Vladimir Vapnik: Statistical Learning

The following is a conversation with Vladimir Vapnik. He's the co inventor of 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 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. LexFriedman 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 good play dice. But we should describe 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 true always and everywhere or it is law which allow you to predict position of moving element? What you believe? You believe that it is God's law that God created the world which adhere 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 realist. So you're trying to really understand the God thought.

So the way you see the world is as an instrumentalist, you know, I'm.

Working for some models model 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 in two different ways from the point of view of instrumentalism. And that's what everybody does now because they say the goal of machine learning is to find the rule for classification. That is true, but it is instrument for prediction. But I can say the God of machine learning is to to learn about conditional probability. So How God played use 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 rule. 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, parts of the 1960 paper by Eugene Wigner, Unreasonable effectiveness of mathematics and natural sciences. Such a. Such a beautiful paper. Yeah, 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, the poetry of that for a second. How do you see the role of math in your life? Is it, 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 saying that math is language which use God.

So I believe speak to God or use God.

Or use God.

Use God, yeah.

So I believe that this article about effectiveness unreasonable effectiveness of mass is that if you're looking in 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 trying 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 talk yesterday about least square method and people had a lot of fun as you have to improve least square method. But if you look going step by step by solving some equations, you. You suddenly will get some term which after thinking you understand that it describes position of observation point. In least square method, we throw out a lot of information. We don't look in composition of point of observations, we look only on residuals. But when you understood that that's very simple and idea, but which not too simple to understand. And you can derive this just from equations.

So some simple algebras a few steps will take you to something surprising that when you think about.

And that is proof that human intuition not too rich and very primitive and it does not see very simple situations.

So let me take a step back in general. Yes, right. But what about human as opposed to Intuition, ingenuity, moments of brilliance. So are you. 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.

See where the axioms take you.

Yeah, but if they correctly take axioms, but it axion 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.

That's. 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?

You it is not description what's going on. It is interpretation. It is your interpretation. Your vision can be wrong. You know, when guy invent microscope, Lewing Gook for the first time only he got this instrument and nobody or he kept secret about microscope. But he wrote report in London Academy of Science. In his report, when he looking at the blood, he look everywhere. On the water, on the blood, on the spin. But he described blood like fight between queen and king. So he saw blood cells, red cells. And he imagines that it is army fighting each other. And it was his interpretation of situation. And he sent this report. In academia of science, they very carefully look because they believe that he's. He's right, he saw something. Yes, but he gave wrong interpretation. And I believe the same can happen his brain with brain. Yeah, the most important part, you know, I believe in human language, in some proverb, in so much wisdom. For example, people say that it is better than thousand days of diligent studies. One day this great teacher. But if You I will ask you what teacher does. Nobody knows. And that is intelligence. And. But we know from history and now from. 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 awesome question.

But we can say what teacher can do. He can introduce some invariance, some predicate for creating invariants. How he doing it? I don't know because teacher knows reality and can describe from this reality predicate invariance. But he knows that when you're using invariant, you can decrease number of observations hundred 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. Right? I played piano, I played guitar for a long time. And it. Yeah, that's. There's. Maybe it's romantic, poetic, but it feels like there's a lot of truth in that statement. Like there's. There is a lot of instruction in that statement. And so can you pull that apart? What is what, what is that the language itself may not contain this information.

Not blah blah blah.

Because it is not blah blah blah.

Yeah, effective. It's what affect you affect your playing.

Yes, it does. But what it's not the laying. 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 in machine learning should be because the rest 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 predicates looks like a duck, what it means. So you saw many ducks that you're training data. So you. You have description of how. How looks integral looks docs.

Yeah, the visual characteristics of a duck.

Yeah, yeah, but you won't. And you have model for recognition now. So you would like so that theoretical description from model coincide with this empirical description which you saw on tear text. So about looks like a dog. It is general. But what about swims like a duck? You should know that duck swims. You can't 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 chess teacher can recognize not useless predicate. So up to now we don't use this predicate in existing machine learning.

And you think so why?

We need zillions of data. But in this English proverbs 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.

Let's talk about swim like a duck. It does not say jumps jump 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 relevant to say swim like a jack.

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.

Yeah.

How far are we from integrating predicates?

