So, tweaking the weights works beautifully 00:14:40.299 --> 00:14:43.899 for basic linear equations, but modern AI uses 00:14:43.899 --> 00:14:46.659 deep neural networks with millions of interconnected 00:14:46.659 --> 00:14:48.820 nodes. Right, much more complex. You can't just 00:14:48.820 --> 00:14:50.879 shrink weights in a web that complex, can you? 00:14:51.120 --> 00:14:53.120 Well... Let's shift and look at the architecture 00:14:53.120 --> 00:14:55.580 of the models themselves. Because the source 00:14:55.580 --> 00:14:58.340 talks about an implicit technique used in neural 00:14:58.340 --> 00:15:00.539 networks called dropout, and I gotta be honest, 00:15:00.659 --> 00:15:03.830 this... absolutely blew my mind. Dropout is wild. 00:15:04.070 --> 00:15:07.210 During training, Dropout repeatedly ignores random 00:15:07.210 --> 00:15:09.669 subsets of neurons. Like you are intentionally 00:15:09.669 --> 00:15:12.230 blinding the network. You make a system smarter 00:15:12.230 --> 00:15:15.110 by randomly giving it selective amnesia. It sounds 00:15:15.110 --> 00:15:18.210 chaotic, right? But the mechanical logic is actually 00:15:18.210 --> 00:15:21.370 brilliant. How so? In a deep neural network, 00:15:21.710 --> 00:15:24.070 neurons can start to rely way too heavily on 00:15:24.070 --> 00:15:27.460 each other. If neuron A always corrects the mistakes 00:15:27.460 --> 00:15:31.080 of neuron B, they create this complex co -adaptation 00:15:31.080 --> 00:15:33.580 that only works for that specific training data. 00:15:33.840 --> 00:15:37.639 They become a clique. Yes, a clique. So by randomly 00:15:37.639 --> 00:15:39.360 dropping neurons out of the network during a 00:15:39.360 --> 00:15:42.580 training pass, literally temporarily erasing 00:15:42.580 --> 00:15:45.519 them and their connections, you force every single 00:15:45.519 --> 00:15:47.620 neuron to learn how to be useful on its own. 00:15:47.980 --> 00:15:50.440 Because a neuron never knows which of its coworkers 00:15:50.440 --> 00:15:52.700 are going to show up to work that day. Exactly. 00:15:52.940 --> 00:15:55.679 If the neuron responsible for detecting a cat's 00:15:55.679 --> 00:15:58.139 ear drops out, the rest of the network has to 00:15:58.139 --> 00:16:00.480 work harder to identify the cat based on its 00:16:00.480 --> 00:16:03.389 tail or its whiskers. The mechanism completely 00:16:03.389 --> 00:16:06.289 shatters those co -adaptations. It does. And 00:16:06.289 --> 00:16:08.929 mathematically, dropout simulates the training 00:16:08.929 --> 00:16:11.149 of multiple different neural network architectures 00:16:11.149 --> 00:16:13.490 at once. Because it's constantly changing shape. 00:16:13.610 --> 00:16:15.649 Right. Because the structure is slightly different 00:16:15.649 --> 00:16:18.549 in every single training iteration. The final 00:16:18.549 --> 00:16:21.070 model is essentially an average of all these 00:16:21.070 --> 00:16:24.230 thousands of subnetworks, which vastly improves 00:16:24.230 --> 00:16:27.100 generalization. That is incredibly clever. But 00:16:27.100 --> 00:16:29.700 dealing with these complex structures and mathematical 00:16:29.700 --> 00:16:31.879 penalties brings up some major computational 00:16:31.879 --> 00:16:35.039 hurdles, too. The text mentions a specific issue 00:16:35.039 --> 00:16:38.519 with computing L1 involving calculus. Ah, yes. 00:16:38.740 --> 00:16:41.840 The infamous L1 kink. The kink. Remember how 00:16:41.840 --> 00:16:44.299 L1 uses the absolute value of the weights? Yeah. 00:16:44.500 --> 00:16:46.159 Well, if you graph an absolute value function, 00:16:46.200 --> 00:16:48.559 it looks like a perfect V. It's convex. It forms 00:16:48.559 --> 00:16:51.460 a bowl. But at the very bottom of the V, at exactly 00:16:51.460 --> 00:16:54.379 zero, there is a sharp corner. A kink. And if 00:16:54.379 --> 00:16:57.500 I remember high school math? Calculus hates sharp 00:16:57.500 --> 00:17:01.240 corners. Calculus absolutely hates sharp corners. 