My Criticism of Singular Learning Theory

[Joar Skalse]

In this post, I will briefly give my criticism of Singular Learning Theory (SLT), and explain why I am skeptical of its significance. I will especially focus on the question of generalisation—I do not believe that SLT offers any explanation of generalisation in neural networks. I will also briefly mention some of my other criticisms of SLT, describe some alternative solutions to the problems that SLT aims to tackle, and describe some related research problems which I would be more excited about.

(I have been meaning to write this for almost 6 months now, since I attended the SLT workshop last June, but things have kept coming in the way.)

For an overview of SLT, see this sequence. This post will also refer to the results described in this post, and will also occasionally touch on VC theory. However, I have tried to make it mostly self-contained.

The Mystery of Generalisation

First of all, what is the mystery of generalisation? The issue is this; neural networks are highly expressive, and typically overparameterised. In particular, when a real-world neural network is trained on a real-world dataset, it is typically the case that this network is able to express many functions which would fit the training data well, but which would generalise poorly. Moreover, among all functions which do fit the training data, there are more functions (by number) that generalise poorly, than functions that generalise well. And yet neural networks will typically find functions that generalise well. Why is this?

To make this point more intuitive, suppose we have a 500,000-degree polynomial, and that we fit this to 50,000 data points. In this case, we have 450,000 degrees of freedom, and we should by default expect to end up with a function which generalises very poorly. But when we train a neural network with 500,000 parameters on 50,000 MNIST images, we end up with a neural network that generalises well. Moreover, adding more parameters to the neural network will typically make generalisation better, whereas adding more parameters to the polynomial is likely to make generalisation worse. Why is this?

A simple hypothesis might be that some of the parameters in a neural network are redundant, so that even if it has 500,000 parameters, the dimensionality of the space of all functions which it can express is still less than 500,000. This is true. However, the magnitude of this effect is too small to solve the puzzle. If you get the MNIST training set, and assign random labels to the test data, and then try to fit the network to this function, you will find that this often can be done. This means that while neural networks have redundant parameters, they are still able to express more functions which generalise poorly, than functions which generalise well. Hence the puzzle.

The anwer to this puzzle must be that neural networks have an inductive bias towards low-complexity functions. That is, among all functions which fit a given training set, neural networks are more likely to find a low-complexity function (and such functions are more likely to generalise well, as per Occam’s Razor). The next question is where this inductive bias comes from, and how it works. Understanding this would let us better understand and predict the behaviour of neural networks, which would be very useful for AI alignment.

I should also mention that generalisation only is mysterious when we have an amount of training data that is small relative to the overall expressivity of the learning machine. Classical statistical learning theory already tells us that any sufficiently well-behaved learning machine will generalise well in the limit of infinite training data. For an overview of these results, see this post. Thus, the question is why neural networks can generalise well, given small amounts of training data.

The SLT Answer

SLT proposes a solution to this puzzle, which I will summarise below. This summary will be very rough—for more detail, see this sequence and this post.

First of all, we can decompose a neural network into the following components:

1. A parameter space, $\Theta$, corresponding to all possible assignments of values to the weights.
2. A function space, $F$, which contains all functions that the neural network can express.
3. A parameter-function map, $m:\Theta \to F$, which associates each parameter assignment $\theta \in \Theta$ with a function $f\in F$.

SLT models neural networks (and other learning machines) as Bayesian learners, which have a prior over $\Theta$, and update this prior based on the training data in accordence with Bayes’ theorem. Moreover, it also assumes that the loss (ie, the likelihood) of the network is analytic in $\Theta$. This essentially amounts to a kind of smoothness assumption. Moreover, as is typical in the statistical learning theory literature, SLT assumes that the training data (and test data) is drawn i.i.d. from some underlying data distribution (though there are some ways to relax this assumption).

We next observe that a neural network typically is overparameterised, in the sense that there for a particular function $f\in F$ typically will be many different parameter assignments $\theta \in \Theta$ such that $m\left(\theta \right)=f$. For example, we can almost always rescale different parameters without affecting the input-output behaviour of a neural network. There can also be symmetries—for example, we can always re-shuffle the neurons in a neural network, which would also never affect its input-output behaviour. Finally, there can be redundancies, where some parameters or parts of the network are not used for any piece of training data.

These symmetries and redundancies lead to paths or valleys of constant loss through the loss landscape, which are created by the fact that the parameters $\theta$ can be continuously moved in some direction without affecting the function $f$ expressed by the network.

Next, note that if we are in such a valley, then we can think of the neural network as having a local “effective parameter dimension” that is lower than the full dimensionality of $\Theta$. For example, if $\Theta$ has 10 dimensions, but there is a (one-dimensional) valley along which $m$ (and hence $f$, and hence the loss) is constant, then the effective parameter dimension of the network is only 9. Moreover, this “effective parameter dimension” can be different in different regions of $\Theta$; if two one-dimensional valleys intersect, then the effective parameter dimension at that point is only 8, since we can move in two directions without changing $f$. And so on. SLT proposes a measurement, called the RLCT, which provides a continuous quantification of the “effective parameter dimension” at different points. If a point or region has a lower RLCT, then that means that there are more redundant parameters in that region, and hence that the effective parameter dimension is lower.

SLT provides a theorem which says that, under the assumptions of the theory, points with low RLCT will eventually dominate the Bayesian posterior, in the limit of infinite data. It is worth noting that this theorem does not require any strong assumptions about the Bayesian prior over $\Theta$. This should be reasonably intuitive. Very loosely speaking, regions with a low RLCT have a larger “volume” than regions with high RLCT, and the impact of this fact eventually dominates other relevant factors. (However, this is a very loose intuition, that comes with several caveats. For a more precise treatment, see the posts linked above.)

