Vladimir Vapnik: Statistical Learning | Lex Fridman Podcast #5
 The following is a conversation with Vladimir Vapnik. He's the co-inventor of the support vector machines, support vector clustering, VC theory, and many foundational ideas in statistical learning. He was born in the Soviet Union and worked at the Institute of Control Sciences in Moscow. Then, in the United States, he worked at AT&T, NEC Labs, Facebook Research, and now is a professor at Columbia University. His work has been cited over 170,000 times. He has some very interesting ideas about artificial intelligence and the nature of learning, especially on the limits of our current approaches and the open problems in the field. This conversation is part of an MIT course on artificial general intelligence and the artificial intelligence podcast. If you enjoy it, please subscribe on YouTube or rate it on iTunes or your podcast provider of choice. Or simply connect with me on Twitter or other social networks at Lex Friedman, spelled F-R-I-D. And now, here's my conversation with Vladimir Vapnik. Einstein famously said that God doesn't play dice. Yeah. You have studied the world through the eyes of statistics, so let me ask you in terms of the nature of reality, fundamental nature of reality, does God play dice? We don't know some factors. And because we don't know some factors, which could be important, it looks like God plays dice. But we should describe it. In philosophy, they distinguish between two positions, positions of instrumentalism, where you're creating theory for prediction and position of realism, where you're trying to understand what God did. Can you describe instrumentalism and realism a little bit? For example, if you have some mechanical laws, what is that? Is it law which is true always and everywhere? Or it is law which allows you to predict the position of a moving element? What you believe? You believe that it is God's law, that God created the world, which obey to this physical law? Or it is just law for predictions? And which one is instrumentalism? For predictions. If you believe that this is law of God, and it's always true everywhere, that means that you're a realist. So you're trying to really understand the God's thought. So the way you see the world is as an instrumentalist? You know, I'm working for some models, models of machine learning. So in this model, we can see setting, and we try to solve, resolve the setting, to solve the problem. And you can do it in two different ways, from the point of view of instrumentalists. And that's what everybody does now, because they say that goal of machine learning is to find the role for classification. That is true, but it is instrument for prediction. But I can say the God of machine learning is to learn about conditional probability. So how God played youth, and is he play? What is probability for one? What is probability for another given situation? But for prediction, I don't need this. I need the role. But for understanding, I need conditional probability. So let me just step back a little bit first to talk about, you mentioned, which I read last night, the parts of the 1960 paper by Eugene Wigner, Unreasonable Effectiveness of Mathematics and Natural Sciences. Such a beautiful paper, by the way. Made me feel, to be honest, to confess my own work in the past few years on deep learning, heavily applied, made me feel that I was missing out on some of the beauty of nature in the way that math can uncover. So let me just step away from the poetry of that for a second. How do you see the role of math in your life? Is it a tool? Is it poetry? Where does it sit? And does math for you have limits of what it can describe? Some people say that math is language which uses God. So I believe... Speak to God or use God? Use God. Use God. Yeah. So I believe that this article about effectiveness, unreasonable effectiveness of math, is that if you're looking at mathematical structures, they know something about reality. And the most scientists from natural science, they're looking on equation and trying to understand reality. So the same in machine learning. If you try very carefully look on all equations which define conditional probability, you can understand something about reality more than from your fantasy. So math can reveal the simple underlying principles of reality perhaps. You know what means simple? It is very hard to discover them. But then when you discover them and look at them, you see how beautiful they are. And it is surprising why people did not see that before you're looking on equation and derive it from equations. For example, I talked yesterday about least square net. And people had a lot of fantasy how to improve least square net. But if you're going step by step by solving some equations, you suddenly will get some term which after thinking, you understand that it describes position of observation point. In least square net, we throw out a lot of information. We don't look in composition of point of observations, we're looking only on residuals. But when you understood that, that's very simple idea. But it's not too simple to understand. And you can derive this just from equations. So some simple algebra, a few steps will take you to something surprising that when you think about, you understand. And that is proof that human intuition not to reach and very primitive. And it does not see very simple situations. So let me take a step back. In general, yes. But what about human as opposed to intuition, ingenuity, moments of brilliance? So do you have to be so hard on human intuition? Are there moments of brilliance in human intuition that can leap ahead of math and then the math will catch up? I don't think so. I think that the best human intuition, it is putting in axioms. And then it is technical way how to... See where the axioms take you. Yeah. But if they correctly take axioms. But it axiom polished during generations of scientists. And this is integral wisdom. So that's beautifully put. But if you maybe look at... When you think of Einstein and special relativity, what is the role of imagination coming first there in the moment of discovery of an idea? So there's obviously a mix of math and out of the box imagination there. But that I don't know. Whatever I did, I exclude any imagination. Because whatever I saw in machine learning that come from imagination, like features, like deep learning, they are not relevant to the problem. When you're looking very carefully from mathematical equations, you're deriving very simple theory which goes far beyond theoretically than whatever people can imagine. Because it is not good fantasy. It is just interpretation. It is just fantasy. But it is not what you need. You don't need any imagination to derive, say, main principle of machine learning. When you think about learning and intelligence, maybe thinking about the human brain and trying to describe mathematically the process of learning, that is something like what happens in the human brain. Do you think we have the tools currently? Do you think we will ever have the tools to try to describe that process of learning? It is not description what's going on. It is interpretation. It is your interpretation. Your vision can be wrong. You know, when Guy invented microscope, Levenguk, for the first time, only he got this instrument and nobody, only he kept secret about microscope. But he wrote reports in London Academy of Science. And his report, when he looking at the blood, he looked everywhere, on the water, on the blood, on the spleen. But he described blood like fight between queen and king. So he saw blood cells, red cells, and he imagined that it is army fighting each other. And it was his interpretation of situation. And he sent this report in Academy of Science. They very carefully looked because they believed that he is right. He saw something. Yes. But he gave wrong interpretation. And I believe the same can happen to his brain. With brain, yeah. The most important part, you know, I believe in human language. In some proverb, is so much wisdom. For example, people say that it is better than thousand days of diligent studies one day with great teacher. But if I will ask you what teacher does, nobody knows. And that is intelligence. But we know from history and now from math and machine learning that teacher can do a lot. So what, from a mathematical point of view, is the great teacher? I don't know. That's an open question. No, no. But we can say what teacher can do. He can introduce some invariants, some predicate for creating invariants. How he doing it, I don't know. Because teacher knows reality and can describe from this reality predicate invariants. But we know that when you're using invariant, you can decrease number of observations 100 times. So but maybe try to pull that apart a little bit. I think you mentioned like a piano teacher saying to the student, play like a butterfly. I played piano, I played guitar for a long time. Yeah, that's, maybe it's romantic, poetic, but it feels like there's a lot of truth in that statement. Like there is a lot of instruction in that statement. And so can you pull that apart? What is that? The language itself may not contain this information. Language not blah, blah, blah. Because it's not blah, blah, blah. It's effective. It's what? Effective. Effective playing. Yes, it does. But it's not the language. It feels like what is the information being exchanged there? What is the nature of