00:17:02.059 --> 00:17:04.519 Gradient descent relies on calculating derivatives, 00:17:04.960 --> 00:17:07.440 which are essentially slopes. Right? At a smooth 00:17:07.440 --> 00:17:10.599 curve, you can easily calculate the slope, but 00:17:10.599 --> 00:17:13.700 at the exact pointy bottom of a V, the slope 00:17:13.700 --> 00:17:16.819 is undefined. It's just a point. Exactly. The 00:17:16.819 --> 00:17:19.099 function is not strictly differentiable at that 00:17:19.099 --> 00:17:21.240 point, meaning standard gradient descent breaks 00:17:21.240 --> 00:17:23.819 down completely. So how do you fix it? Well, 00:17:23.900 --> 00:17:26.259 to solve this efficiently, the text notes we 00:17:26.259 --> 00:17:28.519 use proximal methods. Proximal methods. OK, how 00:17:28.519 --> 00:17:30.640 does a proximal method smooth out that kink? 00:17:30.720 --> 00:17:33.279 How does it bypass a mathematical impossibility? 00:17:33.680 --> 00:17:35.579 So instead of trying to calculate the slope of 00:17:35.579 --> 00:17:37.500 the whole messy function at that sharp point, 00:17:38.039 --> 00:17:40.839 a proximal method splits the problem into two 00:17:40.839 --> 00:17:43.440 distinct steps. OK, two steps. First, it performs 00:17:43.440 --> 00:17:46.019 a standard gradient descent step on the smooth, 00:17:46.259 --> 00:17:48.039 easy -to -calculate part of the loss function, 00:17:48.539 --> 00:17:50.720 basically ignoring the L1 penalty for microsecond. 00:17:50.740 --> 00:17:52.930 Just pretending it's not there. Right. Then it 00:17:52.930 --> 00:17:55.190 applies a proximal operator to project that result 00:17:55.190 --> 00:17:57.650 back into the space permitted by the L1 penalty. 00:17:57.710 --> 00:17:59.869 Okay, give me a physical analogy for that, because 00:17:59.869 --> 00:18:03.190 my brain is straining. Sure. Imagine you are 00:18:03.190 --> 00:18:05.829 walking down a valley, trying to reach the absolute 00:18:05.829 --> 00:18:08.890 bottom, but the very bottom is a dangerously 00:18:08.890 --> 00:18:11.349 sharp spike. Okay. Instead of stepping directly 00:18:11.349 --> 00:18:13.930 on the spike, you step slightly to the side where 00:18:13.930 --> 00:18:17.529 the ground is smooth. Smart. Then the proximal 00:18:17.529 --> 00:18:20.579 operator acts like a mathematical magnet. If 00:18:20.579 --> 00:18:23.059 your step lands close enough to the spike, it 00:18:23.059 --> 00:18:25.119 just snaps your position directly to the center, 00:18:25.359 --> 00:18:30.200 precisely to zero. For L1, this operator is effectively 00:18:30.200 --> 00:18:33.400 a soft thresholding function. It evaluates the 00:18:33.400 --> 00:18:35.480 smooth slope, and if a weight is small enough, 00:18:35.819 --> 00:18:39.240 it elegantly forces it precisely to zero, bypassing 00:18:39.240 --> 00:18:41.740 the need to ever calculate the impossible derivative 00:18:41.740 --> 00:18:44.559 at the kink. That is so elegant. It really is. 00:18:44.740 --> 00:18:46.759 So we've covered a lot of ground on how a single 00:18:46.759 --> 00:18:49.140 model learns from isolated data. But the source 00:18:49.140 --> 00:18:51.359 material pushes into more complex territory. 