Moreover, since regions with a low RLCT roughly can be thought of as corresponding to sub-networks with fewer effective parameter dimensions, we can (perhaps) think of these regions as corresponding to functions with low complexity. Putting this together, it seems like SLT says that overparameterised Bayesian learning machines in the limit will pick functions that have a low complexity, and such functions should also generalise well. Indeed, SLT also provides a theorem which says that the Bayes generalisation loss eventually will be low, where the Bayes generalisation loss is a Bayesian formalisation of prediction error. So this solves the mystery, does it not?

Why the SLT Answer Fails

The SLT answer does not succeed at explaining why neural networks generalise. The easiest way to explain why this is the case will probably be to provide an example. Suppose we have a Bayesian learning machine with 15 parameters, whose parameter-function map is given by

$f\left(x\right)={\theta }_1{x}^4+{\theta }_2{\theta }_3{x}^3+{\theta }_4{\theta }_5{\theta }_6{x}^2+{\theta }_7{\theta }_8{\theta }_9{\theta }_{10}x+{\theta }_{11}{\theta }_{12}{\theta }_{13}{\theta }_{14}{\theta }_{15},$

and whose loss function is the KL divergence. This learning machine will learn 4-degree polynomials. Moreover, it is overparameterised, and its loss function is analytic in its parameters, etc, so SLT will apply to it.

For the sake of simplicity suppose we have a single data point, $(1,1)$. There is an infinite number of functions which $f$ can express that fits this training data, but some natural solutions include:

$f\left(x\right)=1$
$f\left(x\right)=x$
$f\left(x\right)=x^2$
$f\left(x\right)=x^3$
$f\left(x\right)=x^4$

Intuitively, the best solution is $f\left(x\right)=1$, because this solution has the lowest complexity. However, the solution with the lowest RLCT is in this case $f\left(x\right)=x^4$. This is fairly easy to see; around $f\left(x\right)=x^4$, there are many directions in the parameter-space along which the function remains unchanged (and hence, along which loss is constant), whereas around $f\left(x\right)=1$, there are fewer such directions. Hence, this learning machine will in fact be biased towards solutions with high complexity, rather than solutions with low complexity. We should therefore expect it to generalise worse, rather than better, than a simple non-overparameterised learning machine.

To make this less of a toy problem, we can straightforwardly extend the example to a polyfit algorithm with (say) 500,000 parameters, and a dataset with 50,000 data points. The same dynamic will still occur.

Of course, if we get enough training data, then we will eventually rule out all functions that generalise poorly, and start to generalise well. However, this is not mysterious, and is already fully adequately explained by classical statistical learning theory. The question is why we can generalise well, when we have relatively small amounts of training data.

The fundamental thing that goes wrong here is the assumption that regions with a low RLCT correspond to functions that have a low complexity. There is no necessary connection between these two things—as demonstrated by the example above, we can construct learning machines where a low RLCT corresponds to high complexity, and (using an analogous stategy), we can also construct learning machines where a low RLCT corresponds to low complexity. We can therefore not go between these two things, without making any further assumptions.

To state this differently, the core problem we are running up against is this:

1. To understand the generalisation behaviour of a learning machine, we must understand its inductive bias.

2. The inductive bias of a parameterised Bayesian learning machine is determined by (a) its Bayesian prior over the parameter-space, and (b) its parameter-function map.

3. SLT abstracts away from both the prior and the parameter-function map.

4. Hence, SLT is at its core unable to explain generalisation behaviour.

The generalisation bound that SLT proves is a kind of Bayesian sleight of hand, which says that the learning machine will have a good expected generalisation relative to the Bayesian prior that is implicit in the learning machine itself. However, this does basically not tell us anything at all, unless we also separately believe that the Bayesian prior that is implicit in the learning machine also corresponds to the prior that we, as humans, have regarding the kinds of problems that we believe that we are likely to face in the real world.

Thus, SLT does not explain generalisation in neural networks.

The Actual Solution

I worked on the issue of generalisation in neural networks a few years ago, and I believe that I can provide you with the actual explanation for why neural networks generalise. You can read this explanation in depth here, together with the linked posts. In summary:

1. The parameter-function map of neural networks is exponentially biased towards functions with low complexity, such that functions in $F$ with low (Kolmogorov) complexity have exponentially more parameter assignments in $\Theta$. Very roughly, to a first-order approximation, if we randomly sample a parameter assignment $\theta$, then the probability that we will obtain a particular function $f$ is approximately on the order of $2^{-K\left(f\right)}$, where $K\left(f\right)$ is the complexity of $f$.

2. Training with SGD is (to a first-order approximation) similar to uniform sampling from all parameters that fit the training data (considering only parameters sufficiently close to the origin).

Together, these two facts imply that we are likely to find a low-complexity function that fits the training data, even though the network can express many high-complexity functions which also fit the training data. This, in turn, explains why neural networks generalise well, even when they are overparameterised.

To fix the explanation provided by SLT, we need to add the assumption that the parameter-function map $m$ is biased towards low-complexity functions, and that regions with low RLCT typically correspond to functions with low complexity. Both of these assumptions are likely to be true. However, if we add these assumptions, then the machinery of SLT is no longer needed, as demonstrated by the explanation above.