information? What is the representation of that information? I believe that it is sort of predicate, but I don't know. That is exactly what intelligence and machine learning should be. Yes. But that is just mathematical technique. I think that what was discovered recently is that there is two mechanisms of learning. One called strong convergence mechanism and weak convergence mechanism. Before people use only one convergence. In weak convergence mechanism, you can use predicate. That's what play like butterfly. And it will immediately affect your playing. You know, there is English proverb, great. If it looks like a duck, swims like a duck, and quack like a duck, then it is probably duck. Yes. But this is exact about predicate. Looks like a duck, what it means. So you saw many ducks that you're training data. So you have description of how looks integral, looks ducks. Yeah, the visual characteristics of a duck. Yeah. But you want and you have model for recognition doubts. So you would like so that theoretical description from model coincide with empirical description, which you saw on the text there. So about looks like a duck, it is general. But what about swims like a duck? You should know the duck swims. You can say it play chess like a duck. Okay, duck doesn't play chess. And it is completely legal predicate, but it is useless. So half teacher can recognize not useless predicate. So up to now, we don't use this predicate in existing machine learning. And you think that's... So why we need zillions of data. But in this English proverb, they use only three predicate. Looks like a duck, swims like a duck, and quack like a duck. So you can't deny the fact that swims like a duck and quacks like a duck has humor in it, has ambiguity. Just talk about swim like a duck. It doesn't say jumps like a duck. Why? Because it's not relevant. But that means that you know ducks, you know different birds, you know animals. And you derive from this that it is really want to say swim like a duck. So underneath, in order for us to understand swims like a duck, it feels like we need to know millions of other little pieces of information. We pick up along the way. You don't think so. There doesn't need to be this knowledge base. In those statements carries some rich information that helps us understand the essence of duck. How far are we from integrating predicates? You know that when you consider complete machine learning. So what it does, you have a lot of functions and then you're talking it looks like a duck. You see your training data. From training data, you recognize like expected duck should look. Then you remove all functions which does not look like you think it should look from training data. So you decrease the amount of function from which you pick up one. Then you give a second predicate and then again decrease the set of function. And after that, you pick up the best function you can find. It is standard machine learning. So why you need not too many examples. Your predicates aren't very good. Because every predicate is invented to decrease admissible set of function. So you talk about admissible set of functions and you talk about good functions. So what makes a good function? So admissible set of function is set of function which has small capacity or small diversity, small VC dimension. Which contain good function inside. By the way, for people who don't know, VC, you're the V in the VC. So how would you describe to a lay person what VC theory is? How would you describe VC? You have a machine. So machine capable to pick up one function from the admissible set of function. But set of admissibles function can be big. So it contain all continuous functions and it's useless. You don't have so many examples to pick up function. But it can be small. We call it capacity, but maybe better call it diversity. So not very different function in the set. It's infinite set of function, but not very diverse. So it is small VC dimension. When VC dimension is small, you need small amount of training date. So the goal is to create admissible set of functions which have small VC dimension and contain good function. Then you will be able to pick up the function using small amount of observations. So that is the task of learning. Yeah. So the goal is creating a set of admissible functions that has a small VC dimension. Then you've figured out a clever way of picking up. That is goal of learning which I formulated yesterday. Statistical learning theory does not involve in creating admissible set of function. In classical learning theory, everywhere, 100% in textbook, the set of function that admissible set of function is given. But this is science about nothing because the most difficult problem to create admissible set of functions given say a lot of functions, continuum set of function, create admissible set of functions that means that it has finite VC dimension, small VC dimension and contain good function. So this was out of consideration. So what's the process of doing that? I mean it's fascinating. What is the process of creating this admissible set of functions? That is invariant. That's invariance. Can you describe invariance? Yeah. You're looking of properties of training data and properties means that you have some function and you just count what is average value of function on training data. You have model and what is expectation of this function on the model and they should coincide. So the problem is about how to pick up functions. It can be any function. In fact, it is true for all functions. But because when I talking set say duck does not jumping, so you don't ask question jump like a duck because it is trivial, it does not jumping and doesn't help you to recognize jump. But you know something, which question to ask and you asking it swims like a duck. But looks like a duck at its general situation. Looks like say guy who have this illness, this disease, it is legal. So there is a general type of predicate looks like and special type of predicate which related to this specific problem. And that is intelligence part of all this business. And that where teachers involved. Incorporating the specialized predicates. Okay. What do you think about deep learning as neural networks, these arbitrary architectures as helping accomplish some of the tasks you are thinking about? Their effectiveness or lack thereof? What are the weaknesses and what are the possible strengths? You know I think that this is fantasy. Something which like deep learning, like features. Let me give you this example. One of the greatest book is Churchill book about history of second world war. And he starting this book describing that in old time when war is over, so the great kings they gathered together, almost all of them were relatives. And they discussed what should be done, how to create peace. And they came to agreement. And when happens first world war, the general public came in power. And they were so greedy that robbed Germany. And it was clear for everybody that it is not peace. That peace will last only 20 years because they was not professionals. And the same I see in machine learning. There are mathematicians who are looking for the problem from very deep point of view, mathematical point of view. And there are computer scientists who mostly does not know mathematics. They just have interpretation of that. And they invented a lot of blah, blah, blah interpretations like deep learning. Why you did deep learning? Mathematic does not know deep learning. Mathematic does not know neurons. It is just function. If you like to say piecewise linear function, say that. And do in class of piecewise linear function. But they invent something. And then they try to prove advantage of that through interpretations, which mostly wrong. And when it is not enough, they appeal to brain which they know nothing about that. Nobody knows what is going on in the brain. So I think that more reliable, look on math. This is mathematical problem. Do your best to solve this problem. Try to understand that there is not only one way of convergence, which is strong way of convergence. There is a weak way of convergence, which requires predicate. And if you will go through all this stuff, you will see that you don't need deep learning. Even more, I would say one of the theorem which called the representative theory. It says that optimal solution of mathematical problem, which is described learning is on shadow network, not on deep learning. And a shallow network. Yeah. Yeah, shallow network. The problem is there. Absolutely. So in the end, what you're saying is exactly right. The question is you have no value for throwing something on the table, playing with it, not math. It's like in your network where you said throwing something in the bucket or the biological example and looking at kings and queens with the cells or the microscope. You don't see value in imagining the cells or kings and queens and using that as inspiration and imagination for where the math will eventually lead you. You think that interpretation basically deceives you in a way that's not productive. I think that if you're trying to analyze this... The nature of learning....of learning and especially discussion about deep learning, it is discussion about interpretation, not about things, about what you can say about things. That's right. But aren't you surprised by the beauty of it? Not mathematical beauty, but the fact that it works at all. Or you're criticizing that