Andrew Gordon Wilson: belief

Transcript context

…It's a great question. So it's a really important observation that neural nets tend to learn structure before they learn noise and things like this. There's a question around to what extent being able to fit noise is hurting their generalization capabilities. So there's this other phenomenon called benign overfitting, where the model fits typically a mixture of signal and noise, but the noise being fitting the noise doesn't significantly degrade its generalization performance. And this is often seen as something that's specific to deep learning and in the face of everything that we know about generalization. And that's partly because classical frameworks for trying to understand generalization like VC dimension and Rademacher complexity are essentially measuring a model's ability to fit noise. However, are other generalization frameworks like PAC Bayes and countable hypothesis bounds that we explore, which don't penalize an expressive hypothesis space, and instead try to understand what sorts of soft preferences the model has for certain solutions over others. And we've been able to achieve fairly tight bounds on the generalization performance of these large models using something called a Solomonoff prior. And so a Solomonoff prior says that we actually have a maximally over parameterized model. We can represent every possible program on a computer, that we have exponentially stronger preferences for solutions with that have what are called low Kolmogorov complexity, and so they're very compressible. The Kolmogorov complexity is the the shortest possible program that can generate our hypothesis. And the fact that we're able to get these tight generalization bounds for these very large models, and in fact that the models get sort of better, sorry that the generalization bounds get better as we make the models larger, suggests that this is not a bad description of how these models are actually behaving. So doing induction with a Solomonov prior is called Solomonov induction. And so it seems that when we make these transformers for instance very large, we're combining this expressiveness with this strong preference for low Kolmogorov complexity solutions. Another observation that's that's been made is that models are becoming increasingly general purpose. And so 20 or 30 years ago, the typical prescription was to encode as much expert knowledge as possible into the model you're constructing and tailor it very specifically to the problem that you're considering. Because of results like the no free lunch theorems that say that every model is equally good in expectation over all problems sampled uniformly from this distribution over all problems. There are several no free lunch theorems, another 1 says that a single learner is not gonna be good on all problems. Now these results are mathematically correct, but they don't really correspond to the real world data generating distribution. nother 1 says that a single learner is not gonna be good on all problems. Now these results are mathematically correct, but they don't really correspond to the real world data generating distribution. The real world is a small corner of all possible data sets, it's not drawn uniformly from that distribution. If you were to draw data sets from that distribution, you would mostly get noise. And so I guess the question is, well what is the real world data generating distribution really like? And it seems like there is a bias towards generating data with low Kolmogorov complexity and our models share that bias. And this is why we've seen increasingly general systems. So we've moved from feature engineering, basically hard coding structure into our models, to more modality specific models and architectures. So convolutional neural nets for vision, for current neural nets, for sequences, language, etcetera. MLPs for tabular data and regression to transformers for almost everything. And so this isn't to say that transformers have achieved general intelligence, absolutely not. But they're relatively speaking more general than the predecessors. And we have seen this kind of movement towards increasingly general models, and our contention is that this has been made possible by aligning with the real world data generating distribution, which seems to have a bias for low Kolmogorov complexity. And we had this paper on no free lunch theorems in inductive biases in Comagraal complexity. I think 1 of the most surprising findings in that paper was that convolutional neural nets, which were clearly designed for image recognition, so they have locality and translation equivariance and so on, provably have inductive biases for tabular data shaped as And an the only possible reason that could be the case is because they both sort of share this bias for low Kolmogorov complexity. And that bias gets stronger as we make the model bigger. Okay. So I have I have actually 2 questions about this, and it's about the Kolmogorov complexity. So I think if I heard you correctly, on the 1 hand, you're saying that just stock neural network training of today, so just transformers, SGD, batch norm, whatever people are doing, seems to empirically exhibit bias towards lower Kolmogorov complexity models. And this shows up by their generalization kind of falling within, you know, this bound, know, that you found. And then you also mentioned Kolmogorov induction, which I I I believe would be explicitly introducing, you know, a kind of penalty term or a risk term and objective, part of the objective function that has to do with Kolmogorov complexity and kind of maybe pushing the model a little bit further towards, you know, simple. I think you're talking about both. So maybe if you could elaborate and also, you know, where is this this bias towards simplicity coming from? Like, everybody knows the algorithms. Like, which part of the algorithm is is inducing this simplicity bias? Is it because we're using floating point numbers, I triple e, or what? Like, where is it coming from?…

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