Vladimir Vapnik: Predicates, Invariants, and the Essence of Intelligence

The following is a conversation with Vladimir Vapnik Part 2 the second time we spoke on the podcast. 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, worked at the Institute of Control Sciences in Moscow, then in the US, worked at AT&T, NEC Labs, Facebook, AI Research, and now it's a professor at Columbia University. His work has been cited over 200,000 times. The first time we spoke on the podcast was just over a year ago, one of the early episodes. This time we spoke after a lecture he gave titled Complete Statistical Theory of Learning as part of the MIT series of lectures on deep learning and AI that I organized. I'll release the video of the lecture in the next few days. This podcast and lecture are independent from each other, so you don't need one to understand the other. The lecture is quite technical and math heavy, so if you do watch both, I recommend listening to this podcast first since the podcast is probably a bit more accessible. This is the Artificial Intelligence podcast. If you enjoy it, subscribe on YouTube, give it five stars on Apple Podcasts, support it on Patreon, or simply connect with me on Twitter lexfriedman spelled F R I D M A N. As usual, I'll do one or two minutes of ads now and never any ads in the middle that can break the flow of the conversation. I hope that works for you and doesn't hurt the listening experience. This show is presented by Cash app, the 1 finance app in the App Store. When you get it, use code lexpodcast Cash App lets you send money to friends, buy Bitcoin, and invest in the stock market with as little as $1. Brokerage services are provided by Cash App Investing, a subsidiary of Square and member sipc. Since Cash App allows you to send and receive money digitally, peer to peer and security in all digital transactions is very important. Let me mention the PCI Data Security Standard PCI DSS Level 1 that Cash App is compliant with. I'm a big fan of standards for safety and security and PCI DSS is a good example of that where a bunch of competitors got together and agreed that there needs to be a global standard around the security of transactions. Now we just need to do the same for autonomous vehicles and AI systems in general. So again, if you get Cash App from the App Store or Google Play and use the code lexpodcast, you get $10 and cash app will also donate $10 to first. One of my favorite organizations that is helping to advance robotics and STEM education for young people around the world. And now here's my conversation with Vladimir Vapnik. You and I talked about Alan Turing yesterday a little bit.

Yes.

And that he, as the father of artificial intelligence, may have instilled in our field an ethic of engineering and not science, seeking more to build intelligence rather than to understand it. What do you think is the difference between these two paths of engineering intelligence and the science of intelligence?

It's completely different story. Engineering is imitation of human activity. You have to make a device which behave as human behave have all the functions of human. It does not matter how you do it. But to understand what is intelligence about is quite different problem. So I think, I believe that it's somehow related to predicate we talked yesterday about. Because look at the Vladimir Props idea. He just found 31. He predicates here call it units which can explain human behavior. At least in Russian tales. You look at the Russian tales and derive from that and then people realize that it more wide than in Russian tales. It is in tv, in movie serials and so on and so on.

So you're talking about Vladimir Prop.

Right.

Who in 1928 published a book, Morphology of the Folk tale.

Exactly.

Describing 31 predicates that have this kind of sequential structure that a lot of the stories narratives follow in Russian folklore and in other context. We'll talk about it. I'd like to talk about predicates in a focused way. But let me, if you allow me to stay zoomed out on our friend Alan Turing. And you know, he inspired a generation with the. The Imitation Game.

Yes.

Do you think if we can linger in a little bit longer, do you think we can learn? Do you think learning to imitate intelligence can get us closer to the science to understanding intelligence? So why do you think imitation is so far from understanding?

I think that it is different between. You have different goals. So your goal is to create something. Something useful. And that is great. And you can see how much things was done. And I believe that it will be done even more. It's self driving cars and also this business. It is great. And it was inspired by Turing vision. But understanding is very difficult. It more or less philosophical category. What means understand the world. I believe in him, which starts from Plato, that there exists vault of ideas. I believe that intelligence it is vault of ideas, but it is world of pure ideas. And when you combines it with reality things it creates, as in my case, invariance, which is very specific. And that's, I believe, the combination of ideas in Way to construct invariant is intelligence. But first of all, predicate. If you know predicate and hopefully then not. Not too much predicate exists. For example, 31 predicate for human behavior. It is not a lot.

Vladimir Prop used 31, you can even call them predicates 31 predicates to describe stories, narratives. Do you think human behavior, how much of human behavior, how much of our world, our universe, all the things that matter in our existence can be summarized in predicates of the kind that Prop was working with?

I think that we have a lot of form of behavior, but I think the predicate is much less. Because even in these examples which I gave you yesterday, you saw that predicate can be. One predicate can construct many different invariants depending on your data. They're applying to different data and they give different invariants. But pure ideas, maybe not so much, not so many. I don't know about that. But my guess, I hope that's why challenge about digital recognition. How much you need?

I think we'll talk about computer vision and 2D images a little bit in your challenge.

That's exactly about intelligence.

That's exactly. That's exactly about. No, that hopes to be exactly about the spirit of intelligence in the simplest possible way.

Yeah, absolutely. You should start the simplest way, otherwise you will not be able to do it.

Well, there's an open question whether starting at the mnist digit recognition is a step towards intelligence or it's an entirely different thing.

I think that to beat records using say 100, 200 times less examples, you need intelligence.

You need intelligence. So let's. Because you use this term and it would be nice. I'd like to ask simple, maybe even dumb questions. Let's start with a predicate in terms of terms and how you think about it. What is a predicate?

I don't know. I have a feeling formally they exist. But I believe that predicate for 2D images, one of them is symmetry.

Hold on a second. Sorry. Sorry to interrupt and pull you back. At the simplest level, we're not even. We're not being profound. Currently, a predicate is a statement of something that is true.

Yes.

Do you think of predicates as somehow probabilistic in nature? Or is this binary? This is truly constraints of logical statements about the world.

In my definition, the simplest predicate is function function. And you can use this function to make inner product that is predicate.

What's the input and what's the output of the function?

Input is X something which is input in reality. Say if you Consider digit recognition. It pixel space input. But it is function which in pixel space. But it can be any function from pixel space. And you choose. And I believe that there are several functions which is important for understanding of images. One of them is symmetry. It's not so simple construction as I described with derivative with all this stuff. But another I believe, I don't know how many is how. Well, structurized is picture.

Structurized?

Yeah.

What do you mean by structurized?

It is formal definition. Say something happened heavy on the left corner, not so heavy in the middle, and so on. You describe in general concept of what.

You assume concepts, some kind of universal concepts.

Yeah, but I don't know how to formalize this.