very beauty, our human desire to interpret, to find our silly interpretations in these constructs. Let me ask you this. Are you surprised and does it inspire you? How do you feel about the success of a system like AlphaGo at beating the game of Go? Using neural networks to estimate the quality of a board and the quality of the... That is your interpretation, quality of the board. Yeah, yes. Yeah. But it's not our interpretation. The fact is a neural network system doesn't matter. A learning system that we don't, I think, mathematically understand that well beats the best human player. That's something that was thought impossible. That means that it's not a very difficult problem. We've empirically have discovered that this is not a very difficult problem. Yeah. It's true. So maybe it can argue. So... Even more, I would say that if they use deep learning, it is not the most effective way of learning theory. And usually when people use deep learning, they're using zillions of training data. Yeah, but you don't need this. So I describe challenge, can we do some problems which do well, deep learning method, this deep net, using 100 times less training data. Even more, some problems deep learning cannot solve because it's not necessary. They create admissible set of function. To create deep architecture means to create admissible set of functions. You cannot say that you're creating good admissible set of functions. You just, it's your fantasy. It is not comes from mass. But it is possible to create admissible set of functions because you have your training data. But actually for mathematicians, when you consider invariant, you need to use law of large numbers. When you're making training in existing algorithm, you need uniform law of large numbers, which is much more difficult. It requires VC dimension and all this stuff. But nevertheless, if you use both, weak and strong way of convergence, you can decrease a lot of training data. You could do the three, the swims like a duck and quacks like a duck. So let's step back and think about human intelligence in general. Clearly that has evolved in a non-mathematical way. It wasn't, as far as we know, God or whoever didn't come up with a model in place in our brain of admissible functions. It kind of evolved. I don't know. Maybe you have a view on this. But so Alan Turing in the 50s in his paper asked and rejected the question, can machines think is not a very useful question, but can you briefly entertain this useful, useless question? Can machines think, so talk about intelligence and your view of it? I don't know that. I know that Turing described imitation. If computer can imitate human being, let's call it intelligent. And he understands that it is not thinking computer. He completely understand what he doing. Like he set up problem of imitation. So now we understand that the problem not in imitation. I'm not sure that intelligence just inside of us. It may be also outside of us. I have several observations. So when I prove some theorem, it's very difficult theorem. In a couple of years in several places, people prove the same theorem. Say Sawyer-Lemma after us was done, then another guys prove the same theorem. In the history of science, it's happened all the time. For example, geometry. It's happened simultaneously. At first it did Lobachevsky and then Gauss and Boyayi and then other guys. And it approximately in 10 times period, take 10 years period of time. And I saw a lot of examples like that. And many mathematicians thinks that when they develop something, they develop something in general which affect everybody. So maybe our models of intelligence only inside of us is incorrect. It's our interpretation. Maybe there exists some connection with world intelligence. I don't know. You're almost like plugging in into… Yeah, exactly. And contributing to this… Into a big network. Into a big… Maybe in your own network. No, no, no. On the flip side of that, maybe you can comment on big O complexity and how you see classifying algorithms by worst-case running time in relation to their input. So that way of thinking about functions. Do you think P equals NP? Do you think that's an interesting question? Yeah, it is an interesting question. But let me talk about complexity in about worst-case scenario. There is a mathematical setting. When I came to United States in 1991, people did not know this theory. They did not know statistical learning theory. So in Russia, it was published to monographs, our monographs, but in America, they did not know. Then they learned. And somebody told me that it is worst-case theory and they will create real-case theory, but till now, it did not. Because it is mathematical too. You can do only what you can do using mathematics, which has clear understanding and clear description. And for this reason, we introduce complexity. And we need this because using actually this diversity like this one more, this dimension, you can prove some theorems. But we also create theory for case when you know probability measure. And that is the best case which can happen in this entropy theory. So from a mathematical point of view, you know the best possible case and the worst possible case. You can derive different model in medium. But it's not so interesting. Do you think the edges are interesting? The edges are interesting because it is not so easy to get good bound, exact bound. It's not many cases where you have the bound is not exact, but interesting principles which discover the mass. Do you think it's interesting because it's challenging and reveals interesting principles that allow you to get those bounds? Or do you think it's interesting because it's actually very useful for understanding the essence of a function of an algorithm? So it's like me judging your life as a human being by the worst thing you did and the best thing you did versus all the stuff in the middle. It seems not productive. I don't think so because you cannot describe situation in the middle or it will be not general. So you can describe edges cases and it is clear it has some model, but you cannot describe model for every new case. So you will be never accurate when you're using model. But from a statistical point of view, the way you've studied functions and the nature of learning and the world, don't you think that the real world has a very long tail? That the edge cases are very far away from the mean, the stuff in the middle or no? I don't know that. I think that from my point of view, if you will use formal statistic, you need uniform law of large numbers. If you will use this invariance business, you will need just law of large numbers. There's a huge difference between uniform law of large numbers and large numbers. Is it useful to describe that a little more or should we just take it to... For example, when I'm talking about dark, I gave three predicates and it was enough. But if you will try to do formal distinguish, you will need a lot of observation. So that means that information about looks like a dark contain a lot of bit of information, formal bits of information. So we don't know that how much bit of information contain things from intelligence. And that is the subject of analysis. Until now, all business, I don't like how people consider artificial intelligence. They consider as some codes which imitate activity of human being. It is not science. It is applications. You would like to imitate go ahead. It is very useful and a good problem. But you need to learn something more. How people can to develop, say, predicates, swims like a duck or play like butterfly or something like that. Not the teacher says you how it came in his mind. How he chooses this image. So that process... That is the problem of intelligence. That is the problem of intelligence. And you see that connected to the problem of learning? Absolutely. Because you immediately give this predicate like specific predicates, swims like a duck or quack like a duck. It was chosen somehow. So what is the line of work, would you say? If you were to formulate as a set of open problems, that will take us there, will play like a butterfly, will get a system to be able to... Let's separate two stories. One mathematical story that if you have predicate, you can do something. And another story how to get predicate. It is intelligence problem and people even did not start to understand intelligence. Because to understand intelligence, first of all, try to understand what doing teachers. How teacher teach. Why one teacher better than another one? Yeah. So you think we really even haven't started on the journey of generating the predicates? No. We don't understand. We even don't understand that this problem exists. Because did you hear... You do. Yeah. No, I just know name. I want to understand why one teacher better than another and how affect teacher, student. It is not because he repeating the problem which is in textbook. He makes some remarks. He makes some philosophy of reasoning. Yeah, that's a beautiful... It is a formulation of a question that is the open problem. Why is one teacher better than another? Right. What he does better. Yeah. What... Why in every level? How do they get better? What does it mean to be better? The whole... Yeah. From whatever model I have, one teacher can give a very good predicate. One teacher can say, swims like a dog and another can say, jump like a dog. And jump like a dog carries zero information. So what is the most exciting problem