Other Issues With SLT

Besides the fact that SLT does not explain generalisation, I think there are also other issues with SLT. First of all, SLT is largely is based on examining the behaviour of learning machines in the limit of infinite data. I claim that this is fairly uninteresting, because classical statistical learning theory already gives us a fully adequate account of generalisation in this setting which applies to all learning machines, including neural networks. Again, I have written an overview of some of these results here. In short, generalisation in the infinite-data limit is not mysterious.

Next, one of the more tantalising promises of SLT is that the posterior will eventually be dominated by singular points with low RLCT, which correspond to places where multiple valleys of low loss intersect. If this is true, then that suggests that we may be able to understand the possible generalisation behaviours of a neural network by examining a finite number of highly singular points. However, I think that this claim also is false. In particular, the result which says that these intersections eventually will dominate the posterior is heavily dependent on the assumption that the loss landscape is analytic, together with the fact that we are examining the behaviour of the system in the limit of infinite data. If we drop one or both of these assumptions, then the result may no longer hold. This is easy to verify. For example, suppose we have a two-dimensional parameter space, with two parameters $x$, $y$, and that the loss is given by $min(|x|,|y|)$. Here, the most singular point is $(0,0)$. However, if we do a random walk in this loss landscape, we will not be close to $(0,0)$ most of the time. It is true that we will be close to$(0,0)$ more often than we will be close to any other individual point with low loss, but $(0,0)$ will not dominate the posterior. This fact is compatible with the mathematical results of SLT, because if we create an analytic approximation of $min(|x|,|y|)$, then this approximation with be “flatter” around $(0,0)$, and this flatness creates a basin of attraction that it is easy to get stuck in, but difficult to leave. Moreover, if a neural network uses ReLu activations, then its loss function will not be analytic in the parameters. It is thus questionable to what extent this dynamic will hold in practice.

I should also mention a common piece of criticism of SLT which I do not agree with. SLT models neural networks as Bayesian learning machines, whereas they in reality are trained by gradient descent, rather than Bayesian updating. I do not think that this is a significant issue, because gradient descent is empirically quite similar to Bayesian sampling. For details, see again this post.

SLT is also sometimes criticised because it assumes that the loss function is analytic in the parameters of the network. Moreover, if we use ReLu activations, then this is not the case. I do not think that this necessarily is too concerning, because ReLu functions can be approximated by some analytic activation functions. However, when this assumption is combined with the infinite-data assumption, then we may run into issues, as I outlined above.

Some Research I Would Be Excited About

I worked on issues related to generalisation and the science of deep learning a few years ago. I have not actively worked on it very much since, and instead prioritised research around reward learning and outer alignment. However, there are still multiple research directions in this area that I think are promising, and which I would encourage people to explore, but which do not fall under the umbrella of SLT.

First of all, the results presented in this line of research suggest that the generalisation behaviour (and hence the inductive bias) of neural networks is primarily determined by their parameter-function map. This should in fact be quite intuitive, because;

1. If we change the architecture of a neural network, then this can often have a very large impact on its generalisation behaviour and inductive bias. For example, fully connected networks and CNNs have very different generalisation behaviour. Moreover, changing the architecture is equivalent to changing the parameter-function map.

2. If we change other components of a neural network, such as what type of optimiser we are using, then this will typically have a comparatively small impact on its generalisation behaviour and inductive bias.

Therefore, if we want to understand generalisation and inductive bias, then we should study the properties of the parameter-function map. Moreover, there are many open questions on this topic, that it would be fairly tractable to tackle. For example, the results in this post suggest that the parameter-function maps of many different neural network architectures are exponentially biased towards low complexity. However, they do not give a detailed answer to the question of precisely which complexity measure they minimise—they merely show that this result holds for many different complexity measures. For example, I would expect that fully connected neural networks are biased towards functions with low Boolean circuit complexity, or something very close to that. Verifying this claim, and deriving similar results about other kinds of network architectures, would make it easier to reason about what kinds of functions we should expect a neural network to be likely or unlikely to learn. This would also make it easier to reason about out-of-distribution generalisation, etc.

Another area which I think is promising is the study of random networks and random features. The results in this post suggest that training a neural network is functionally similar to randomly initialising it until you find a function that fits the training data. This suggests that we may be able to draw conclusions about what kinds of features a neural network is likely to learn, based on what kinds of features are likely to be created randomly. There are also other results that are relevant to this topic. For example, this paper shows that a randomly initialised neural network with high probability contains a sub-network that fits the training data well and generalises well. Specifically, if we randomly initialise a neural network, and then use a traning method which only allows us to set weights to 0, then we will find a reasonably good network. Note that this result is not the same as the result from the Lottery Ticket paper. Similarly, the fact that sufficiently large extreme learning machines attain similar performance to neural networks that have been trained normally, also suggests that this may be a fruitful angle of attack.

Another interesting question might be to try to quantify exactly how much of the generalisation behaviour is determined by different sources. For example, how much data augmentation would we need to do, before the effect of that data augmentation is comparable to the effect that we would get from changing the optimiser, or changing the network architecture? Etc.

What I Think SLT Can Do

This being said, I think there are many questions that are relevant to generalisation and inductive bias, which I think SLT may be able to help with. For example, it seems like it would be difficult to get a good angle on the question of phase shifts based on studying the parameter-function map. Moreover, SLT is a good candidate for a framework from which to analyse these questions. Therefore, I would not be that surprised if SLT could provide a good account of phenomena such as double descent or grokking, for example.

Closing Remarks

In summary, I think that the significance of SLT is somewhat over-hyped at the moment. I do not think that SLT will provide anything like a “unified theory of deep learning”, and specifically, I do not think that it can explain generalisation or inductive bias in a satisfactory way. I still think there are questions that SLT can help to solve, but I see its significance as being more limited to certain specific problems.