So this is the thing. There's a million ways we can talk about this. I'll keep bringing it up. But we humans have such concepts when we look at digits. But it's hard to put them just like you're saying now. It's hard to put them into words.

You know, that is example. When critics in music trying to describe music, they use predicate and not too many predicate, but in different combination. But they have some special words for describing music. And the same should be for images. But maybe there are critics who understand essence of what this image is about.

Do you think there exists critics who can summarize the essence of images, human beings?

I hope so.

Yes, but explicitly state them on paper. The fundamental question I'm asking is, do you think there exists a small set of predicates that will summarize images? It feels to our mind like it does that the concept of what makes a two and a three and a four.

No, no, no. It's not on this level. It should not describe 2, 3, 4. It's. It describes some construction which allow you.

To create invariance and invariance. Sorry to stick on this, but terminology.

Invariance, it is property of your image. I can say, looking on my image, it is more or less symmetric. And I can give you a value of symmetry, say, level of symmetry using this function which I gave yesterday. And you can describe that your image have these characteristics exactly in the way how musical critics describe music. So. But this is invariant applied to specific data, to specific music, to something. I strongly believe in this plot ideas that there exists world of predicate and world of reality. And predicate and reality is somehow connected. And you have to know that.

Let's talk about Plato a little bit. So you draw a line from Plato to Hegel to Wigner to today. Yes, so Plato has forms, the. The theory of forms. There's a world of ideas and a world of things as you talk about. And there's a connection. And presumably the world of ideas is very small and the world of things is arbitrarily big. But they're all what Plato calls them like it's a shadow. The real world is a shadow from the world of form.

Yeah, you have projection. Projection of world of idea.

Yeah, very poetic.

In reality you can realize this projection using. Using this invariance because it is projection for. On specific examples which create specific features of specific objects. So.

And so the essence of intelligence is while only being able to observe the world of things, try to come up with a world of ideas.

Exactly like in this music story. Intelligent musical critics knows this, all this work and have a feeling about what.

I feel like that's a contradiction. Intelligent music critics. But I think. I think music is to be enjoyed in all its forms. The notion of critic like a food critic.

No, I don't won't touch emotion.

That's an interesting question. Does emotion. There's certain elements of the human psychology of the human experience which seem to almost contradict intelligence and reason. Like emotion, like fear, like. Like love. All of those things. Are those not connected in any way to the space of ideas.

Thus. I don't know. I just want to be concentrate on very simple story on digit recognition.

So you don't think you have to love and fear death in order to recognize digits.

I don't know because it's so complicated. It involved a lot of stuff which I never consider. But I know about digit recognition and I know that for digit recognition to get records from small number of observations, you need predicate, but not special predicate for this problem, but universal predicate which understand world of images of visual information. Yeah. But on the first step they understand say world of handwritten digits or characters or something simple.

So like you said, symmetry is an interesting.

No, that's what I think. One of the predicated related to symmetry. But the level of symmetry.

Okay. Degree of symmetry. So you think symmetry at the bottom is a universal notion. And there's. There's degrees of a single kind of symmetry. Or is there many kinds of symmetries?

Many kinds of symmetries. There is a symmetry anti symmetry, say letter S. So it has vertical anti symmetry. And it could be diagonal symmetry. Vertical symmetry.

So when you cut vertically the letter S. Yeah.

Then the upper part and low part in different directions.

Yeah. Inverted along the Y axis. Yeah, but that's just like one Example of symmetry. Right. Isn't there like.

Right, but there is a degree of symmetry. If you play all this lead derivative stuff to to do tangent distance, whatever I describe, you can have a degree of symmetry. And that is what describing reason of image. It is the same as you will describe this image. Saying about digites has anti symmetry. Digit 3 Symmetric, more or less look for symmetry.

Do you think such concepts like symmetry predicates like symmetry. Is it a hierarchical set of concepts? Or are these independent distinct predicates that we want to discover a subset of?

No, there is idea of symmetry. And you can this idea of symmetry make very general like degree of symmetry. The degree of symmetry can be zero. No symmetry at all. Or degree of symmetry say more or less symmetrical. But you have one of this description and symmetry can be different. As I told horizontal, vertical, diagonal and anti symmetry is also concept of symmetry.

What about shape in general? I mean symmetry is a fascinating notion.

But no, no, I'm talking about digit. I would like to concentrate on all I would like to know predicate for digit recognition.

Yes, but symmetry is not enough for digit recognition, right?

It is not necessarily for digit recognition. It helps to create invariant which will. Which you can use when you will have examples for digit recognition. You have regular problem of digit recognition. You have examples of the first class or second class. Plus you know that there is this. There exists concept of symmetry and you apply. When you looking for decision rule. You will apply concept of symmetry of this level of symmetry which you estimate from me. So let's. Let's talk. Everything is comes from a big convergence.

What is convergence? What is weak convergence? What is strong convergence? I'm sorry, I'm going to do this to you. What are we converging from into.

You're converging. You would like to have a function the function which say indicator function which.

Indicate your digit 5 for example, a classification task.

Let's talk only about classification.

So classification means you will say whether this is a 5 or not or say which of the 10 digits it is.

Right? Right. I would like to have these functions. Then I have some examples. I can consider property of these examples. Say symmetry. And I can measure level of symmetry for every digit. And then I can take average and I from. From my training data. And I will consider only functions of conditional probability which I'm looking for my decision rule which applying to digits will give me the same average as I absorb on training data. So actually this is different level of description of what you want. You want not just you show not one digit. You show this. This predicate. Show general property of all digits which you have in mind. If you have in mind Digit 3, it gives you property of Digit 3. And you select as admissible set of function. Only function which keeps this property. You will not consider other functions. So you're immediately looking for smaller subset of function.

That's what you mean by admissible functions.

Admissible function. Exactly.

Which is still a pretty large for the number three.

It's pretty large. But if you have one predicate, but according to there is a strong and weak convergence. Strong convergence is convergence and function. You're looking at the function on one function and you're looking on another function. And square difference from them should be small. If you take difference in any points, make a square, make an integral, and it should be small. That is convergence in function. Suppose you have some function, any function. So I would say, I say that some function converge to this function if integral from square difference between them is small.

That's the definition of strong convergence. That definition of two functions. The integral of the difference.