in statistical learning you've ever worked on or are working on now? I just finished this invariant story and I'm happy that I believe that it is ultimate learning story. At least I can show that there are no another mechanism, only two mechanisms. But they separate statistical part from intelligent part. And I know nothing about intelligent part. And if we will know the intelligent part, so it will help us a lot in teaching, in learning. In learning. Yeah. You will know it when we see it? So for example, in my talk, the last slide was a challenge. So you have, say, NIST, digital recognition problem. And deep learning claims that they did it very well, say, 99.5% of correct answers. But say you use 60,000 observations. Can you do the same using 100 times less? But incorporating invariants, what it means, you know, digit one, two, three. Just looking at that, explain me which invariant I should keep to use 100 examples. Or say 100 times less examples to do the same job. Yeah, that last slide, unfortunately, your talk ended quickly, but that last slide was a powerful open challenge in a formulation of the evidence here. That is exact problem of intelligence. Because everybody, when machine learning started, it was developed by mathematicians, they immediately recognize that we use much more training data than human needed. But now again, we came to the same story. Have to decrease. And that is the problem of learning. It is not like in deep learning, they use zillions of training data because maybe zillions are not enough if you have a good invariant. Maybe you will never collect some number of observations. But now it is a question to intelligence. Have to do that because statistical part is ready. As soon as you supply us with predicate, we can do good job with small amount of observations. And the very first challenges will know digit recognition. And you know digits. And please tell me invariants. I think about that, I can say. For digit three, I would introduce concept of horizontal symmetry. So the digit three has horizontal symmetry, say more than say digit two or something like that. But as soon as I get the idea of horizontal symmetry, I can mathematically invent a lot of measure of horizontal symmetry or vertical symmetry or diagonal symmetry, whatever, if I have idea of symmetry. But what else? I think on digit I see that it is meta predicate, which is not shape. It is something like symmetry, like how dark is whole picture, something like that, which can self-rise a predicate. You think such a predicate could rise out of something that's not general, meaning it feels like for me to be able to understand the difference between a two and a three, I would need to have had a childhood of 10 to 15 years playing with kids, going to school, being yelled by parents, all of that, walking, jumping, looking at ducks. And now then I would be able to generate the right predicate for telling the difference between two and a three. Or do you think there's a more efficient way? I don't know. I know for sure that you must know something more than digits. Yes. And that's a powerful statement. Yeah. But maybe there are several languages of description, these elements of digits. So I'm talking about symmetry, about some properties of geometry. I'm talking about something abstract. I don't know that. But this is a problem of intelligence. So in one of our article, it is trivial to show that every example can carry not more than one bit of information in real. Because when you show example and you say this is one, you can remove, say, a function which does not tell you one. Say it's the best strategy. If you can do it perfectly, it's remove half of the functions. But when you use one predicate, which looks like a duck, you can remove much more functions than half. And that means that it contains a lot of bit of information from formal point of view. But when you have a general picture of what you want to recognize, a general picture of the world, can you invent this predicate? And that predicate carries a lot of information. Beautifully put. Maybe just me, but in all the math you show, in your work, which is some of the most profound mathematical work in the field of learning AI and just math in general, I hear a lot of poetry and philosophy. You really kind of talk about philosophy of science. There's a poetry and music to a lot of the work you're doing and the way you're thinking about it. So do you, where does that come from? Do you escape to poetry? Do you escape to music or not? I think that there exists ground truth. There exists ground truth? Yeah. And that can be seen everywhere. The smart guy, philosopher, sometimes I surprise how they deep see. Sometimes I see that some of them are completely out of subject. But the ground truth, I see in music. Music is the ground truth? Yeah. In poetry, in many poetry, they believe that they take dictation. So what piece of music as a piece of empirical evidence gave you a sense that they are touching something in the ground truth? It is structure. The structure of it. The math of music. Yeah, because when you're listening to Bach, you see the structure. Very clear, very classic, very simple. And the same in Bach when you have axioms in geometry, you have the same feeling. And in poetry, sometimes you see the same. And if you look back at your childhood, you grew up in Russia. You maybe were born as a researcher in Russia. You've developed as a researcher in Russia. You've came to the United States and a few places. If you look back, what were some of your happiest moments as a researcher? Some of the most profound moments, not in terms of their impact on society, but in terms of their impact on how damn good you feel that day and you remember that moment. You know, every time when you found something, it is great in the life. Every simple things. But my general feelings at that time, most of my time was wrong. You should go again and again and again and try to be honest in front of yourself. Not to make interpretation, but try to understand that it related to ground truth. It is not my blah, blah, blah interpretation and something like that. But you're allowed to get excited at the possibility of discovery. Oh, yeah. You have to double check it. No, but how it related to the other ground truth? Is it just temporary or it is forever? You know, you always have a feeling when you found something, how big is that? So 20 years ago when we discovered statistical learning theory, nobody believed except for one guy, Dudley from MIT. And then in 20 years it became fashion. And the same with support vector machines, kernel machines. So with support vector machines and learning theory, when you were working on it, you had a sense that you had a sense of the profundity of it, how this seems to be right. This seems to be powerful. Right. Absolutely. Immediately. I recognize that it will last forever. And now when I found this invariant story, I have a feeling that it is completely wrong because I have proved that there are no different mechanisms. We can have some, say, cosmetic improvement you can do, but in terms of invariants, we need both invariants and statistical learning and they should work together. But also I'm happy that we can formulate what is intelligence from that and to separate from technical part. That is completely different. Absolutely. Well, Vladimir, thank you so much for talking today. Thank you. It's an honor.
 The following is a conversation with Vladimir Vapnik.
 He's the co-inventor of the support vector machines, support vector clustering,
 VC theory, and many foundational ideas in statistical learning.
 He was born in the Soviet Union and worked at the Institute of Control Sciences in Moscow.
 Then, in the United States, he worked at AT&T, NEC Labs, Facebook Research,
 and now is a professor at Columbia University.
 His work has been cited over 170,000 times.
 He has some very interesting ideas about artificial intelligence and the nature of learning,
 especially on the limits of our current approaches and the open problems in the field.
 This conversation is part of an MIT course on artificial general intelligence
 and the artificial intelligence podcast.
 If you enjoy it, please subscribe on YouTube or rate it on iTunes or your podcast provider of choice.
 Or simply connect with me on Twitter or other social networks at Lex Friedman, spelled F-R-I-D.
 And now, here's my conversation with Vladimir Vapnik.
 Einstein famously said that God doesn't play dice.
 Yeah.
 You have studied the world through the eyes of statistics,
 so let me ask you in terms of the nature of reality, fundamental nature of reality,
 does God play dice?
 We don't know some factors.
 And because we don't know some factors, which could be important,
 it looks like God plays dice.
 But we should describe it.
 In philosophy, they distinguish between two positions, positions of instrumentalism,
 where you're creating theory for prediction and position of realism,
 where you're trying to understand what God did.
 Can you describe instrumentalism and realism a little bit?
 For example, if you have some mechanical laws, what is that?
 Is it law which is true always and everywhere?
 Or it is law which allows you to predict the position of a moving element?
 What you believe?
 You believe that it is God's law, that God created the world,
 which obey to this physical law?
 Or it is just law for predictions?
 And which one is instrumentalism?
 For predictions.