00:18:51.400 --> 00:18:53.299 It definitely scales up. What happens when we 00:18:53.299 --> 00:18:55.559 have complex systems analyzing interconnected 00:18:55.559 --> 00:18:58.859 tasks, or worse, data without labels? I mean, 00:18:59.160 --> 00:19:01.079 gathering labeled data is incredibly expensive 00:19:01.079 --> 00:19:03.079 in the real world. It's the bottleneck of AI. 00:19:03.400 --> 00:19:06.279 Right. If I have a database of a million pictures 00:19:06.279 --> 00:19:09.740 of cats, but I only paid a human to label 10 00:19:09.740 --> 00:19:13.140 of them as cat. How does regularization help 00:19:13.140 --> 00:19:15.940 the AI figure out the rest? Well, that brings 00:19:15.940 --> 00:19:19.259 us to semi -supervised learning. You have a tiny 00:19:19.259 --> 00:19:22.279 bit of labeled data and a massive amount of unlabeled 00:19:22.279 --> 00:19:25.220 data. The text outlines how regularizers can 00:19:25.220 --> 00:19:27.740 actually guide the learning algorithm by respecting 00:19:27.740 --> 00:19:29.759 the hidden structure of that unlabeled data. 00:19:29.900 --> 00:19:31.720 You use a symmetric weight matrix and something 00:19:31.720 --> 00:19:35.119 called a graph Laplacian matrix. Laplacian matrix? 00:19:35.539 --> 00:19:37.240 That sounds intimidating. Explain that for someone 00:19:37.240 --> 00:19:38.980 who doesn't build supercomputers for a living. 00:19:39.339 --> 00:19:41.140 Honestly, it's just a way to represent a web 00:19:41.140 --> 00:19:43.900 or a graph mathematically. A web? Imagine all 00:19:43.900 --> 00:19:46.220 your million pictures are nodes on a giant spider 00:19:46.220 --> 00:19:49.839 web. The Laplacian matrix just maps out how tightly 00:19:49.839 --> 00:19:52.420 connected every node is to its neighbors based 00:19:52.420 --> 00:19:53.920 on their features. Okay, I can picture that. 00:19:54.180 --> 00:19:56.140 The regular riser relies on a simple intuition. 00:19:56.829 --> 00:19:59.470 If a distance metric tells us that two input 00:19:59.470 --> 00:20:01.269 points are very similar to each other, meaning 00:20:01.269 --> 00:20:04.089 they are close together on the web, then the 00:20:04.089 --> 00:20:06.750 regularizer enforces a massive penalty if the 00:20:06.750 --> 00:20:08.670 model predicts vastly different outputs for them. 00:20:08.809 --> 00:20:12.549 It's guilt by association, but for math. Exactly. 00:20:12.569 --> 00:20:14.990 Think of it like a rubber band network. If an 00:20:14.990 --> 00:20:17.289 unlabeled picture looks mathematically similar 00:20:17.289 --> 00:20:20.910 to one of my 10 labeled cat pictures, the rubber 00:20:20.910 --> 00:20:23.849 bands pull them together. Yes. The regularizer 00:20:23.849 --> 00:20:26.950 gently forces the model to treat unlabeled picture 00:20:26.950 --> 00:20:29.690 like a cat because it's caught in the same part 00:20:29.690 --> 00:20:31.990 of the web. And the beauty of this is that the 00:20:31.990 --> 00:20:34.069 unlabeled part of the function can actually be 00:20:34.069 --> 00:20:36.849 solved analytically using the pseudo inverse 00:20:36.849 --> 00:20:40.170 of the Laplacian submatrices. Meaning? You are 00:20:40.170 --> 00:20:43.150 calculating the reverse paths on the graph to 00:20:43.150 --> 00:20:46.329 seamlessly propagate those 10 labels across the 00:20:46.329 --> 00:20:49.269 entire million picture data set. Mind -blowing. 00:20:49.750 --> 00:20:52.210 So what does this all mean for you, the listener, 00:20:52.349 --> 00:20:55.200 in your daily life? It affects everything. The 00:20:55.200 --> 00:20:57.740 text gives a fantastic real -world example of 00:20:57.740 --> 00:21:00.279 this interconnected math, and it has to do with 00:21:00.279 --> 00:21:03.460 how we binge watch television. Multitask learning. 