Yeah, small. It is convergence in functions. Yeah, but you have different convergence in functionals. You take any function, you take some function phi and take inner product, this function, this f function f0 function which you want to find. And that gives you some value. So you say that set of functions converge in inner product to this function. If this value of inner product converge to value f0, that is for 1 Phi. But v convergence requires that it converge for any function of Hilbert space. If it converge for any function of Hilbert space, then you will say that this is weak convergence. You can think that when you take integral, that is property integral property of function. For example, if you will take sine or cosine, it is coefficient of say Fourier expansion. So if it converge for all coefficients of Fourier expansion. So under some condition it converges to to function you're looking for. But weak convergence means any property convergence. Not point wise, but integral property of function. So weak convergence means integral property of functions. When I talking about predicate, I would like to formulate which integral properties I would like to have for convergence. So, and if I will take one predicated function which I measure property. If I will use one predicate and say I will consider only function which give me the same value as with this predicate I selecting set of functions from functions which is admissible in the sense that function which I looking for in this set of functions. Because I checking in training data, it gives the same.

Yes, it always has to be connected to the training data in terms Of.

Yeah, but, but property you can know independent on training data and this guy prop. Yeah, so that there is formal property 31 property and you fairy tale, the Russian fairy tale. But Russian fairy tale is not so interesting. More interesting that people apply this to movies, to theater, to different things and the same works. They're universal.

Well, so I would argue that there's a little bit of a difference between the kinds of things that were applied to which are essentially stories and digit recognition.

It is the same story.

You're saying digits. There's a story within the digit.

Yeah, but my point is why I hope that it's possible to beat a record using not 60,000, but say 100 times less. Because instead you will give predicates and you will select your decision not from wide set of functions, but from set of function which keeps this predicate. But predicate is not related just to digit recognition.

Right.

So like in Plato's case.

Do you think it's possible to automatically discover the predicates? This. So you basically said that the essence of intelligence is the discovery of good predicates.

Yeah.

Now the natural question is, you know, that's what Einstein was good at doing in physics. Can we make machines do these kinds of discovery of good predicates or is this ultimately a human endeavor?

Thus I don't know. I don't think that machine can do. Because according to theory about weak convergence, any function from Hilbert space can be predicated. So you have infinite number of predicate in upper and before you don't know which predicate is good on which. But whatever prop show and why people call it breakthrough that there is not too many predicate which cover most of situation happened in the world.

So there's a sea of predicates and most of the only a small amount are useful for the kinds of things that happen in the world.

I think that I would say only small part of predicate. Very useful. Useful all of all of them.

Only very few are what we should let's call them good predicates.

Very good predicates.

Right. Very good predicates. So can we linger on it? What's your intuition? Why is it hard for a machine to discover good predicates?

I even in my talk described how to do predicate how to find new predicate. I'm not sure that it is very good.

What did you propose in your talk?

No, in my talk I gave example for diabetes. Diabetes when we achieve some percentage. So then we're looking for area where some sort of predicate which I formulate does not keeps invariant. So if it doesn't keep, I retrain my data. I select only function which keeps this invariant. And the way I did it, I improve my performance. I can looking for this predicate. I know technically how to do that. And you can of course do it using machine. But I'm not sure that we will construct the smartest predicate.

But this is the. Allow me to linger on it, because that's the essence. That's the challenge. That is artificial. That's the human level intelligence that we seek is the discovery of these good predicates. You've talked about deep learning as a way to the predicates they use. And the functions are mediocre. We can find better ones.

Let's talk about deep learning.

Sure, let's do it.

I know only Jan Sli Kun convolutional network. And what else I don't know. And it's a very simple convolution.

There's not much else to know left and right.

Yes, I can do it like that with one predicate. It is.

Convolution is a single predicate.

It's single. It's. It's single predicate.

Yes, it is.

You know exactly. You take lead derivative for translation and predicate. This should. Should be kept.

So that's a single predicate. But humans discovered that one or at.

Least not that is a risk. Not too many predicates. This and that is big story because Jan did it 25 years ago and nothing so clear was added to the deep network. Then I don't understand why we should talk about deep network instead of talking about piecewise linear functions which keeps this predicate.

Well, the. You know, a counter argument is that maybe the amount of predicates necessary to solve general intelligence, say in space of images doing efficient recognition of handwritten digits is very small. And so we shouldn't be so obsessed about finding will find other good predicates like convolution, for example. You know, there has been other advancements like if you look at the work with attention there's attentional mechanisms in especially used in natural language, focusing the network's ability to learn which part of the input to look at. The thing is there's other things besides predicates that are important for the actual engineering mechanism of showing how much you can really do given such these predicates. I mean that's essentially the work of deep learning is constructing architectures that are able to be given the training data to be able to converge towards.

A.

Function that can approximate can generalize well it's an engineering problem.

Yeah, I understand. But let's talk not on emotional level, but on a mathematical level. You have set of piecewise linear functions. It is all possible neural networks. It's just piecewise linear functions. There's many, many pieces large, large number of piecewise linear. Okay, exactly.

But very large.

Very large almost. But it feels like too large. It's still simpler than say convolution, than reproducing kernel Hilbert space, which have a Hilbert set of functions.

What's Hilbert space?

It's space with infinite number of coordinates, function for expansion, something like that. So it's much richer. So and when I talking about closed form solution, I talking about this set of function, not piecewise linear set, which is particular case. It is small part.

So neural networks is a small part of the space. You're talking of functions.

You're talking about say small set of functions. Let me take it. But it is fine. It is fine. I don't want to discuss the small or big. Take advantage. So you have some set of functions. So now when you're trying to create architecture, you would like to create admissible set of functions. All your tricks to use not all functions, but some subset of this set of functions. Say, when you're introducing convolutional net, it is way to make this subset useful for you. But from my point of view, convolutional it is something you want to keep some invariance, say translation invariance. But now, if you understand this and you cannot explain on the level of ideas, what neural network does, you should agree that it is much better to have a set of functions. And they say this set of functions should be admissible. It must keep this invariant, this invariant and that invariant. You know that as soon as you incorporate new invariance set of function, because smaller and smaller and smaller.

But all the invariants are specified by you, the human.

Yeah, but what I hope that there is a standard predicate like Prop show that what that's what I want to find for digit recognition. If we start, it is completely new area. What is intelligence about on the level, starting from Plata's idea. What is vault of ideas? And I believe that it's not too many. But you know, it is amusing that mathematician doing something in neural network in general function. But people from literature, from art, the uses of the time.

That's right.

In variance saying it is great how people describe music. We should learn from that and something on this level. But so why Vladimir Prop, who was just theoretical, who studied theoretical literature. He Found that.

You know what? Let me throw that right back at you because there's a little bit of a. That's less mathematical and more emotional philosophical Vladimir prop. I mean, he wasn't doing math.