If there are any important errors in this post, then please let me know in the comments.

EDIT: For those reading on the Alignment Forum, I want to highlight this conversation thread, and especially this comment; Daniel Murfet clarified my point very well.

[Daniel Murfet]

The easiest way to explain why this is the case will probably be to provide an example. Suppose we have a Bayesian learning machine with 15 parameters, whose parameter-function map is given by

$f\left(x\right)={\theta }_1+{\theta }_2{\theta }_{3}x+{\theta }_4{\theta }_5{\theta }_6{x}^2+{\theta }_7{\theta }_8{\theta }_9{\theta }_{10}{x}^3+{\theta }_{11}{\theta }_{12}{\theta }_{13}{\theta }_{14}{\theta }_{15}{x}^4,$

and whose loss function is the KL divergence. This learning machine will learn 4-degree polynomials. Moreover, it is overparameterised, and its loss function is analytic in its parameters, etc, so SLT will apply to it.

In your example there are many values of the parameters that encode the zero function (e.g.${\theta }_1={\theta }_2={\theta }_4={\theta }_7={\theta }_{11}=0$ and all other parameters free) in addition to there being many parameters that encode the function $x^4$ (e.g. ${\theta }_1={\theta }_2={\theta }_4={\theta }_7=0$, variables ${\theta }_3,{\theta }_5,{\theta }_6,{\theta }_8,{\theta }_9,{\theta }_{10}$ free and ${\theta }_{11}{\theta }_{12}{\theta }_{13}{\theta }_{14}{\theta }_{15}=1$). Without thinking about it more I’m not sure which is actually has local learning coefficient (RLCT) and therefore counts as “more simple” from an SLT perspective.

However, if I understand correctly it’s not this specific example that you care about. We can agree that there is some way of coming up with a simple model which (a) can represent both the functions $x\mapsto 0$ and $x\mapsto x^2$ and (b) has parameters $w_0^{\ast }$ and $w_{square}^{\ast }$ respectively representing these functions with local learning coefficients $\lambda \left(w_0^{\ast }\right)>\lambda \left(w_{square}^{\ast }\right)$. That is, according to the local learning coefficient as a measure of model complexity, the neighbourhood of the parameter $w_0^{\ast }$ is more complex than that of $w_{square}^{\ast }$. I believe your observation is that this contradicts an a priori notion of complexity that you hold about these functions.

Is that a fair characterisation of the argument you want to make?

Assuming it is, my response is as follows. I’m guessing you think $x\mapsto 0$ is simpler than $x\mapsto x^2$ because the former function can be encoded by a shorter code on a UTM than the latter. But this isn’t the kind of complexity that SLT talks about: the local learning coefficient $\lambda \left(w^{\ast }\right)$ that appears in the main theorems represents the complexity of representing a given probability distribution $p\left(x|w^{\ast }\right)$ using parameters from the model, and is not some intrinsic model-free complexity of the distribution itself.

One way of saying it is that Kolmogorov complexity is the entropy cost of specifying a machine on the description tape of a UTM (a kind of absolute measure) whereas the local learning coefficient is the entropy cost per sample of incrementally refining an almost true parameter in the neural network parameter space (a kind of relative measure). I believe they’re related but not the same notion, as the latter refers fundamentally to a search process that is missing in the former.

We can certainly imagine a learning machine set up in such a way that it is prohibitively expensive to refine an almost true parameter nearby a solution that looks like $x\mapsto 0$ and very cheap to refine an almost true parameter near a solution like $x\mapsto x^2$, despite that being against our natural inclination to think of the former as simpler. It’s about the nature of the refinement /​ search process, not directly about the intrinsic complexity of the functions.

So we agree that Kolmogorov complexity and the local learning coefficient are potentially measuring different things. I want to dig deeper into where our disagreement lies, but I think I’ll just post this as-is and make sure I’m not confused about your views up to this point.

[Joar Skalse]

In your example there are many values of the parameters that encode the zero function

Ah, yes, I should have made the training data be (1,1), rather than (0,0). I’ve fixed the example now!

Is that a fair characterisation of the argument you want to make?

Yes, that is exactly right!

Assuming it is, my response is as follows. I’m guessing you think $x\mapsto 0$ is simpler than $x\mapsto x^2$ because the former function can be encoded by a shorter code on a UTM than the latter.

The notion of complexity that I have in mind is even more pre-theoretic than that; it’s something like ”$x^2$ looks like an intuitively less plausible guess than $0$”. However, if we want to keep things strictly mathematical, then we can substitute this for the definition in terms of UTM codes.

But this isn’t the kind of complexity that SLT talks about

I’m well aware of that—that is what my example attempts to show! My point is that the kind of complexity which SLT talks about does not allow us to make inferences about inductive bias or generalisation behaviour, contra what is claimed e.g. here and here.

So we agree that Kolmogorov complexity and the local learning coefficient are potentially measuring different things. I want to dig deeper into where our disagreement lies, but I think I’ll just post this as-is and make sure I’m not confused about your views up to this point.

As far as I can tell, we don’t disagree about any object-level technical claims. Insofar as we do disagree about something, it may be more methodolocical meta-questions. I think that what would probably be the most important thing to understand about neural networks is their inductive bias and generalisation behaviour, on a fine-grained level, and I don’t think SLT can tell you very much about that. I assume that our disagreement must be about one of those two claims?

[Daniel Murfet]

I think that what would probably be the most important thing to understand about neural networks is their inductive bias and generalisation behaviour, on a fine-grained level, and I don’t think SLT can tell you very much about that. I assume that our disagreement must be about one of those two claims?

