Video · 2026-08-10 · 1h 19m · 6 moments

Why Deep Networks Don’t Need to Memorize Everything — Matthieu Wyart

✦ AI generated

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Fact

The loss landscape of neural networks undergoes a jamming phase transition identical to that of granular materials like sand: underparameterized systems are stuck in metastable states while overparameterized systems flow freely to flat minima.

Wyart draws a direct analogy between the physics of sand and machine learning: both are constraint-satisfaction problems with continuous degrees of freedom, and both exhibit the same jamming phase transition — the peak of the 'double descent' curve corresponds to the critical point where the system transitions from a rough, trapped landscape to a smooth, flowing one.

transcript

Matthieu Wyart: when you actually underparameterized when you don't have enough parameters you have a rough landscape with many metastable state and if you train your machine and you train it many times it will end up in different position where it's actually stuck but if you have enough parameters then suddenly the system can flow I in your landscape has many flat valleys and you can which have essentially zero energy. So there is really a close analogy we disco we discovered that like 9 years ago at the same time others find a very similar I mean the same phenomenon and called it double descent. So now this name has stuck but this double descent this peak of the double descent is really for physicists a jamming transition.

Claim◆

Deep architectures have an implicit bias that enables them to learn hierarchical generative structure from exponentially fewer examples than Chomsky's poverty of stimulus argument predicts is possible.

Wyart's group showed that while shallow networks memorize and fail to learn grammar, deep architectures leverage their hierarchical structure as an implicit bias, learning to generate novel grammatical sentences from only polynomial (not exponential) amounts of training data — directly contradicting Chomsky's poverty of stimulus argument.

transcript

Matthieu Wyart: Chumsky gave this uh poverty of stimulus argument uh you know arguing that it was actually impossible to learn to become creative from example and essentially okay this would be a very crude way of of summarizing his argument but I described the fact that you have this sort of generative treel like rich contextf free grammarss you know assuming that that and really capturing the fact that the the world as a yarchy of abstract concepts. Um and but you have other possible generative grammar. Some are much simpler. One is called regular grammar. So this will be a carature but essentially the idea that maybe a group of words will fix the probability of the next word essentially. And Chumsky's argument is to say that well even if you give me 1 million sentences okay I can fit those you know those sentences by a contextf free grammar but I can also fit them by a much simpler I mean a regular grammar simpler in it classification but to fit those sentences it would have to be awfully complicated you know many many many rules... So in our idealized world where the true world is can machine learn to be creative or not and what we find is that if you have a shallow network what Shamsky worried about is completely true. You learn some you don't learn this sort of interesting generative grammar. You essentially memorize and you can't do anything. But if you have a deep architecture, there's a huge implicit bias to build those course grain variables. Thisarchical architecture leads very easily to some iterative calculation. And so what we what we found is that uh indeed you can learn to be creative by having being exposed to very small number of sentences. So let's say in our model if D is an is you know the size of the sentence the number of sentences is huge with D is exponential in D but the number of sentences you need to see to be creative is only polomial in D. So these models are really a counter example to uh to his argument.

supports · 1

Mechanism

Concepts and abstractions in deep generative models emerge from statistics alone by grouping together configurations that predict similar contexts around them.

In LLMs and diffusion models, abstract concepts like 'street' or 'house' are not predefined but instead emerge when the network groups together diverse low-level configurations (passersby, cars, sidewalks) that all predict similar surrounding contexts.

transcript

Matthieu Wyart: if you think about uh LLMs or diffusion models, the way they build concepts, they emerge from statistics alone. Those abstraction they emerge they are there in the data they emerge and those concept emerge if you group together configuration that predict similar context around them. So maybe if you have a street typically you have houses nearby and maybe the houses have colors or edges and so you would predict color and edges and with that in those models at least you find that if you have enough data you can learn all the abstractions but as you get more and more abstract you have a problem because you're always trying to build those abstraction by saying how they are predictive but at a very low level and when you're very abstract you how you predict pixels or colors and so on is a super noisy signal. So essentially that's why in those models we find and we have empirical evidence and I'm happy to talk about empirical evidence that that the more abstract concept are you know the toughest to learn because essentially your signal as you get more and more abstract your signal gets diluted.

explains mechanism · 1extends · 1

Mechanism◆

Deep networks solve the curse of dimensionality by discovering hierarchical abstract variables in data, allowing them to generalize with polynomial rather than exponential amounts of data.