No.

And you just said another emotional statement, which is you believe that this Plato world of ideas is small.

I hope.

I hope. Do. Do you do. What's your intuition though? If we can linger on it.

Yeah. You know, it is not just small or big. I know, exactly. Then when I introducing some predicate, I decrease set of functions. But my goal to degree set of.

Function much by as much as possible.

By as much as possible good predicate which. Which does this. Then I should choose next predicate which does which decrease set as much as possible. So set of good predicate it is such that they decrease this amount of admissible functions.

So if each good predicate significantly reduces the set of admissible functions that there naturally should not be that many good predicates.

No, but if you reduce very well the VC dimension of the function of admissible set of function is small and you need not too much training data to do well.

And VC dimension, by the way, is some measure of capacity of this set of functions, right?

Roughly speaking, how many functions function in this set? So you're decreasing, decreasing and it makes easy for you to find function you're looking for. But the most important part to create good admissible set of functions. And it probably there are many ways but the good predicated such that can do that for this duck. You should know a little bit about dog because what are the.

What are the three fundamental laws of ducks?

Looks like a duck, swims like a duck.

You should know something about ducks to.

Be able to not necessarily looks like say horse. It's also good.

It's just not. It generalizes.

Yes, and talk like. Like. And make sound like horse. Yeah, something and run like horse and moves like horse. It is general. It is general predicate that this applied to. To duck. But for Doc, you can say play.

Chess like that you cannot say play chess.

Why not?

So you're saying you can, but that would not be a good.

No, you. You. You will not reduce A lot of fun.

You will not do. Yeah, you would not reduce the setting of functions.

So you can. The. The story is formal story mathematical story is that you can use any function you want as a predicate. But some of them are good, some of them are not, because some of them reduce a lot of functions to admissible set of some of them.

So. But the question is. And I'll probably keep asking this question, but how do we find such. What's your intuition? So handwritten. Handwritten recognition. How do we find the. The answer to your challenge?

Yeah, yeah, I understand. It's like that. I understand what. What.

What defined.

What it means. I knew. Predicate.

Yeah.

Like guy who understand music can say this word which he described when he listened to music. He understand music. He use not too many different. Or you can do like prop. You can make collection. What he talking about music about this, about that. It's not too many different situations he described.

Because we mentioned Vladimir prop a bunch. Let me just mention there's a. So there's a sequence of. Of 31 structural notions that are common in stories.

And I think you call it units.

Units. And I think they resonate. I mean, it starts just to give an example. Absention. A member of the hero's community, a family leaves the security of the home environment. Then it goes to the interdiction. A forbidding edict or command is passed upon the hero. Don't go there. Don't do this. The heroes warn against some action. Then step three. Violate violation of interdiction. Break the rules, break out on your own. Then reconnaissance. The villain makes an effort to attain knowledge needing to fulfill their plot. So on it goes on like this. Ends. Ends in a wedding. Number 31.

Yeah.

Happily ever after.

No, he. He just gave description of all situation. He understands this vault of folk tales. Yeah, not folk stories, stories. And this story is not in just folktales. This story is in detective serials as well.

And probably in our lives we probably live.

Read this at the end. They wrote that this predicate is good for different situation from movie for movie. For theater.

By the way, there's also criticism, right? There's an other way to interpret narratives from Claude Levy Strauss.

I don't. I. I am not in this business.

No, I know it's theoretical literature, but it's looking at paradise.

It's always the. The. The philosophy. Yeah, yeah, but at least there is a units. It's not too many units that can describe, but this guy probably gives another units or another way of.

Exactly. Another set of units, another set of predicates.

It doesn't matter. But they exist. Probably.

My question is whether given those units, whether without our human brains to interpret these units, they would still hold as much power as they have meaning. Are those units enough when we give them to the alien species?

Let me ask you. Do you understand digit recogn. Digit images?

No, I don't understand.

No, no, no. When you can Recognize these digit images. It means that you understand. Yes. You understand characters, you understand.

No, no, no, no. I, I, it's the, it's the imitation versus understanding question. Because I don't understand the mechanism by which I.

No, no, I'm not talking about, I'm talking about predicates. You understand that it involves symmetry, maybe structure, maybe something else. I cannot formulate. I just was able to find symmetries.

So degree of symmetries, that's really good. So this is a good line. I feel like I understand the basic elements of what makes a good hand recognition system. My own like symmetry connects with me. It seems like that's a very powerful predicate. My question is, is there a lot more going on that we're not able to introspect? Maybe I need to be able to understand a huge amount in the world of ideas, thousands of predicates, millions of predicates in order to do hand recognition.

I don't think so.

So you're, you're both your hope and your intuition.

Let me explain.

Enough.

You're using digits, you're using examples as well. Theory says that if you will use all possible functions from Hilbert space, all possible predicate, you don't need training data, you just will have admissible set of functions which contain one function.

Yes. So the trade off is when you're not using all predicates, you're only using a few good predicates. You need to have some training data.

Yes, exactly.

The more, the more good predicates you have, the less training data.

Exactly. That is intelligent learning still.

Okay, I'm going to keep asking the same dumb question, handwritten recognition to solve the challenge. You kind of propose a challenge that says we should be able to get state of the art mnist error rates by using very few 60, maybe fewer examples per digit. What kind of predicates do you think it will solve?

That is the challenge. So people who will solve this problem, they will answer. They will answer.

Do you think they'll be able to answer it in a human explainable way?

They just need to write function. That's it.

But so can that function be written? I guess by an automated reasoning system. Whether we're talking about a neural network learning a particular function or another mechanism.

No, Meryl, I'm not against neural network. I am against admissible set of functions which create neural network. You did it by hand. You don't do it by invariance, by predicate, by reason.

But neural networks can then reverse. Do the reverse step of helping you find a function Just the task of a neural network is to find a disentangled representation. For example they call is to find that one predicate function that's really captures some kind of essence. One not the entire essence, but one very useful essence of this particular visual space. Do you think that's possible? Like listen, I'm grasping, hoping there's an automated way to find good predicates. Right. So the question is what are the mechanisms of finding good predicates? Ideas that you think we should pursue.

A young grad student listening right now I gave example. So find situation where predicate which you suggesting don't create invariant. It's like in physics, find situation where existing theory cannot explain it.

Find situation where the existing theory cannot explain this. So you're finding contradictions.

Find contradiction and then remove this contradiction. But in my case, what means contradiction? You find function which if you will use this function, you're not keeping converts.

So really the process of discovering contradictions.