You know that when you consider complete theory machine learning. So what it does, you have a lot of functions and then you're talking. It looks like a dog. You see your training data from training data, you recognize light expected DAG should look. Then you remove all functions which does not look like you think it should look from training data. So you decrease amount of function from which you pick up one. Then you give a second predicate and 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?

Because your predicates aren't very good.

That means that predicate very well. 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. Exactly. Which contain good function inside.

So by the way, for people who don't know vc, you're the V in the vc. So how would you describe to a layperson what VC theory is? How would you describe these?

You have a machine so machine capable to pick up one function from the admissible set of functions. But set of admissibles function can be big, say contain all continuous functions and useless. You don't have so many examples to pick up function. But it can be small. Small. We call it capacity, but maybe better called diversity. So not very different function in the set is infinite set of function, but not very diverse. So it is small VC dimension. When VC dimension is small, you need not. You need small amount of training data. 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.

Is creating a set of admissible functions that has a small VC dimension. And then you figure 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 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 dimensions, 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 value average value of functional 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 that say duck does not jumping. So you don't ask question jump like a duck because it is trivial. It does the jumping and doesn't help you to recognize jump. But you know something which question to ask. And you asking. It swims like a jar like a duck, but looks like a duck at this general situation. Looks like say guy who have this illness. Is this disease. It is legal. Yeah. 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're thinking about their effectiveness or lack thereof. What are. What are the weaknesses and what are the possible strengths?

You know, I think that this is fantasy. Everything which like deep learning, like features. Let me give you this example. One of the greatest books is Churchill book about history of second World War. And he started this book describing that in all 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 happened 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 a very deep point of view, mathematical point of view. And there are computer scientists with 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, but 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. In class of piecewise linear function. But they invent something and then they try to. To. To. To prove advantage of that through interpretations, which mostly wrong. And when not enough, they. They appeal to brain, which they know nothing about that nobody knows what's going on in the brain. So I think that more reliable look on mass. This is mathematical problem. Do your best to solve this problem. Try to understand that there is no 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 representative theory. It says that optimal solution of mathematical problem, which is. Which describe learning is on shadow network, not on deep learning and a shallow network.

Yeah, yeah. 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 a neural network. Or you said throwing something in the bucket and. Or by 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 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? So not mathematical beauty, but the fact that it works at all? Or are you criticizing that very beauty our human desire to. To interpret, to. To find our silly, silly interpretations in these constructs. Like 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. The quality of a. Of a board and, and the quality of the position?

Your interpretation quality of the board.

Yeah. Yes. Yeah, but it works. So 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 does something that was thought impossible.

That means that it's not very difficult problem.

That's it. That's so you empiric. We've empirically have discovered that this is not a very difficult problem. Yeah, yeah, it's true. So maybe can't 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 with deep net using hundred times list 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're 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. Actually for mathematicians when you consider invariant you need to use law of large numbers. When you making training in existing algorithm you need uniform law of large numbers which is much more difficult. It requires 3C dimension and all this stuff. But nevertheless, if you use both VIC and stroke way of convergence, you can decrease a lot of training data.

Yeah, you could do the three that swims like a duck and quacks like a duck. But our. 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 and place it 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 describe imitation. If computer can imitate human being, let's call it intelligent. And he understands that it is not thinking computer. Yes, he. He completely understand what he doing. But 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 theory. In couple of years, in several places, people prove the same theorem. Say soy lemma. After us was done, then another guys proved the same theory. In the history of science, it happened all the time. For example, geometry, it happened simultaneously. First it did Lobachevsky and then Gauss and Boyai and another guys. And it approximately in 10 times period, 10 years period of time. And I saw a lot of examples like that. And many mathematicians think that when they develop something, they develop something in general which affect everybody. So maybe our model that intelligence is only inside of us is incorrect.

It's our interpretation.

Yeah, it might be there exists some connection. Yes. World intelligence. I don't know.

You're almost like plugging in into.

Yeah, exactly.

And contributing to this into the big network, into a big. Maybe in your own network. 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 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 1990, first people did not know this is theories. They did not know statistical learning theory. So in Russia it was published two monographs, our monographs. But in America they didn't 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 tool. You can do only what you can do using mathematics and which has a clear understanding and clear description. And for this reason we introduce complexity. And we need this because using actually it is diversity, like this one more DC 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. It is entropy theory. So from mathematical point of view, you know the best possible case and the worst possible case. You can derive different model in medium medium. But it's not so interesting.