That seems probable. Maybe it’s useful for me to lay out a more or less complete picture of what I think SLT does say about generalisation in deep learning in its current form, so that we’re on the same page. When people refer to the “generalisation puzzle” in deep learning I think they mean two related but distinct things:

(i) the general question about how it is possible for overparametrised models to have good generalisation error, despite classical interpretations of Occam’s razor like the BIC
(ii) the specific question of why neural networks, among all possible overparametrised models, actually have good generalisation error in practice (saying this is possible is much weaker than actually explaining why it happens).

In my mind SLT comes close to resolving (i), modulo a bunch of questions which include: whether the asymptotic limit taking the dataset size to infinity is appropriate in practice, the relationship between Bayesian generalisation error and test error in the ML sense (comes down largely to Bayesian posterior vs SGD), and whether hypotheses like relative finite variance are appropriate in the settings we care about. If all those points were treated in a mathematically satisfactory way, I would feel that the general question is completely resolved by SLT.

Informally, knowing SLT just dispels the mystery of (i) sufficiently that I don’t feel personally motivated to resolve all these points, although I hope people work on them. One technical note on this: there are some brief notes in SLT6 arguing that “test error” as a model selection principle in ML, presuming some relation between the Bayesian posterior and SGD, is similar to selecting models based on what Watanabe calls the Gibbs generalisation error, which is computed by both the RLCT and singular fluctuation. Since I don’t think it’s crucial to our discussion I’ll just elide the difference between Gibbs generalisation error in the Bayesian framework and test error in ML, but we can return to that if it actually contains important disagreement.

Anyway I’m guessing you’re probably willing to grant (i), based on SLT or your own views, and would agree the real bone of contention lies with (ii).

Any theoretical resolution to (ii) has to involve some nontrivial ingredient that actually talks about neural networks, as opposed to general singular statistical models. The only specific results about neural networks and generalisation in SLT are the old results about RLCTs of tanh networks, more recent bounds on shallow ReLU networks, and Aoyagi’s upcoming results on RLCTs of deep linear networks (particularly that the RLCT is bounded above even when you take the depth to infinity).

As I currently understand them, these results are far from resolving (ii). In its current form SLT doesn’t supply any deep reason for why neural networks in particular are often observed to generalise well when you train them on a range of what we consider “natural” datasets. We don’t understand what distinguishes neural networks from generic singular models, nor what we mean by “natural”. These seem like hard problems, and at present it looks like one has to tackle them in some form to really answer (ii).

Maybe that has significant overlap with the critique of SLT you’re making?

Nonetheless I think SLT reduces the problem in a way that seems nontrivial. If we boil the “ML in-practice model selection” story to “choose the model with the best test error given fixed training steps” and allow some hand-waving in the connection between training steps and number of samples, Gibbs generalisation error and test error etc, and use Watanabe’s theorems (see Appendix B.1 of the quantifying degeneracy paper for a local formulation) to write the Gibbs generalisation error as

$G_g\left(n\right)=L_0+\frac{1}{n}(\lambda +\nu )$

where $\lambda$ is the learning coefficient and $\nu$ is the singular fluctuation and $L_0$ is roughly the loss (the quantity that we can estimate from samples is actually slightly different, I’ll elide this) then (ii), which asks why neural networks on natural datasets have low generalisation error, is at least reduced to the question of why neural networks on natural datasets have low $L_0,\lambda ,\nu$.

I don’t know much about this question, and agree it is important and outstanding.

Again, I think this reduction is not trivial since the link between $\lambda ,\nu$ and generalisation error is nontrivial. Maybe at the end of the day this is the main thing we in fact disagree on :)

[Joar Skalse]

Anyway I’m guessing you’re probably willing to grant (i), based on SLT or your own views, and would agree the real bone of contention lies with (ii).

Yes, absolutely. However, I also don’t think that (i) is very mysterious, if we view things from a Bayesian perspective. Indeed, it seems natural to say that an ideal Bayesian reasoner should assign non-zero prior probability to all computable models, or something along those lines, and in that case, notions like “overparameterised” no longer seem very significant.

Maybe that has significant overlap with the critique of SLT you’re making?

Yes, this is basically exactly what my criticism of SLT is—I could not have described it better myself!

Again, I think this reduction is not trivial since the link between $\lambda$, $\nu$ and generalisation error is nontrivial.

I agree that this reduction is relevant and non-trivial. I don’t have any objections to this per se. However, I do think that there is another angle of attack on this problem that (to me) seems to get us much closer to a solution (namely, to investigate the properties of the parameter-function map).

[Daniel Murfet]

However, I do think that there is another angle of attack on this problem that (to me) seems to get us much closer to a solution (namely, to investigate the properties of the parameter-function map)

Seems reasonable to me!

[Daniel Murfet]

Re: the articles you link to. I think the second one by Carroll is quite careful to say things like “we can now understand why singular models have the capacity to generalise well” which seems to me uncontroversial, given the definitions of the terms involved and the surrounding discussion.

I agree that Jesse’s post has a title “Neural networks generalize because of this one weird trick” which is clickbaity, since SLT does not in fact yet explain why neural networks appear to generalise well on many natural datasets. However the actual article is more nuanced, saying things like “SLT seems like a promising route to develop a better understanding of generalization and the limiting dynamics of training”. Jesse gives a long list of obstacles to walking this route. I can’t find anything in the post itself to object to. Maybe you think its optimism is misplaced, and fair enough.