Yeah, it is like in physics. Find situation where you have contradiction for one of the property for one of the predicate. Then include this predicate making invariance and solve again this problem. Now you don't have contradiction, but it is not the best way probably I don't know to looking for predicate.

It's just one way.

Okay, that. No, no, it is brute force way.

The brute force way. What about the ideas of some what big umbrella term of symbolic AI there's what in the 80s with expert systems, sort of logic reasoning based systems. Is there hope there to find some through sort of deductive reasoning to find good predicates?

I don't think so. I think that just logic is not enough.

It's kind of a compelling notion though, you know that when smart people sit in a room and reason through things, it seems compelling. And making our machines do the same is also compelling.

So everything is very simple. When you have infinite number of predicate, you can choose the function you want. You have invariance and you can choose the function you want. But you have to have not too many invariance to solve the problem. So and have from infinite number of function to select finite number and hopefully small finite number of functions which is good enough to extract small set of admissible functions. So they will be admissible, it's for sure because every function just decrease set of function and leaving it admissible. But it will be small.

But why do you think logic based systems don't can't help intuition.

Not because you you should know reality, you should know life. This guy like prop he knows something and he tried to. To put in invariant his understanding.

So. But that's the human. Yeah, but see you're. You're putting too much value into Vladimir Prop knowing something.

No, it is what means, you know, life, what it means.

Comments? You know common sense.

No, no. You know something common sense. It is some rules.

You think so? Common sense is simply rules. Common sense is every. It's mortality, it's. No, it's. It's fear of death, it's love, it's spirituality, it's happiness and sadness. All of it is tied up into understanding gravity, which is what we think of as common sense.

I don't really to discuss. So why I want to discuss understand digital. Understand digital recognition.

Anytime I bring up love and death, you bring it back to digital recognition.

No, you know, it is durable because there is a challenge.

Yeah.

Which I see how to solve it. If I will have a student concentrate on this work, I will suggest something to solve.

You mean handwritten recognition? Yeah. It's a beautifully simple, elegant and yet.

I think that I know invariance which will solve this.

You do?

I think so. You think so. But it is not universal. It is maybe I want some universal invariants which are good not only for digit recognition, for image understanding.

So let me ask, how hard do you think is 2D image understanding? So if we. We can kind of intuit handwritten recognition, how big of a step leap journey is it from that? If I gave you good, if I solved your challenge for handwritten recognition, how long would my journey then be from that to understanding more general natural images?

Immediately. You will understand this as soon as you will make a record, because it is not for free. As soon as you will create several invariants which will help you to get the same performance that the best neural net did using hundred and maybe more than 100 times less examples. You have to have something smart to do that.

And you're saying that it is invariant?

It is predicated because you should put some idea how to do that.

But. Okay, let me just pause. Maybe it's a trivial point, maybe not. But handwritten recognition feels like a 2D two dimensional problem. And it seems like how much complicated is the fact that most images are projection of a three dimensional world onto a 2D plane. It feels like for a three dimensional world we need to start understanding common sense in order to understand an image. It's no longer visual shape and symmetry, it's having to start to understand concepts of understand life.

Yeah, you're talking that there are different environment, different Predicates. Yeah.

And potentially much larger number.

You know, maybe. But let's start from simple.

Yeah, but you said that, you know.

I, I, I cannot think.

Yes.

About things which I don't understand. This I understand, but I'm sure that I don't understand everything there.

Yeah, that's the difference.

I do. As simple as possible, but not simpler. And that is exact case with handwritten. With handwritten.

Yeah, but never. That's the difference between you and I, I, I, I welcome and enjoy thinking about things I completely don't understand. Because to me it's a natural extension without having solved handwritten recognition to wonder how difficult is the next step of understanding 2D 3D images? Because ultimately, while the science of intelligence is fascinating, it's also fascinating to see how that maps to the engineering of intelligence. And recognizing handwritten digits is not, doesn't help you. It might, it may not help you with the problem of general intelligence. We don't know. It'll help you a little bit.

We don't know.

It's unclear. Yeah, it might very much.

But I would like to make a remark. Yes. I start not from very primitive problem. Make challenge problem. I start with very general problem with Plato. So you understand. And it comes from Plato to digit recognition.

So you basically took Plato and the world of forms and ideas and mapped and projected into the clearest simplest formulation of that big world world.

I would say that I did not understand Plato until recently and until I consider weak convergence and then predicate and then oh, this is what Plato told.

So you linger on that like why how do you think about this world of ideas and world of things in Plato?

No, it is metaphor.

It is, it's a metaphor for sure. It's a compelling, it's a poetic and a beautiful metaphor. But what can you.

But it is a way how you should try to understand how attack ideas in the world. So from my point of view it is very clear. But it is lying all the time. People looking for that say Plato, then Hegel. Whatever reasonable it exists. Whatever exists it is reasonable. I don't know what he have in mind. Reasonable.

Right. This philosophers again.

No, no, no, no, no, no, no. It is, it is next stop of Wigner that mathematics understand something of reality. It is the same plot line. And then it comes suddenly to Vladimir Prop. Look, 31 ideas, 31 units and describes everything.

There's abstractions, ideas that represent our world and we should always try to reach into that.

Yeah, but you should make a projection on reality. But understanding is, it is abstract ideas you have in your mind several abstract ideas which you can apply to reality.

And reality in this case. So if we look at machine learning.

Is data, example data.

Data. Okay, let me put this on you. Because I'm an emotional creature. I'm not a mathematical creature like you. I find compelling the idea, forget the space, the sea of functions. There's also a sea of data in the world. And I find compelling that there might be, like you said, teacher, small examples of data that are most useful for discovering good. Whether it's predicates or good functions, that the selection of data may be a powerful journey, a useful mechanism. You know, coming up with a mechanism for selecting good data might be useful too. Do you find this idea of finding the right data set interesting at all? Or do you kind of take the data set as a given?

I think that it is. You know, my scheme is very simple. You have huge set of functions, if you will apply and you have not too many data. Right. If you will pick up function which describes this data, you will do not very well.

You will randomly pick.

Yeah. You will ever fit. Yeah. It will be overfitting. So you should decrease set of function from which you picking up one. So. So you should go some half to admissible set of function. And this. What about weak conversions? But from another point of view, to make admissible set of function you need just a dig. You just function which you will take in inner product which you will measure property of your function. And that is how it works.

No, I get it, I get it, I understand it. But do you? The reality is.

But let's think about examples. You have huge set of functions and you have several examples. If you just trying to keep take function which satisfies these examples, you still will overfit. You need decreases, you need admissible set of functions.