You think the edge, the edges are interesting.

The edges is 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 edge 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, because I think that. But from my point of view, if you will use formal statistic, you need uniform law of large numbers. If you will use this in variance business, you will need just law of large numbers. And 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. No.

For example, when I talking about dac, I gave three predicates, that was enough. But if you will try to do formal distinguish, you will need a lot of observation. And so that means that information about looks like a duck contain a lot of bit of information, formal bits of information. So we don't know that how much bit of information contained things from artificially from intelligence intelligence. And that is the subject of analysis till now all business. I don't like how people consider artificial intelligence. They consider us 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 and good problem. But you need to learn something more. How people try to. How people can to develop say predicate swims like a dog or play like butterfly or something like that. Then not the teacher says you how it came in his mind. Have he chooses image.

So that process.

That is problem of intelligence.

That is the problem of intelligence. And you see that connected to the problem of learning.

Absolutely. Are they because you immediately give this predicate like specific predicate swims like a dog 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 to play like a butterfly? We'll 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 you have to get predicate. It is intelligence problem. And people even did not start understanding intelligence. Because to understand intelligence, first of all, try to understand what doing teachers half 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 predators?

You don't understand. We even don't understand that this problem exist because did you you do?

Yeah.

No, I, I, I just don't name. Yeah, I, I want to understand why one teacher better than another and how effect teacher student. It is not because he repeating the problem which is in textbook. Yes. He makes some remarks, he makes some philosophy of reasoning.

You know, that's a beautiful. So 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. Why at every level, how do they get better? What does it mean to be better? The whole.

Yeah, yeah. From. From whatever model I have.

Yeah.

One teacher can give a very good predicate when teacher can say swims like a dog and another can say jump like a duck. And jump like a duck carries zero information.

Yeah. 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 this intelligent part, so it will help us a lot in teaching. In learning.

In learning.

Yeah.

You know, we'll know it when we see it.

So for example, in, in my talk, the last slide was the challenge. So you have say NIST digital cognition Problem. And deep learning claims that they did it very well. Say 99.5% of correct answers, but they use 60,000 observations. Yeah, can you do the same using hundred times less, but incorporating invariance what it means, you know, digit one to seven? Yeah, just looking at that explain me which invariant I should keep to use hundred 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 and a formula simulation of the essence.

That is exact problem of intelligence. Because everybody when machine learning started, it was developed by mathematician. 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. They use zillions of training data. Because maybe zillions are not enough. If you have a good invariance, maybe you will never collect some number of observations. But now it is question to intelligence how 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 challenge is well known digit recognition. And you know, digits. And please tell me invariance. I think it about that. I can say for digit 3 I would introduce concept of horizontal symmetry. So the digit 3 has horizontal symmetry, say more than say digit 2 or something like that. But as soon as I get the 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 Looking 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 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 know for sure that you must know something more than digits.

Yes, that's a powerful statement.

Yeah, but maybe there are several languages of description these elements of digits. So I talking about symmetry, about some symmetry, right? Properties of geometry. I 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 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 in 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 and general picture of the world, can you invent this predicate and that predicate carry 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. 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. 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. Sometime 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. And in poetry, many poets, they. They believe that they take dictation.

So what, what piece of music as a piece of a empirical evidence gave you a sense that they are, they're touching something in the ground truth.

It is structure.

The structure.

You're listening to Bach.

Yeah.

You see the structure.

Yeah.

Very clear, very classic, very simple. And the same in Mars. When you have axioms in geometry, you have the same feeling.

Yeah.

Even poetry, sometimes you see the same.

Yeah. And if you look back 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 in a few places. If you look back, what was 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 just. But My general feeling that I mostly 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 grand truth. It is not my blah, blah, blah interpretation and something like. Exactly.

But you're allowed to get excited at the. At the possibility of discovery. Oh yeah, you'll double. You have to double check it.

But no but how it related to another ground throws. Is it just temporary or it is for forever? You know, you always have a feeling when you found something how big is that? So 20 years ago, when we discovered statistical load in 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, that's kernel machines.

So with support vector machines, with learning theory, with. 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 invariance story, you feel the same. I have a feeling that it is complete learning, because I have proof that there are no different mechanism. You can have some say cosmetic improvement you can do. But in term of invariance, you 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, that's an honor.