 If you believe that this is law of God, and it's always true everywhere,
 that means that you're a realist.
 So you're trying to really understand the God's thought.
 So the way you see the world is as an instrumentalist?
 You know, I'm working for some models, models of machine learning.
 So in this model, we can see setting, and we try to solve, resolve the setting,
 to solve the problem.
 And you can do it in two different ways, from the point of view of instrumentalists.
 And that's what everybody does now, because they say that goal of machine learning
 is to find the role for classification.
 That is true, but it is instrument for prediction.
 But I can say the God of machine learning is to learn about conditional probability.
 So how God played youth, and is he play?
 What is probability for one?
 What is probability for another given situation?
 But for prediction, I don't need this.
 I need the role.
 But for understanding, I need conditional probability.
 So let me just step back a little bit first to talk about, you mentioned,
 which I read last night, the parts of the 1960 paper by Eugene Wigner,
 Unreasonable Effectiveness of Mathematics and Natural Sciences.
 Such a beautiful paper, by the way.
 Made me feel, to be honest, to confess my own work in the past few years on deep learning,
 heavily applied, made me feel that I was missing out on some of the beauty of nature
 in the way that math can uncover.
 So let me just step away from the poetry of that for a second.
 How do you see the role of math in your life?
 Is it a tool? Is it poetry?
 Where does it sit?
 And does math for you have limits of what it can describe?
 Some people say that math is language which uses God.
 So I believe...
 Speak to God or use God?
 Use God.
 Use God.
 Yeah.
 So I believe that this article about effectiveness,
 unreasonable effectiveness of math, is that if you're looking at mathematical structures,
 they know something about reality.
 And the most scientists from natural science,
 they're looking on equation and trying to understand reality.
 So the same in machine learning.
 If you try very carefully look on all equations which define conditional probability,
 you can understand something about reality more than from your fantasy.
 So math can reveal the simple underlying principles of reality perhaps.
 You know what means simple?
 It is very hard to discover them.
 But then when you discover them and look at them, you see how beautiful they are.
 And it is surprising why people did not see that before you're looking on equation
 and derive it from equations.
 For example, I talked yesterday about least square net.
 And people had a lot of fantasy how to improve least square net.
 But if you're going step by step by solving some equations,
 you suddenly will get some term which after thinking,
 you understand that it describes position of observation point.
 In least square net, we throw out a lot of information.
 We don't look in composition of point of observations,
 we're looking only on residuals.
 But when you understood that, that's very simple idea.
 But it's not too simple to understand.
 And you can derive this just from equations.
 So some simple algebra, a few steps will take you to something surprising
 that when you think about, you understand.
 And that is proof that human intuition not to reach and very primitive.
 And it does not see very simple situations.
 So let me take a step back.
 In general, yes.
 But what about human as opposed to intuition, ingenuity, moments of brilliance?
 So do you have to be so hard on human intuition?
 Are there moments of brilliance in human intuition
 that can leap ahead of math and then the math will catch up?
 I don't think so.
 I think that the best human intuition, it is putting in axioms.
 And then it is technical way how to...
 See where the axioms take you.
 Yeah.
 But if they correctly take axioms.
 But it axiom polished during generations of scientists.
 And this is integral wisdom.
 So that's beautifully put.
 But if you maybe look at...
 When you think of Einstein and special relativity,
 what is the role of imagination coming first there in the moment of discovery of an idea?
 So there's obviously a mix of math and out of the box imagination there.
 But that I don't know.
 Whatever I did, I exclude any imagination.
 Because whatever I saw in machine learning that come from imagination,
 like features, like deep learning, they are not relevant to the problem.
 When you're looking very carefully from mathematical equations,
 you're deriving very simple theory which goes far beyond theoretically
 than whatever people can imagine.
 Because it is not good fantasy.
 It is just interpretation.
 It is just fantasy.
 But it is not what you need.
 You don't need any imagination to derive, say, main principle of machine learning.
 When you think about learning and intelligence,
 maybe thinking about the human brain and trying to describe mathematically the process of learning,
 that is something like what happens in the human brain.
 Do you think we have the tools currently?
 Do you think we will ever have the tools to try to describe that process of learning?
 It is not description what's going on.
 It is interpretation.
 It is your interpretation.
 Your vision can be wrong.
 You know, when Guy invented microscope, Levenguk, for the first time,
 only he got this instrument and nobody, only he kept secret about microscope.
 But he wrote reports in London Academy of Science.
 And his report, when he looking at the blood, he looked everywhere,
 on the water, on the blood, on the spleen.
 But he described blood like fight between queen and king.
 So he saw blood cells, red cells, and he imagined that it is army fighting each other.
 And it was his interpretation of situation.
 And he sent this report in Academy of Science.
 They very carefully looked because they believed that he is right.
 He saw something.
 Yes.
 But he gave wrong interpretation.
 And I believe the same can happen to his brain.
 With brain, yeah.
 The most important part, you know, I believe in human language.
 In some proverb, is so much wisdom.
 For example, people say that it is better than thousand days of diligent studies one day
 with great teacher.
 But if I will ask you what teacher does, nobody knows.
 And that is intelligence.
 But we know from history and now from math and machine learning that teacher can do a lot.
 So what, from a mathematical point of view, is the great teacher?
 I don't know.
 That's an open question.
 No, no.
 But we can say what teacher can do.
 He can introduce some invariants, some predicate for creating invariants.
 How he doing it, I don't know.
 Because teacher knows reality and can describe from this reality predicate invariants.
 But we know that when you're using invariant, you can decrease number of observations 100 times.
 So but maybe try to pull that apart a little bit.
 I think you mentioned like a piano teacher saying to the student, play like a butterfly.
 I played piano, I played guitar for a long time.
 Yeah, that's, maybe it's romantic, poetic, but it feels like there's a lot of truth in that statement.
 Like there is a lot of instruction in that statement.
 And so can you pull that apart?
 What is that?
 The language itself may not contain this information.
 Language not blah, blah, blah.
 Because it's not blah, blah, blah.
 It's effective.
 It's what?
 Effective.
 Effective playing.
 Yes, it does.
 But it's not the language.
 It feels like what is the information being exchanged there?
 What is the nature of information?
 What is the representation of that information?
 I believe that it is sort of predicate, but I don't know.
 That is exactly what intelligence and machine learning should be.
 Yes.
 But that is just mathematical technique.
 I think that what was discovered recently is that there is two mechanisms of learning.
 One called strong convergence mechanism and weak convergence mechanism.
 Before people use only one convergence.
 In weak convergence mechanism, you can use predicate.
 That's what play like butterfly.
 And it will immediately affect your playing.
 You know, there is English proverb, great.
 If it looks like a duck, swims like a duck, and quack like a duck, then it is probably
 duck.
 Yes.
 But this is exact about predicate.
 Looks like a duck, what it means.
 So you saw many ducks that you're training data.
 So you have description of how looks integral, looks ducks.
 Yeah, the visual characteristics of a duck.
 Yeah.
 But you want and you have model for recognition doubts.