00:21:03.900 --> 00:21:06.220 Right. Sometimes, instead of solving one isolated 00:21:06.220 --> 00:21:08.180 problem, you want to solve multiple problems 00:21:08.180 --> 00:21:10.140 simultaneously because the tasks are inherently 00:21:10.140 --> 00:21:12.240 related. You want to borrow strengths from that 00:21:12.240 --> 00:21:14.140 relatedness. Makes sense. The text introduces 00:21:14.140 --> 00:21:16.960 a concept called clustered mean -constrained 00:21:16.960 --> 00:21:20.740 regularization. Clustered mean -constrained regularization. 00:21:20.839 --> 00:21:22.960 Say that three times fast. It is a mouthful. 00:21:23.130 --> 00:21:25.390 but the application is something everyone interacts 00:21:25.390 --> 00:21:29.269 with. Netflix recommendations. Ah, Netflix. Think 00:21:29.269 --> 00:21:32.630 of predicting what movie one specific user will 00:21:32.630 --> 00:21:36.369 like as a single isolated task. Okay. Netflix 00:21:36.369 --> 00:21:38.529 has hundreds of millions of users, so they have 00:21:38.529 --> 00:21:41.269 hundreds of millions of simultaneous tasks. And 00:21:41.269 --> 00:21:44.170 predicting my incredibly niche movie tastes is 00:21:44.170 --> 00:21:47.420 a single task. But I'm not entirely unique. I 00:21:47.420 --> 00:21:50.859 belong to a cluster of people who, say, love 00:21:50.859 --> 00:21:53.539 gritty sci -fi thrillers, but absolutely hate 00:21:53.539 --> 00:21:56.339 romantic comedies. Right. The text explains that 00:21:56.339 --> 00:21:58.720 a cluster corresponds to a group of people sharing 00:21:58.720 --> 00:22:01.740 similar preferences. OK. The clustered mean -constrained 00:22:01.740 --> 00:22:04.700 regularizer enforces a mathematical penalty if 00:22:04.700 --> 00:22:06.940 the function learned for your specific profile 00:22:06.940 --> 00:22:09.299 strays too far from the overall average function 00:22:09.299 --> 00:22:12.000 of your cluster. So it's regularizing my recommendations 00:22:12.000 --> 00:22:14.180 by tethering them to the people mathematically 00:22:14.980 --> 00:22:18.799 Similar to me. Exactly. It enforces similarity 00:22:18.799 --> 00:22:21.900 between tasks meaning users within the same cluster. 00:22:22.900 --> 00:22:25.160 This captures complex prior information about 00:22:25.160 --> 00:22:28.319 human behavior. Like shared tastes. Yeah. If 00:22:28.319 --> 00:22:31.259 10 people in your cluster loved an obscure new 00:22:31.259 --> 00:22:34.720 sci -fi documentary, the regularizer ensures 00:22:34.720 --> 00:22:37.400 your predictive model borrows that strength. 00:22:37.859 --> 00:22:40.420 It penalizes the algorithm if it doesn't recommend 00:22:40.420 --> 00:22:43.180 that documentary to you, because doing so would 00:22:43.180 --> 00:22:45.859 deviate from the cluster's mean. So the math 00:22:45.859 --> 00:22:48.799 of regularization is literally actively shaping 00:22:48.799 --> 00:22:50.839 what you and I choose to watch on a Friday night. 00:22:50.960 --> 00:22:53.319 It really is. That is wild. It's everywhere. 00:22:53.500 --> 00:22:55.759 If we take a step back and synthesize this whole 00:22:55.759 --> 00:22:58.799 journey we've been on. Regularization is clearly 00:22:58.799 --> 00:23:01.680 not just an obscure calculus trick for statisticians. 