Absolutely. But what say you have more data than functions? So consider the. I mean, maybe not more data than functions, because that's impossible. Impossible. But I was trying to be poetic for a second. I mean, you have a huge amount of data, a huge amount of examples.

But amount of function can be even bigger. I understand everything is.

There's always a bigger boat.

Paul Hilbert space.

I gotcha. Okay, but you don't find the world of data to be an interesting optimization space like the. The optimization should be in the space of functions.

In creating admissible set of functions.

Admissible set of functions.

No. You know, even from the classical basis theory, from structure risk minimization, you should organize function in the way that they will be useful for you. Right? And that is.

Yeah, but the way you're thinking about useful is you're given a small set.

Of functions small set of functions which contain function by looking for.

Yeah, but as looking for. Based on the empirical set of small examples.

Yeah, but that is another story. I don't touch it because I believe that this small example is not too small, say 60 per class. That law of large numbers works. I don't need uniform law. The story is that in statistics there are two law. Law of large numbers and uniform law of large numbers. So I want to be in situation where I use law of large numbers but not uniform law large numbers.

Right. So 60 is law of large. It's large enough, I hope no sounds.

It still need some evaluation, some bounce. So it's. But idea is the following, that if you trust that say this average gives you something close to expectation. So you can talk about about this predicate and that is basis of human intelligence.

Good predicates is the discovery of good predicates is the basis of human intelligence.

No, no, it is discovery of your. Of your understanding world, of your methodology of understanding world. Because you have several functions which you will apply to reality.

Can you say that again? So you're.

You have several functions. Yeah, predicate. But they abstract. Yes. Then you will apply them to reality, to your data and you will create in this way predicate which is useful for your task. But predicate are not related specifically to your task, to this, your task. It is abstract functions which being applied.

Applied to any tasks that you might be interested in.

It might be many tasks. I don't know. Well, different tasks.

Well, they should be many tasks, right?

Yeah. I believe like in prop case it was for fairy tales, but it's happened everywhere.

Okay, so we talked about images a little bit, but can we talk about Noam Chomsky for a second?

I believe. I don't know him personally. Well, not personally.

I don't know his ideas.

His ideas?

Well, let me just say. Do you think language, human language is essential to expressing ideas, as Noam Chomsky believes so language is at the core of our formation of predicates, the human language.

For me, language and all the story of language is very complicated. I don't understand this and I'm not. I thought about.

Nobody does.

I'm not ready to work on that because it's so huge. It is not for me and I believe not for our century.

The 21st.

Century, not for 21st century. So you should learn something. A lot of stuff from simple task like digit recognition.

So you think okay, you think digital recognition, 2D image. What. How would you more abstractly define digit recognition? It's 2D image symbol recognition essentially. I mean, I'm trying to get a sense sort of thinking about it now, having worked with MNIST forever. How small of a subset is this of the general vision recognition problem and the general intelligence problem? Is it? Yeah. Is it a giant subset? Is it not? And how far away is language?

You know, let me refer to take the simplest problem as simple as possible, but not simpler. And this is challenge is simple problem, but it's simple by idea, but not simple to get it. When you will do this, you will find some predicament it a bit.

Well, yeah, I mean with Einstein you can. You. You look at general relativity, but that doesn't help you with quantum mechanics, that's another story.

You. You don't have any universal instrument.

Yes. So I'm. I'm trying to wonder if. Which space we're in, whether the. Whether handwritten recognition is like general relativity and then language is like quantum mechanics. So you're still going to have to do a lot of mess to universalize it. But I'm trying to see one. So what's your intuition? Why handwritten recognition is easier than language. Just I think a lot of people would agree with that. But if you could elucidate sort of the. The intuition of why.

I don't know. No, I don't think in this direction. I just think in direction that this is problem which if we will solve it well, we will create some abstract understanding of images. Maybe not all images. I would like talk to guys who doing unreal images in Columbia University.

What kind of images? Unreal.

Real images.

Real images?

Yeah. What their idea is there are predicate what can be predicated. I still symmetry will play role in real life images. In any real life images. 2D images. Let's talk about 2D images because that's what we know a neural network was created for 2D images.

So the people I know in vision science, for example, the people who study human vision, that they usually go to the world of symbols and like handwritten recognition, but not really it's other kinds of symbols to study our visual perception system. As far as I know, not much predicate type of thinking is understood about our vision system.

They did not think in this direction.

They don't. Yeah, they. But how do you even begin to think in that direction?

That's. There's so much I would like to discuss with them.

Yeah.

Because if we will be able to show that it is what working and theoretical scheme is not so bad.

So the unfortunate. So if we compare to language, language has like letters, finite set of letters and a finite set of ways you can put together those letters. So it feels more amenable to kind of analysis. With natural images, there is so many pixels.

No, no, no litter. Language is much, much more complicated. It involves a lot of different stuff. It's not just understanding of very simple class of tasks. I would like to see lists of tasks with language involved.

Yes. So there's a lot of nice benchmarks now in natural language processing. From the very trivial, like understanding the elements of a sentence to question answering to much more complicated, where you talk about open domain dialogue. The natural question is with handwritten recognition, it's really the first step of understanding visual information.

Right. But even our records show that we go in wrong direction because we need 60,000 digits.

So even this first step. So forget about talking about the full journey. This first step should be taking in the right direction.

No, no, wrong direction because 60,000 is unacceptable.

No, I'm saying it should be taken in the right direction because 60,000 is not acceptable.

You can talk. It's great. We have half percent of error and.

Hopefully the step four from doing hand recognition using very few examples. The step towards what babies do when they crawl and understand their physical environment. I know you don't know about babies.

If you will do from very small examples, you will find principles that will fight a baby. From what we use now and theoretically it's small, less clear. That means that you. You will use weak convergence, not just strong convergence.

Do you think these principles are. Will naturally be human interpretable?

Oh yeah.

So like when we'll be able to explain them and have a nice presentation to show what those principles are or are they very. Going to be very kind of abstract kinds of functions.

For example, I talk yesterday about symmetry.

Yes.

And I gave very simple examples. The same will be like that you.

Gave like a predicate of a basic.

For symmetries.

Yes, for different symmetries. And you have for degree of symmetries.

That is important, not just symmetry. Existence doesn't exist. Degree of symmetry?

Yeah. For handwritten recognition.

No, it's not for handwritten, it's for any images. But I would like apply to handwritten.