So I don’t really understand what claims about inductive bias or generalisation behaviour in these posts you think is invalid?

[Joar Skalse]

I think the second one by Carroll is quite careful to say things like “we can now understand why singular models have the capacity to generalise well” which seems to me uncontroversial, given the definitions of the terms involved and the surrounding discussion.

The title of the post is Why Neural Networks obey Occam’s Razor! It also cites Zhang et al, 2017, and immediately after this says that SLT can help explain why neural networks have the capacity to generalise well. This gives the impression that the post is intended to give a solution to problem (ii) in your other comment, rather than a solution to problem (i).

Jesse’s post includes the following expression:

$\text{Complex Singularities}⟺\text{Fewer Parameters}⟺\text{Simpler Functions}⟺\text{Better Generalization}$

I think this also suggests an equivocation between the RLCT measure and practical generalisation behaviour. Moreover, neither post contains any discussion of the difference between (i) and (ii).

[Daniel Murfet]

Well neural networks do obey Occam’s razor, at least according to the formalisation of that statement that is contained in the post (namely, neural networks when formulated in the context of Bayesian learning obey the free energy formula, a generalisation of the BIC which is often thought of as a formalisation of Occam’s razor).

I think that expression of Jesse’s is also correct, in context.

However, I accept your broader point, which I take to be: readers of these posts may naturally draw the conclusion that SLT currently says something profound about (ii) from my other post, and the use of terms like “generalisation” in broad terms in the more expository parts (as opposed to the technical parts) arguably doesn’t make enough effort to prevent them from drawing these inferences.

I have noticed people at the Berkeley meeting and elsewhere believing (ii) was somehow resolved by SLT, or just in a vague sense thinking SLT says something more than it does. While there are hard tradeoffs to make in writing expository work, I think your criticism of this aspect of the messaging around SLT on LW is fair and to the extent it misleads people it is doing a disservice to the ongoing scientific work on this important subject.

I’m often critical of the folklore-driven nature of the ML literature and what I view as its low scientific standards, and especially in the context of technical AI safety I think we need to aim higher, in both our technical and more public-facing work. So I’m grateful for the chance to have this conversation (and to anybody reading this who sees other areas where they think we’re falling short, read this as an invitation to let me know, either privately or in posts like this).

I’ll discuss the generalisation topic further with the authors of those posts. I don’t want to pre-empt their point of view, but it seems likely we may go back and add some context on (i) vs (ii) in those posts or in comments, or we may just refer people to this post for additional context. Does that sound reasonable?

At least right now, the value proposition I see of SLT lies not in explaining the “generalisation puzzle” but in understanding phase transitions and emergent structure; that might end up circling back to say something about generalisation, eventually.

[Joar Skalse]

Well neural networks do obey Occam’s razor, at least according to the formalisation of that statement that is contained in the post (namely, neural networks when formulated in the context of Bayesian learning obey the free energy formula, a generalisation of the BIC which is often thought of as a formalisation of Occam’s razor).

Would that not imply that my polynomial example also obeys Occam’s razor?

However, I accept your broader point, which I take to be: readers of these posts may naturally draw the conclusion that SLT currently says something profound about (ii) from my other post, and the use of terms like “generalisation” in broad terms in the more expository parts (as opposed to the technical parts) arguably doesn’t make enough effort to prevent them from drawing these inferences.

Yes, I think this probably is the case. I also think the vast majority of readers won’t go deep enough into the mathematical details to get a fine-grained understanding of what the maths is actually saying.

I’m often critical of the folklore-driven nature of the ML literature and what I view as its low scientific standards, and especially in the context of technical AI safety I think we need to aim higher, in both our technical and more public-facing work.

Yes, I very much agree with this too.

Does that sound reasonable?

Yes, absolutely!

At least right now, the value proposition I see of SLT lies not in explaining the “generalisation puzzle” but in understanding phase transitions and emergent structure; that might end up circling back to say something about generalisation, eventually.

I also think that SLT probably will be useful for understanding phase shifts and training dynamics (as I also noted in my post above), so we have no disagreements there either.

[Liam Carroll]

I would argue that the title is sufficiently ambiguous as to what is being claimed, and actually the point of contention in (ii) was discussed in the comments there too. I could have changed it to Why Neural Networks can obey Occam’s Razor, but I think this obscures the main point. Regular linear regression could also obey Occam’s razor (i.e. “simpler” models are possible) if you set high-order coefficients to 0, but the posterior of such models does not concentrate on those points in parameter space.

At the time of writing, basically nobody knew anything about SLT, so I think it was warranted to err on the side of grabbing attention in the introductory paragraphs and then explaining in detail further on with “we can now understand why singular models have the capacity to generalise well”, instead of caveating the whole topic out of existence before the reader knows what is going on.

As we discussed at Berkeley, I do like the polynomial example you give and this whole discussion has made me think more carefully about various aspects of the story, so thanks for that. My inclination is that the polynomial example is actually quite pathological and that there is a reasonable correlation between the RLCT and Kolmogorov complexity in practice (e.g. the one-node subnetwork preferred by the posterior compared to the two-node network in DSLT4), but I don’t know enough about Kolmogorov complexity to say much more than that.

[Joar Skalse]

I could have changed it to Why Neural Networks can obey Occam’s Razor, but I think this obscures the main point.

I think even this would be somewhat inaccurate (in my opinion). If a given parametric Bayesian learning machine does obey (some version of) Occam’s razor, then this must be because of some facts related to its prior, and because of some facts related to its parameter-function map. SLT does not say very much about either of these two things. What the post is about is primarily the relationship between the RLCT and posterior probability, and how this relationship can be used to reason about training dynamics. To connect this to Occam’s razor (or inductive bias more broadly), further assumptions and claims would be required.