00:23:02.299 --> 00:23:04.480 It is the fundamental principle of preventing 00:23:04.480 --> 00:23:06.960 artificial intelligence from becoming an obsessive 00:23:06.960 --> 00:23:10.099 over -thinker. It stops AI from memorizing the 00:23:10.099 --> 00:23:13.160 practice test. Whether it's the ruthless minimalism 00:23:13.160 --> 00:23:15.680 of Lasso throwing out useless medical markers, 00:23:16.220 --> 00:23:18.279 or Dropout giving a neural network selective 00:23:18.279 --> 00:23:21.539 amnesia so it builds better architecture, or 00:23:21.539 --> 00:23:24.140 Netflix clustering your movie tastes with strangers. 00:23:25.309 --> 00:23:27.910 Regularization is the pursuit of elegant simplicity. 00:23:28.910 --> 00:23:31.930 It is about finding the true signal in a world 00:23:31.930 --> 00:23:35.309 entirely overwhelmed by noisy data. And, you 00:23:35.309 --> 00:23:37.369 know, this raises an important question. Stepping 00:23:37.369 --> 00:23:39.210 away from the math for a second and looking at 00:23:39.210 --> 00:23:42.930 how this applies to us. Oh, lay it on us. We've 00:23:42.930 --> 00:23:44.410 been talking about the math and the text all 00:23:44.410 --> 00:23:46.430 day, but applying this to our own lives raises 00:23:46.430 --> 00:23:49.180 a wild thought. I'm ready. The source notes that 00:23:49.180 --> 00:23:51.880 from a Bayesian point of view, regularization 00:23:51.880 --> 00:23:54.440 is simply imposing certain prior distributions 00:23:54.440 --> 00:23:57.380 on model parameters. Prior distribution. We are 00:23:57.380 --> 00:24:00.059 mathematically telling the AI what we expect 00:24:00.059 --> 00:24:02.059 the world to look like before it even looks at 00:24:02.059 --> 00:24:04.720 the data. We enforce a prior belief that the 00:24:04.720 --> 00:24:07.779 world is inherently simple or sparse or clustered. 00:24:08.220 --> 00:24:10.640 Oh, so we are hard coding our assumptions into 00:24:10.640 --> 00:24:13.099 the machine so it doesn't get overwhelmed. Exactly. 00:24:13.339 --> 00:24:17.440 So if our most advanced logical objective supercomputers 00:24:18.029 --> 00:24:21.130 absolutely require mathematically enforced priors 00:24:21.130 --> 00:24:24.130 and penalties to prevent them from drawing absurd 00:24:24.130 --> 00:24:28.369 overly complex conclusions from noisy data. Yeah. 00:24:28.730 --> 00:24:31.910 Are human assumptions actually just a biological 00:24:31.910 --> 00:24:35.089 form of regularization? Oh, wow. Right. We often 00:24:35.089 --> 00:24:38.049 think of our cognitive priors, our mental shortcuts 00:24:38.049 --> 00:24:41.579 and assumptions as flaws. But are those biological 00:24:41.579 --> 00:24:44.099 priors the only mechanism keeping our brains 00:24:44.099 --> 00:24:46.619 from being completely paralyzed by the infinite 00:24:46.619 --> 00:24:49.680 noise of everyday life? Are we just biologically 00:24:49.680 --> 00:24:52.960 regularized machines? That is a heavy, fascinating 00:24:52.960 --> 00:24:55.240 thought to leave on. Maybe our brains are constantly 00:24:55.240 --> 00:24:57.059 applying early stopping so we can get out of 00:24:57.059 --> 00:24:58.799 bed in the morning without overanalyzing the 00:24:58.799 --> 00:25:00.680 atomic structure of the floor. I know. I definitely 00:25:00.680 --> 00:25:02.539 use early stopping before my morning coffee. 00:25:02.680 --> 00:25:04.279 To everyone listening, thank you for joining 00:25:04.279 --> 00:25:06.680 us on this deep dive into the mathematical engine 00:25:06.680 --> 00:25:08.680 of AI. Next time you're studying for a test, 00:25:08.940 --> 00:25:11.299 remember... Don't memorize the practice questions, 00:25:11.680 --> 00:25:13.920 learn the overarching concepts, stay curious, 00:25:14.180 --> 00:25:16.140 embrace simplicity, and we'll catch you on the 00:25:16.140 --> 00:25:16.779 next deep dive.