Right. In theory, it's more general. Okay, okay. So a lot of the things we've been talking about falls. We've been talking about philosophy a little bit, but also about mathematics and statistics. A lot of it falls into this idea. A universal Idea of statistical theory of learning. What is the most beautiful and sort of powerful or essential idea you've come across, even just for yourself personally in the world of statistics or statistic theory of learning?

Probably uniform convergence, which we did with Alexei Chalvonenkis.

Can you describe universal convergence?

You have law of large numbers. So for any function expectation of function, average of function converge state expectation. But if you have set of functions for any function, it is true, but it should converge simultaneously for all set of functions. And for learning you need uniform convergence. Just convergence is not enough, because when you pick up one which gives minimum, you can pick up one function which does not converge and it will give you the best answer for this function. So you need the uniform convergence to guarantee learning. So learning does not rely on trivial law large numbers. It really on universal. But idea of the convergence existing statistics for a long time. But it is interesting that as I think about myself, how stupid I was 50 years I did not see weak convergence. I work only on strong convergence. But now I think that most powerful is weak convergence because it makes admissible set of functions. And even in all pro in proverbs, when people try to understand recognition about dog law, looks like a dog and so on, they use vic convergence. People in language, they understand this. But when we trying to create artificial intelligence, we want event in different way. We just consider strong convergence arguments.

So reducing a set of admissible functions, you think there should be effort put into understanding the properties of weak convergence?

You know, in classical mathematics, in Gilbert space, there are only two ways to form of convergence. Strong and weak. Now we can use both. That means that we did everything. And it so happened. Then when we use Hilbert space, which is very rich space, space of continuous functions, which has an integral square, so we can apply weak and strong convergence for learning and have closed form solution. So for computationally simple for me it is sign that it is right way. Because you don't need any every stick, you just do whatever you want. But now the only what left is this concept of what is predicate of predicate. But it is not statistics.

By the way, I like the fact that you think the heuristics are a mess that should be removed from the system. So closed form solution is the ultimate?

No, it so happened. Then when you're using right instrument, you have closed form solution.

Do you think intelligence, human level intelligence, when we create it, we'll have something like a closed form solution?

You know, now I'm looking on bonds which I gave bounds for convergence. And when I looking for Bounds I thinking what is the most appropriate kernel for this bound would be. So we know that in say all our businesses we use radial bayes function. But looking on the bound I think that I start to understand that maybe we need to make corrections to radial bayes function to be closer to work better for this bounce. So I'm again trying to understand what type of kernel have best approximation not approximation best fit to this bone.

Sure. So there's a. There's a lot of interesting work that could be done in discovering better functions than radial basis functions for.

Yeah, it still comes from you. You're looking to Mars and trying to.

Understand what from your own mind looking at the.

Yeah, but I don't know then I trying to understand what. What will be good for that.

Yeah, but to me there's still a beauty Again, maybe I'm a descendant valentorian to heuristics. To me ultimately intelligence will be a mess of heuristics. That's the engineering answer, I guess.

Absolutely. When you're doing say self driving cars the great guy who will do this it does not matter what theory behind that who has a better feeling how to applied. But by the way, it is the same story about predicate because you cannot create rule for citation is much more than you have rule for that. But maybe you can have more abstract rule then it will be less than zero. It is the same story about ideas and ideas applied to specific cases.

But still you should.

You cannot avoid this.

Yes, of course. But you should still reach for the ideas to understand the science.

Yeah.

Let me kind of ask. Do you think neural networks or functions can be made to reason? Sort of. What do you think? Been talking about intelligence but this idea of reasoning there's a. There's an element of sequentially disassembling interpreting the. The images. So when. When you think of handwritten recognition we kind of think that there'll be a single. There's an input and output. There's not a recurrence.

Yeah.

What do you think about sort of the idea of recurrence of going back to memory and thinking through this sort of sequentially mangling the different representations over and over until you arrive at a conclusion or is ultimately all that can be wrapped up into a function.

Suggesting that let us use this type of algorithm. When I started thinking, I first of all starting to understand what I want. Can I write down what I want? And then I trying to formalize and when I do that I think I have to solve this problem. And. Till now I did not see Situation.

Where you need recurrence. But do you observe human beings?

Yeah.

Do you try to? It's the imitation question, right? It seems that human beings reason this kind of sequentially, sort of. Does that inspire in you a thought that we need to add that into our intelligent systems? You're saying, okay, you've kind of answered saying until now I haven't seen a need for it. And so because of that, you don't see a reason to think about it.

You know, most of things I don't understand in reasoning in human. It is for me too complicated. For me, the most difficult part is to ask questions, good questions. How it works, how people asking questions. I don't know this.

You said that machine learning is not only about technical things. Speaking of questions, but it's also about philosophy. So what role does philosophy play in machine learning? We talked about Plato, but generally thinking in this philosophical way, does it have. How does philosophy and math fit together in your mind?

First ideas and then their implementation. It's like predicate, like say admissible set of functions. It comes together everything. Because the first iteration of theory was done 50 years ago. I told that Mississauga. So everything there. If you have data, you can and your set of function. Have not big capacity, so low VC dimension. You can do that. You can make structural risk, minimization, control capacity. But you was not able to make admissible set of function. But no, when suddenly realize that we did not use another idea of convergence, which we can. Everything comes together.

But those are mathematical notions. Philosophy plays a role of simply saying that we should be swimming in the space of ideas.

Let's talk what is philosophy? Philosophy means understanding of life. So understanding of life, say people like Plato, they understand on very high abstract level of life. And whatever I doing is just implementation of my understanding of life. But every new step, it is very difficult, for example, to find this idea that we need weak convergence was not simple for me.

So that required thinking about life. A little bit hard to trace. But there was some thought process, you know, I working.

I thinking about the same problem for 50 years. Somehow and again and again and again. I trying to be honest. And that is a very important not to be very enthusiastic.

Yeah.

But concentrate on whatever we was not able to achieve, for example, and understand why. And now I understand that because I believe in math. I believe that in Vigner's idea. But now when I see that there are only two way of convergence and we're using both, that means that we must do as well as people doing. But now Exactly. In philosophy and what we know about predicate, what we. How we understand life can we describe as a predicate? I thought about that and that is more or less obvious level of symmetry. But next I have a feeling it's something about structures. But I don't know how to formulate how to measure. Measure of structure and all this stuff. And guy who will solve this challenge problem. Then when we will looking how he did it. Probably just only symmetry is not enough.

But something like symmetry will be there.