At the time of writing, basically nobody knew anything about SLT

Yes, thank you so much for taking the time to write those posts! They were very helpful for me to learn the basics of SLT.

As we discussed at Berkeley, I do like the polynomial example you give and this whole discussion has made me think more carefully about various aspects of the story, so thanks for that.

I’m very glad to hear that! :)

My inclination is that the polynomial example is actually quite pathological and that there is a reasonable correlation between the RLCT and Kolmogorov complexity in practice

Yes, I also believe that! The polynomial example is definitely pathological, and I do think that low $\lambda$ almost certainly is correlated with simplicity in the case of neural networks. My point is more that the mathematics of SLT does not explain generalisation, and that additional assumptions definitely will be needed to derive specific claims about the inductive bias of neural networks.

[Dmitry Vaintrob]

I think there is some misunderstanding of what SLT says here, and you are identifying two distinct notions of complexity as the same, when in fact they are not. In particular, you have a line

“The generalisation bound that SLT proves is a kind of Bayesian sleight of hand, which says that the learning machine will have a good expected generalisation relative to the Bayesian prior that is implicit in the learning machine itself.”

I think this is precisely what SLT is saying, and this is nontrivial! One can say that a photon will follow a locally fastest route through a medium, even if this is different from saying that it will always follow the “simplest” route. SLT arguments always works relative to a loss landscape, and interpreting their meaning should (ideally) be done relative to the loss landscape. The resulting predictions are, nevertheless, nontrivial, and are sometimes confirmed. For example we have some work on this with Nina Rimsky.

You point at a different notion of complexity, associated to considering the parameter-function map. This also seems interesting, but is distinct from complexity phenomena in SLT (at least from the more basic concepts like the RLCT), and which is not considered in the basic SLT paradigm. Saying that this is another interesting avenue of study or a potentially useful measure of complexity is valid, but is a priori independent of criticism of SLT (and of course ideally, the two points of view could be combined).

Note that loss landscape considerations are more important than parameter-function considerations in the context of learning. For example it’s not clear in your example why f(x) = 0 is likely to be learned (unless you have weight regularization). Learning bias in a NN should most fundamentally be understood relative to the weights, not higher-order concepts like Kolmogorov complexity (though as you point out, there might be a relationship between the two).

Also I wanted to point out that in some ways, your “actual solution” is very close to the definition of RLCT from SLT. The definition of the RLCT is how much entropy you have to pay (in your language, the change in negative log probability of a random sample) to gain an exponential improvement of loss precision; i.e., “bits of specification per bit of loss”. See e.g. this article.

The thing is, the “complexity of f” (your K(f)) is not a very meaningful concept from the point of view of a neural net’s learning (you can try to make sense of it by looking at something like the entropy of the weight-to-function mapping, but then it won’t interact that much with learning dynamics). I think if you follow your intuitions carefully, you’re likely to precisely end up arriving at something like the RLCT (or maybe a finite-order approximation of the RLCT, associated to the free energy).

I have some criticisms of how SLT is understood and communicated, but I don’t think that the ones you mention seem that important to me. In particular, my intuition is that for purposes of empirical measurement of SLT parameters, the large-sample limit of realistic networks is quite large enough to see approximate singularities in the learning landscape, and that the SGD-sampling distinction is much more important than many people realize (indeed, there is no way to explain why generalizable networks like modular addition still sometimes memorize without understanding that the two are very distinct).

My main update in this field is that people should be more guided by empiricism and experiments, and less by competing paradigms of learning, which tend to be oversimplified and to fail to account for messy behaviors of even very simple toy networks. I’ve been pleasantly surprised by SLT making the same update in recent months.

[Joar Skalse]

I think this is precisely what SLT is saying, and this is nontrivial!

It is certainly non-trivial, in the sense that it takes many lines to prove, but I don’t think it tells you very much about the actual behaviour of neural networks.

Note that loss landscape considerations are more important than parameter-function considerations in the context of learning.

One of my core points is, precisely, to deny this claim. Without assumptions about the parameter function map, you cannot make inferences from the characteristics of the loss landscape to conclusions about generalisation behaviour, and understanding generalisation behaviour is crucial for understanding learning. (Unless you mean something like “convergence behaviour” when you say “in the context of learning”, in which case I agree, but then you would consider generalisation to be outside the scope of learning.)

For example it’s not clear in your example why f(x) = 0 is likely to be learned

My point is precisely that it is not likely to be learned, given the setup I provided, even though it should be learned.

Learning bias in a NN should most fundamentally be understood relative to the weights, not higher-order concepts like Kolmogorov complexity (though as you point out, there might be a relationship between the two).

There is a relationship between the two, and I claim that this relationship underlies the mechanism behind why neural networks work well compared to other learning machines.

The thing is, the “complexity of f” (your K(f)) is not a very meaningful concept from the point of view of a neural net’s learning

If we want to explain generalisation in neural networks, then we must explain if and how their inductive bias aligns with out (human) priors. Moreover, our human priors are (in most contexts) largely captured by computational complexity. Therefore, we must somewhere, in some way, connect neural networks to computational complexity.

indeed, there is no way to explain why generalizable networks like modular addition still sometimes memorize without understanding that the two are very distinct

Why not? The memorising solution has some nonzero “posterior” weight, so you would expect it to be found with some frequency. Does the empirical frequency of this solution deviate far from the theoretical prediction?