 So you would like so that theoretical description from model coincide with empirical description,
 which you saw on the text there.
 So about looks like a duck, it is general.
 But what about swims like a duck?
 You should know the duck swims.
 You can say it play chess like a duck.
 Okay, duck doesn't play chess.
 And it is completely legal predicate, but it is useless.
 So half teacher can recognize not useless predicate.
 So up to now, we don't use this predicate in existing machine learning.
 And you think that's...
 So why we need zillions of data.
 But in this English proverb, they use only three predicate.
 Looks like a duck, swims like a duck, and quack like a duck.
 So you can't deny the fact that swims like a duck and quacks like a duck has humor in
 it, has ambiguity.
 Just talk about swim like a duck.
 It doesn't say jumps like a duck.
 Why?
 Because it's not relevant.
 But that means that you know ducks, you know different birds, you know animals.
 And you derive from this that it is really want to say swim like a duck.
 So underneath, in order for us to understand swims like a duck, it feels like we need to
 know millions of other little pieces of information.
 We pick up along the way.
 You don't think so.
 There doesn't need to be this knowledge base.
 In those statements carries some rich information that helps us understand the essence of duck.
 How far are we from integrating predicates?
 You know that when you consider complete machine learning.
 So what it does, you have a lot of functions and then you're talking it looks like a duck.
 You see your training data.
 From training data, you recognize like expected duck should look.
 Then you remove all functions which does not look like you think it should look from training
 data.
 So you decrease the amount of function from which you pick up one.
 Then you give a second predicate and then again decrease the set of function.
 And after that, you pick up the best function you can find.
 It is standard machine learning.
 So why you need not too many examples.
 Your predicates aren't very good.
 Because every predicate is invented to decrease admissible set of function.
 So you talk about admissible set of functions and you talk about good functions.
 So what makes a good function?
 So admissible set of function is set of function which has small capacity or small diversity,
 small VC dimension.
 Which contain good function inside.
 By the way, for people who don't know, VC, you're the V in the VC.
 So how would you describe to a lay person what VC theory is?
 How would you describe VC?
 You have a machine.
 So machine capable to pick up one function from the admissible set of function.
 But set of admissibles function can be big.
 So it contain all continuous functions and it's useless.
 You don't have so many examples to pick up function.
 But it can be small.
 We call it capacity, but maybe better call it diversity.
 So not very different function in the set.
 It's infinite set of function, but not very diverse.
 So it is small VC dimension.
 When VC dimension is small, you need small amount of training date.
 So the goal is to create admissible set of functions which have small VC dimension and
 contain good function.
 Then you will be able to pick up the function using small amount of observations.
 So that is the task of learning.
 Yeah.
 So the goal is creating a set of admissible functions that has a small VC dimension.
 Then you've figured out a clever way of picking up.
 That is goal of learning which I formulated yesterday.
 Statistical learning theory does not involve in creating admissible set of function.
 In classical learning theory, everywhere, 100% in textbook, the set of function that
 admissible set of function is given.
 But this is science about nothing because the most difficult problem to create admissible
 set of functions given say a lot of functions, continuum set of function, create admissible
 set of functions that means that it has finite VC dimension, small VC dimension and contain
 good function.
 So this was out of consideration.
 So what's the process of doing that?
 I mean it's fascinating.
 What is the process of creating this admissible set of functions?
 That is invariant.
 That's invariance.
 Can you describe invariance?
 Yeah.
 You're looking of properties of training data and properties means that you have some function
 and you just count what is average value of function on training data.
 You have model and what is expectation of this function on the model and they should
 coincide.
 So the problem is about how to pick up functions.
 It can be any function.
 In fact, it is true for all functions.
 But because when I talking set say duck does not jumping, so you don't ask question jump
 like a duck because it is trivial, it does not jumping and doesn't help you to recognize
 jump.
 But you know something, which question to ask and you asking it swims like a duck.
 But looks like a duck at its general situation.
 Looks like say guy who have this illness, this disease, it is legal.
 So there is a general type of predicate looks like and special type of predicate which related
 to this specific problem.
 And that is intelligence part of all this business.
 And that where teachers involved.
 Incorporating the specialized predicates.
 Okay.
 What do you think about deep learning as neural networks, these arbitrary architectures as
 helping accomplish some of the tasks you are thinking about?
 Their effectiveness or lack thereof?
 What are the weaknesses and what are the possible strengths?
 You know I think that this is fantasy.
 Something which like deep learning, like features.
 Let me give you this example.
 One of the greatest book is Churchill book about history of second world war.
 And he starting this book describing that in old time when war is over, so the great
 kings they gathered together, almost all of them were relatives.
 And they discussed what should be done, how to create peace.
 And they came to agreement.
 And when happens first world war, the general public came in power.
 And they were so greedy that robbed Germany.
 And it was clear for everybody that it is not peace.
 That peace will last only 20 years because they was not professionals.
 And the same I see in machine learning.
 There are mathematicians who are looking for the problem from very deep point of view,
 mathematical point of view.
 And there are computer scientists who mostly does not know mathematics.
 They just have interpretation of that.
 And they invented a lot of blah, blah, blah interpretations like deep learning.
 Why you did deep learning?
 Mathematic does not know deep learning.
 Mathematic does not know neurons.
 It is just function.
 If you like to say piecewise linear function, say that.
 And do in class of piecewise linear function.
 But they invent something.
 And then they try to prove advantage of that through interpretations, which mostly wrong.
 And when it is not enough, they appeal to brain which they know nothing about that.
 Nobody knows what is going on in the brain.
 So I think that more reliable, look on math.
 This is mathematical problem.
 Do your best to solve this problem.
 Try to understand that there is not only one way of convergence, which is strong way of
 convergence.
 There is a weak way of convergence, which requires predicate.
 And if you will go through all this stuff, you will see that you don't need deep learning.
 Even more, I would say one of the theorem which called the representative theory.
 It says that optimal solution of mathematical problem, which is described learning is on
 shadow network, not on deep learning.
 And a shallow network.
 Yeah.
 Yeah, shallow network.
 The problem is there.
 Absolutely.
 So in the end, what you're saying is exactly right.
 The question is you have no value for throwing something on the table, playing with it, not
 math.
 It's like in your network where you said throwing something in the bucket or the biological
 example and looking at kings and queens with the cells or the microscope.
 You don't see value in imagining the cells or kings and queens and using that as inspiration
 and imagination for where the math will eventually lead you.
 You think that interpretation basically deceives you in a way that's not productive.
 I think that if you're trying to analyze this...
 The nature of learning.
...of learning and especially discussion about deep learning, it is discussion about interpretation,
 not about things, about what you can say about things.
 That's right.
 But aren't you surprised by the beauty of it?
 Not mathematical beauty, but the fact that it works at all.
 Or you're criticizing that very beauty, our human desire to interpret, to find our silly
 interpretations in these constructs.
 Let me ask you this.
 Are you surprised and does it inspire you?