When data has a hierarchical generative structure, deep networks can discover the underlying coarse-grained variables efficiently, reducing effective dimensionality and enabling generalization far beyond what shallow models or kernel methods can achieve.

transcript

Matthieu Wyart: So in physics we know that a volume goes like a length to the exponent of the dimension. So in 3D L cube in 2D L² L is a length. So think about a large dimension. So if you think about an image, D may be the number of pixels at least naively. If you think about text, it may be the number of words in your sentence again naively. So those volumes are huge. They're exponential. You know, they're exponential at large in the dimension. So what it means is that even if you give me one trillion points, because the volume is so huge, they're extremely far away from each other. Extremely far away. And so if you have a machine that's just interpolating and now you ask a question about a new test point and you can prove mathematically that if the data has little structure like you're trying to learn to regress some function that's smooth it's hopeless. I mean the only way you will extrapolate and have you know power to to to generalize is if you know you bring those point together. it means you have an exponentially large number of data. You have more data than atoms in the universe. So it's just impossible. So to me this is completely fundamental question... and so essentially The answer to that in in is that if the world is thearchical, if it has those hidden co grand variables, those machines are super good to discover them and they can discover them with a generically with an number of data that's not huge polomial in the dimension once again.

Mechanism◆

Algorithms that predict in abstract latent space rather than raw token space are significantly more sample-efficient because they build abstractions from stronger, less noisy signals.

Joint embedding prediction architectures — where a student network predicts the teacher's latent representations of occluded data rather than predicting raw tokens — are much more sample-efficient because they build abstract concepts by predicting nearby abstract concepts rather than correlating abstractions with noisy low-level pixels.

transcript

Matthieu Wyart: in those models what we found is that those algorithms that are introspective that learn from their own latent are much more powerful in terms of sample complexity and they will eventually they will learn the same the same abstraction but much faster to build abstraction You need to bring configuration. Think again about the street. All those configuration you need to understand it's one entity a street and your what diffusion or next token prediction do is that they the signal they have to bring those together has to do with pixels around very lowle features and I told you that this correlation between abstract things and things that are not abstract it's it's it's there it exists but it's very noisy but imagine instead that you when you start when you understood the concept of cars and passerby And you also understood the concept of house houses. Then essentially what those methods do is they can build the concept of street by predicting that those configuration have not pixel the painting of the house but just the concept houses nearby and then the signal is much larger and so you need much less data to extract the signal from noise.

provides context · 3

Mechanism

Scaling laws in large language models can be quantitatively predicted from two measurable linguistic quantities: the power-law decay rate of token correlations with distance and the power-law entropy reduction as context length increases.

Wyart's team proposed a theory that explains the empirical scaling laws driving massive AI investment: the training loss exponent can be derived from two measurable properties of natural language — how quickly word-word correlations decay with distance and how quickly entropy drops as more context is provided — both following power laws.

transcript

Matthieu Wyart: the theory predicts that there is a simple recipe to for those exponents to extract those and and essentially this is saying again what's underlying it is the fact that if you give me more data I can learn more abstract concept and that's longer range leads to longer range correlation but at the end of the day the two quantities you need to measure is one the fact that words or tokens are correlated and that this correlation it's it was well known before that this correlation decreases as a polo of the distance between those two words. From that you can measure exponents and they depend on the language you look at as your data set. You can measure them and then there is another key quantity we argue which is related to the entropy of text. So entropy of text has been discussed already by Shannon in the 50s. It's beautiful question. So essentially it's the entropy is a log of the number of possible words that you would have at one location in average. What we argue is very important to look at and that we could finally measure with LLMs or other architecture and we find consistent result is what is the entropy left after a sentence of n token. If you see n token the more token you see the least possibility you have there. What is the entropy of that? And in the toy model of it's a polo and in real life it's also a polo that is found. And so essentially what we argue is that with those two exponents you can combine them in a way that we specify to get the training curve exponent of LLM's acting on those natural languages and it works very well.

Highlight slides

Chomsky's Poverty of Stimulus — Challenged✦ from: Deep architectures have an implicit bias that enables them to learn hierarchical generative structure from exponentially fewer examples than Chomsky's poverty of stimulus argument predicts is possible.Exponential Complexity, Polynomial Data✦ from: Deep architectures have an implicit bias that enables them to learn hierarchical generative structure from exponentially fewer examples than Chomsky's poverty of stimulus argument predicts is possible.The Curse of Dimensionality✦ from: Deep networks solve the curse of dimensionality by discovering hierarchical abstract variables in data, allowing them to generalize with polynomial rather than exponential amounts of data.Deep Networks: Hierarchical Structure Solution✦ from: Deep networks solve the curse of dimensionality by discovering hierarchical abstract variables in data, allowing them to generalize with polynomial rather than exponential amounts of data.Latent Prediction > Token Prediction✦ from: Algorithms that predict in abstract latent space rather than raw token space are significantly more sample-efficient because they build abstractions from stronger, less noisy signals.Why Latent Space Wins✦ from: Algorithms that predict in abstract latent space rather than raw token space are significantly more sample-efficient because they build abstractions from stronger, less noisy signals.Concrete Example: Building 'Street'✦ from: Algorithms that predict in abstract latent space rather than raw token space are significantly more sample-efficient because they build abstractions from stronger, less noisy signals.

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