Oh yes, absolutely. Symmetry will be there. Level of symmetry will be there. And level of symmetry anti symmetry, diagonal, vertical. And I even don't know how you can use in different direction idea of symmetry. It's very general, but it will be there. I think that people very sensitive to idea of symmetry. But there are several ideas like symmetry. As I would like to learn. But you cannot learn. Just thinking about that. You should do challenging problems and then analyze them. Why we was able to solve them. And then we will see very simple things. It's not easy to find even we talking about this every time. Yes, about. I was surprised. I tried to understand is people describe in language strong convergence mechanism for learning. I did not see. I don't know. But we convergence this dark story and story like that. When you will explain to Kit, you will use weak convergence argument. It looks like it does like it is. But when you try to formalize we're just ignoring this. Why? Why 50 years from start of machine learning.

And that's the role of philosophy.

I think that might be. I don't know. Maybe this is theory also we should blame for that. Because empirical risk minimization know this stuff. And if you read now textbooks they just about bound about empirical risk minimization. They don't looking for another problem like admissible set.

But on the topic of life, perhaps we. You could talk in Russian for a little bit. What's your favorite memory from childhood.

Music?

How about. Can you try to answer in Russian? Just so you know, now that we're talking about Bach, let's switch back to English because I like Beethoven and Chopin.

So Chopin it's another amusing story.

But Bach if. If we talk about predicates. Bach probably has the most sort of well defined predicates that underlie.

You know, it is very interesting to read what critics writing about Bach which words they're using. They trying to describe predicates. Yeah, and, and, and. And then Chopin it is very different vocabulary, very different predicate. And I think that if you will make collection of that. So maybe from this you can describe predicate for digit recognition.

Well, from Bach and Chopin.

No, no, no, not from Bach and Chopin.

From the critical interpretation of the music. Yeah.

When they trying to explain you music, what they use this, they describe high level ideas of Plato's ideas. What behind this music.

That's brilliant. So art is not self explanatory in some sense. So you have to try to convert it into ideas.

It is impossible problems. When you go from ideas to the representation it is easy way, but when you're trying to go back, it is. You'll post problems. But nevertheless, I believe that when you looking from that even from art you will be able to find predicate for digital recognition.

That's such a fascinating and powerful notion. Do you ponder your own mortality? Do you think about it? Do you fear it? Do you draw insight from it about mortality?

Oh yeah.

Are you afraid of death?

Not too much. Not too much. It is pity that I will not be able to do something which I think I have a feeling to do that. For example, I will be very happy to work with guys theoretician from music to write this collection of description. How they describe music, how they use that predicate and from art as well. Then take what is in common and try to understand predicate which is absolute for everything.

And then use that for visual recognition. See if there is a connection.

Yeah, exactly.

There's still time. We got time. We got time.

It takes years and years and years. It's a long way.

Well, see, you've got the patient mathematician's mind. I think it could be done very quickly and very beautifully. I think it's a really elegant idea. Yeah, but also some of many.

You know, the. The most time it is not to make this collection to understand what is the common to think about that once.

Again and again and again, again and again and again. But I think sometimes, especially just when you say this idea now, even just putting together the collection and looking at the different sets of data, language, trying to interpret music, criticize music and images, I think there will be sparks of ideas that will come. Of course, again and again you'll come up with better ideas. But even just that notion is a beautiful notion.

I even have some example. So I have four friend who was specialist in Russian poetry. She is professor of Russian poetry. He did not write poems. But she know a lot of stuff. She make book, several books. In one of them is a collection of Russian poetry. She have images of Russian poetry. She collect all images of Russian poetry. And I asked her to do following you Have NIPS digit recognition and you get hundred digits or maybe less than 100. I don't remember, maybe 50 digits digits. And try from poetical point of view describe every image which she see using only words of images of Russian poetry. And she did it. And then we tried to. I call it learning using privileged information. I call it privileged information. You have on two languages. One language is just image of digit. And another language poetic description of this image. And this is privileged information. There is an algorithm when we are working using privileged information. You're doing better, much better.

So there's something there, something there.

And there is a. In nec she unfortunately died the collection of digits in poetic descriptions of these digits.

So there's something there in that poetic description.

But I think that there is an abstract ideas on the plateau level of ideas.

Yeah, that they're there that could be discovered. And music seems to be a good entry point.

But as soon as we start with this challenge problem. The challenge problem immediately connected to all.

This stuff, especially with your talk and this podcast. And I'll do whatever I can to advertise. It's such a clean, beautiful Einstein like formulation of challenge before us.

Right.

Let me ask another absurd question. We talked about mortality, we talked about philosophy of life. What do you think is the meaning of life? What's the predicate for our mysterious existence here on earth?

I don't know. It's very interesting how we have in Russia. I don't know if you know the guy Strugatsky, they writing fixtures, they thinking about human what's going on. And they have a deer that there are just developing two type of people, common people and very smart people. They just started and these two branches of people will go in different direction very soon. So that's what they thinking about life.

So the purpose of life is to create two. Two paths of human societies.

Yes. Simple people and more complicated people.

Which do you like best? The simple people or the complicated ones?

I don't know that he is just his fantasy. But you know, every week we have guy who is just writer and also theoretical literature. And he explained how he understand literature and human relationship, how he see life. And I understood that I'm just small kids comparing to him. He's very smart guy in understanding life. He knows this predicate. He knows big blocks of life. I am used every time when I listen to him and he just talking about literature. And I think that I was surprised. So the managers in big companies, most of them are guys who study English language and English literature. So why? Because they understand life, they understand models. And among them maybe many talented critics which just analyzing this and this is big science like prop did these blocks. Very smart.

It amazes me that you are and continue to be humbled by the brilliance of others.

I'm very modest about myself. I see so smart guys around.

Well, let me be immodest for you. You're one of the greatest mathematicians, statisticians of our time. It's truly an honor. Thank you for talking.

Okay, okay.

And let's talk.

It is not. Yeah, yeah, I know my limits.

Let's. Let's talk again when your challenge is taken on and solved by a grad student. Especially when maybe music will be involved. Vladimir, thank you so much. It's been.

Thank you very much.

Thanks for listening to this conversation with Vladimir Vapnik. And thank you to our presenting sponsor, Cash app. Download it, use code LEXPodcast. You'll get $10 and $10 will go to first, an organization that inspires and educates young minds to become science and technology innovators of tomorrow. If you enjoy this podcast, subscribe on YouTube, give it five stars on Apple Podcast, support it on Patreon, or simply connect with me on Twitter lexfriedman. And now let me leave you with some words from Vladimir. When solving a problem of interest, do not solve a more general problem as an intermediate step. Thank you for listening. I hope to see you next time.

Sa.