 How do you feel about the success of a system like AlphaGo at beating the game of Go?
 Using neural networks to estimate the quality of a board and the quality of the...
 That is your interpretation, quality of the board.
 Yeah, yes.
 Yeah.
 But it's not our interpretation.
 The fact is a neural network system doesn't matter.
 A learning system that we don't, I think, mathematically understand that well beats
 the best human player.
 That's something that was thought impossible.
 That means that it's not a very difficult problem.
 We've empirically have discovered that this is not a very difficult problem.
 Yeah.
 It's true.
 So maybe it can argue.
 So...
 Even more, I would say that if they use deep learning, it is not the most effective way
 of learning theory.
 And usually when people use deep learning, they're using zillions of training data.
 Yeah, but you don't need this.
 So I describe challenge, can we do some problems which do well, deep learning method, this
 deep net, using 100 times less training data.
 Even more, some problems deep learning cannot solve because it's not necessary.
 They create admissible set of function.
 To create deep architecture means to create admissible set of functions.
 You cannot say that you're creating good admissible set of functions.
 You just, it's your fantasy.
 It is not comes from mass.
 But it is possible to create admissible set of functions because you have your training
 data.
 But actually for mathematicians, when you consider invariant, you need to use law of
 large numbers.
 When you're making training in existing algorithm, you need uniform law of large numbers, which
 is much more difficult.
 It requires VC dimension and all this stuff.
 But nevertheless, if you use both, weak and strong way of convergence, you can decrease
 a lot of training data.
 You could do the three, the swims like a duck and quacks like a duck.
 So let's step back and think about human intelligence in general.
 Clearly that has evolved in a non-mathematical way.
 It wasn't, as far as we know, God or whoever didn't come up with a model in place in our
 brain of admissible functions.
 It kind of evolved.
 I don't know.
 Maybe you have a view on this.
 But so Alan Turing in the 50s in his paper asked and rejected the question, can machines
 think is not a very useful question, but can you briefly entertain this useful, useless
 question?
 Can machines think, so talk about intelligence and your view of it?
 I don't know that.
 I know that Turing described imitation.
 If computer can imitate human being, let's call it intelligent.
 And he understands that it is not thinking computer.
 He completely understand what he doing.
 Like he set up problem of imitation.
 So now we understand that the problem not in imitation.
 I'm not sure that intelligence just inside of us.
 It may be also outside of us.
 I have several observations.
 So when I prove some theorem, it's very difficult theorem.
 In a couple of years in several places, people prove the same theorem.
 Say Sawyer-Lemma after us was done, then another guys prove the same theorem.
 In the history of science, it's happened all the time.
 For example, geometry.
 It's happened simultaneously.
 At first it did Lobachevsky and then Gauss and Boyayi and then other guys.
 And it approximately in 10 times period, take 10 years period of time.
 And I saw a lot of examples like that.
 And many mathematicians thinks that when they develop something, they develop something
 in general which affect everybody.
 So maybe our models of intelligence only inside of us is incorrect.
 It's our interpretation.
 Maybe there exists some connection with world intelligence.
 I don't know.
 You're almost like plugging in into…
 Yeah, exactly.
 And contributing to this…
 Into a big network.
 Into a big…
 Maybe in your own network.
 No, no, no.
 On the flip side of that, maybe you can comment on big O complexity and how you see classifying
 algorithms by worst-case running time in relation to their input.
 So that way of thinking about functions.
 Do you think P equals NP?
 Do you think that's an interesting question?
 Yeah, it is an interesting question.
 But let me talk about complexity in about worst-case scenario.
 There is a mathematical setting.
 When I came to United States in 1991, people did not know this theory.
 They did not know statistical learning theory.
 So in Russia, it was published to monographs, our monographs, but in America, they did not
 know.
 Then they learned.
 And somebody told me that it is worst-case theory and they will create real-case theory,
 but till now, it did not.
 Because it is mathematical too.
 You can do only what you can do using mathematics, which has clear understanding and clear description.
 And for this reason, we introduce complexity.
 And we need this because using actually this diversity like this one more, this dimension,
 you can prove some theorems.
 But we also create theory for case when you know probability measure.
 And that is the best case which can happen in this entropy theory.
 So from a mathematical point of view, you know the best possible case and the worst
 possible case.
 You can derive different model in medium.
 But it's not so interesting.
 Do you think the edges are interesting?
 The edges are interesting because it is not so easy to get good bound, exact bound.
 It's not many cases where you have the bound is not exact, but interesting principles which
 discover the mass.
 Do you think it's interesting because it's challenging and reveals interesting principles
 that allow you to get those bounds?
 Or do you think it's interesting because it's actually very useful for understanding the
 essence of a function of an algorithm?
 So it's like me judging your life as a human being by the worst thing you did and the best
 thing you did versus all the stuff in the middle.
 It seems not productive.
 I don't think so because you cannot describe situation in the middle or it will be not
 general.
 So you can describe edges cases and it is clear it has some model, but you cannot describe
 model for every new case.
 So you will be never accurate when you're using model.
 But from a statistical point of view, the way you've studied functions and the nature
 of learning and the world, don't you think that the real world has a very long tail?
 That the edge cases are very far away from the mean, the stuff in the middle or no?
 I don't know that.
 I think that from my point of view, if you will use formal statistic, you need uniform
 law of large numbers.
 If you will use this invariance business, you will need just law of large numbers.
 There's a huge difference between uniform law of large numbers and large numbers.
 Is it useful to describe that a little more or should we just take it to...
 For example, when I'm talking about dark, I gave three predicates and it was enough.
 But if you will try to do formal distinguish, you will need a lot of observation.
 So that means that information about looks like a dark contain a lot of bit of information,
 formal bits of information.
 So we don't know that how much bit of information contain things from intelligence.
 And that is the subject of analysis.
 Until now, all business, I don't like how people consider artificial intelligence.
 They consider as some codes which imitate activity of human being.
 It is not science.
 It is applications.
 You would like to imitate go ahead.
 It is very useful and a good problem.
 But you need to learn something more.
 How people can to develop, say, predicates, swims like a duck or play like butterfly or
 something like that.
 Not the teacher says you how it came in his mind.
 How he chooses this image.
 So that process...
 That is the problem of intelligence.
 That is the problem of intelligence.
 And you see that connected to the problem of learning?
 Absolutely.
 Because you immediately give this predicate like specific predicates, swims like a duck
 or quack like a duck.
 It was chosen somehow.
 So what is the line of work, would you say?
 If you were to formulate as a set of open problems, that will take us there, will play
 like a butterfly, will get a system to be able to...
 Let's separate two stories.
 One mathematical story that if you have predicate, you can do something.
 And another story how to get predicate.
 It is intelligence problem and people even did not start to understand intelligence.
 Because to understand intelligence, first of all, try to understand what doing teachers.
 How teacher teach.
 Why one teacher better than another one?
 Yeah.
 So you think we really even haven't started on the journey of generating the predicates?
 No.
 We don't understand.
 We even don't understand that this problem exists.
 Because did you hear...
 You do.
 Yeah.
 No, I just know name.
 I want to understand why one teacher better than another and how affect teacher, student.
 It is not because he repeating the problem which is in textbook.
 He makes some remarks.
 He makes some philosophy of reasoning.
 Yeah, that's a beautiful...
 It is a formulation of a question that is the open problem.
 Why is one teacher better than another?
 Right.
 What he does better.
 Yeah.
 What...
 Why in every level?
 How do they get better?
 What does it mean to be better?
 The whole...
 Yeah.
 From whatever model I have, one teacher can give a very good predicate.
 One teacher can say, swims like a dog and another can say, jump like a dog.
 And jump like a dog carries zero information.
 So what is the most exciting problem in statistical learning you've ever worked on or are working
 on now?
 I just finished this invariant story and I'm happy that I believe that it is ultimate learning
 story.
 At least I can show that there are no another mechanism, only two mechanisms.
 But they separate statistical part from intelligent part.
 And I know nothing about intelligent part.
 And if we will know the intelligent part, so it will help us a lot in teaching, in learning.
 In learning.
 Yeah.
 You will know it when we see it?
 So for example, in my talk, the last slide was a challenge.
 So you have, say, NIST, digital recognition problem.
 And deep learning claims that they did it very well, say, 99.5% of correct answers.
 But say you use 60,000 observations.
 Can you do the same using 100 times less?
 But incorporating invariants, what it means, you know, digit one, two, three.
 Just looking at that, explain me which invariant I should keep to use 100 examples.
 Or say 100 times less examples to do the same job.
 Yeah, that last slide, unfortunately, your talk ended quickly, but that last slide was
 a powerful open challenge in a formulation of the evidence here.
 That is exact problem of intelligence.
 Because everybody, when machine learning started, it was developed by mathematicians, they immediately
 recognize that we use much more training data than human needed.
 But now again, we came to the same story.
 Have to decrease.
 And that is the problem of learning.
 It is not like in deep learning, they use zillions of training data because maybe zillions
 are not enough if you have a good invariant.
 Maybe you will never collect some number of observations.
 But now it is a question to intelligence.
 Have to do that because statistical part is ready.
 As soon as you supply us with predicate, we can do good job with small amount of observations.
 And the very first challenges will know digit recognition.
 And you know digits.
 And please tell me invariants.
 I think about that, I can say.
 For digit three, I would introduce concept of horizontal symmetry.
 So the digit three has horizontal symmetry, say more than say digit two or something like
 that.
 But as soon as I get the idea of horizontal symmetry, I can mathematically invent a lot
 of measure of horizontal symmetry or vertical symmetry or diagonal symmetry, whatever, if
 I have idea of symmetry.
 But what else?
 I think on digit I see that it is meta predicate, which is not shape.
 It is something like symmetry, like how dark is whole picture, something like that, which
 can self-rise a predicate.
 You think such a predicate could rise out of something that's not general, meaning it
 feels like for me to be able to understand the difference between a two and a three,
 I would need to have had a childhood of 10 to 15 years playing with kids, going to school,
 being yelled by parents, all of that, walking, jumping, looking at ducks.
 And now then I would be able to generate the right predicate for telling the difference
 between two and a three.
 Or do you think there's a more efficient way?
 I don't know.
 I know for sure that you must know something more than digits.
 Yes.
 And that's a powerful statement.
 Yeah.
 But maybe there are several languages of description, these elements of digits.
 So I'm talking about symmetry, about some properties of geometry.
 I'm talking about something abstract.
 I don't know that.
 But this is a problem of intelligence.
 So in one of our article, it is trivial to show that every example can carry not more
 than one bit of information in real.
 Because when you show example and you say this is one, you can remove, say, a function
 which does not tell you one.
 Say it's the best strategy.
 If you can do it perfectly, it's remove half of the functions.
 But when you use one predicate, which looks like a duck, you can remove much more functions
 than half.
 And that means that it contains a lot of bit of information from formal point of view.
 But when you have a general picture of what you want to recognize, a general picture of
 the world, can you invent this predicate?
 And that predicate carries a lot of information.
 Beautifully put.
 Maybe just me, but in all the math you show, in your work, which is some of the most profound
 mathematical work in the field of learning AI and just math in general, I hear a lot
 of poetry and philosophy.
 You really kind of talk about philosophy of science.
 There's a poetry and music to a lot of the work you're doing and the way you're thinking
 about it.
 So do you, where does that come from?
 Do you escape to poetry?
 Do you escape to music or not?
 I think that there exists ground truth.
 There exists ground truth?
 Yeah.
 And that can be seen everywhere.
 The smart guy, philosopher, sometimes I surprise how they deep see.
 Sometimes I see that some of them are completely out of subject.
 But the ground truth, I see in music.
 Music is the ground truth?
 Yeah.
 In poetry, in many poetry, they believe that they take dictation.
 So what piece of music as a piece of empirical evidence gave you a sense that they are touching
 something in the ground truth?
 It is structure.
 The structure of it.
 The math of music.
 Yeah, because when you're listening to Bach, you see the structure.
 Very clear, very classic, very simple.
 And the same in Bach when you have axioms in geometry, you have the same feeling.
 And in poetry, sometimes you see the same.
 And if you look back at your childhood, you grew up in Russia.
 You maybe were born as a researcher in Russia.
 You've developed as a researcher in Russia.
 You've came to the United States and a few places.
 If you look back, what were some of your happiest moments as a researcher?
 Some of the most profound moments, not in terms of their impact on society, but in terms
 of their impact on how damn good you feel that day and you remember that moment.
 You know, every time when you found something, it is great in the life.
 Every simple things.
 But my general feelings at that time, most of my time was wrong.
 You should go again and again and again and try to be honest in front of yourself.
 Not to make interpretation, but try to understand that it related to ground truth.
 It is not my blah, blah, blah interpretation and something like that.
 But you're allowed to get excited at the possibility of discovery.
 Oh, yeah.
 You have to double check it.
 No, but how it related to the other ground truth?
 Is it just temporary or it is forever?
 You know, you always have a feeling when you found something, how big is that?
 So 20 years ago when we discovered statistical learning theory, nobody believed except for
 one guy, Dudley from MIT.
 And then in 20 years it became fashion.
 And the same with support vector machines, kernel machines.
 So with support vector machines and learning theory, when you were working on it, you had
 a sense that you had a sense of the profundity of it, how this seems to be right.
 This seems to be powerful.
 Right.
 Absolutely.
 Immediately.
 I recognize that it will last forever.
 And now when I found this invariant story, I have a feeling that it is completely wrong
 because I have proved that there are no different mechanisms.
 We can have some, say, cosmetic improvement you can do, but in terms of invariants, we
 need both invariants and statistical learning and they should work together.
 But also I'm happy that we can formulate what is intelligence from that and to separate
 from technical part.
 That is completely different.
 Absolutely.
 Well, Vladimir, thank you so much for talking today.
 Thank you.
 It's an honor.