[Cleo Nardo]

My point is precisely that it is not likely to be learned, given the setup I provided, even though it should be learned.

How am I supposed to read this?

What most of us need from a theory of deep learning is a predictive, explanatory account of how neural networks actually behave. If neural networks learn functions which are RLCT-simple rather than functions which are Kolmogorov-simple, then that means SLT is the better theory of deep learning.

I don’t know how to read “x^4 has lower RLCT than x^2 despite x^2 being k-simpler” as a critique of SLT unless there is an implicit assumption that neural networks do in fact find x^2 rather than x^4.

[Joar Skalse]

A few things:

1. Neural networks do typically learn functions with low Kolmogorov complexity (otherwise they would not be able to generalise well).
2. It is a type error to describe a function as having low RLCT. A given function may have a high RLCT or a low RLCT, depending on the architecture of the learning machine.
3. The critique is against the supposition that we can use SLT to explain why neural networks generalise well in the small-data regime. The example provides a learning machine which would not generalise well, but which does fit all assumptions made my SLT. Hence, the SLT theorems which appear to prove that learning machines will generalise well when they are subject to the assumptions of SLT must in fact be showing something else.

My point is precisely that SLT does not give us a predictive account of how neural networks behave, in terms of generalisation and inductive bias, because it abstacts away from factors which are necessary to understand generalisation and inductive bias.

[Daniel Murfet]

I don’t understand the strong link between Kolmogorov complexity and generalisation you’re suggesting here. I think by “generalisation” you must mean something more than “low test error”. Do you mean something like “out of distribution” generalisation (whatever that means)?

[Roman Leventov]

Kolmogorov complexity is definitely a misleading path here, and it’s unfortunate that Joar chose it as the “leading” example of complexity in the post. Note this passage:

However, they do not give a detailed answer to the question of precisely which complexity measure they minimise—they merely show that this result holds for many different complexity measures. For example, I would expect that fully connected neural networks are biased towards functions with low Boolean circuit complexity, or something very close to that. Verifying this claim, and deriving similar results about other kinds of network architectures, would make it easier to reason about what kinds of functions we should expect a neural network to be likely or unlikely to learn. This would also make it easier to reason about out-of-distribution generalisation, etc.

This quote from the above comment is better:

If we want to explain generalisation in neural networks, then we must explain if and how their inductive bias aligns with out (human) priors. Moreover, our human priors are (in most contexts) largely captured by computational complexity. Therefore, we must somewhere, in some way, connect neural networks to computational complexity.

I’ve expressed this idea with some links here:

Bayesian Brain theorists further hypothesise that animal brains do effectively implement something like these “simple” algorithms (adjusted to the level of generality and sophistication of the world model each animal species needs) due to the strong evolutionary pressure on energy efficiency of the brain (“The free energy principle induces neuromorphic development”). The speed-accuracy tradeoffs in brain hardware add another kind of pressure that points in the same direction (“Internal feedback in the cortical perception–action loop enables fast and accurate behavior”).

Then if we combine two claims:

• Joar’s “DNNs are (kind of) Bayesian” (for the reasons that I don’t understand because I didn’t read their papers, so I just take his word here), and

• Fields et al.‘s “brains are ‘almost’ Bayesian because Bayesian learning is information-efficient (= energy-efficient), and there is a strong evolutionary pressure for brains in animals to be energy-efficient”,

is this an explanation explanation of DNNs’ remarkable generalisation ability? Or more quantification should be added to both of these claims to turn this into a good explanation?

[Joar Skalse]

Thank you for this—I agree with what you are saying here. In the post, I went with a somewhat loose equivocation between “good priors” and “a prior towards low Kolmogorov complexity”, but this does skim past a lot of nuance. I do also very much not want to say that the DNN prior is exactly towards low Kolmogorov complexity (this would be uncomputable), but only that it is mostly correlated with Kolmogorov complexity for typical problems.

[Joar Skalse]

Yes, I mostly just mean “low test error”. I’m assuming that real-world problems follow a distribution that is similar to the Solomonoff prior (i.e., that data generating functions are more likely to have low Kolmogorov complexity than high Kolmogorov complexity) -- this is where the link is coming from. This is an assumption about the real world, and not something that can be established mathematically.

[Nina Rimsky]

suppose we have a 500,000-degree polynomial, and that we fit this to 50,000 data points. In this case, we have 450,000 degrees of freedom, and we should by default expect to end up with a function which generalises very poorly. But when we train a neural network with 500,000 parameters on 50,000 MNIST images, we end up with a neural network that generalises well. Moreover, adding more parameters to the neural network will typically make generalisation better, whereas adding more parameters to the polynomial is likely to make generalisation worse.

Only tangentially related, but your intuition about polynomial regression is not quite correct. A large range of polynomial regression learning tasks will display double descent where adding more and more higher degree polynomials consistently improves loss past the interpolation threshold.

Examples from here:

Relevant paper

[Lucius Bushnaq]

IIRC this is probably the case for a broad range of non-NN models. I think the original Double Descent paper showed it for random Fourier features.

My current guess is that NN architectures are just especially affected by this, due to having even more degenerate behavioral manifolds, ranging very widely from tiny to large RLCTs.

[Joar Skalse]

That’s interesting, thank you for this!

[rotatingpaguro]

I’m trying to get a quick intuition of this. I’ve not read the papers.

My attempt:

• On a compact domain, any function can be uniformly approximated by a polynomial (Weierstrass)

• Powers explode quickly, so you need many terms to make a nice function with a power series, to correct the high powers at the edges

• As the domain gets larger, it is more difficult to make the approximation

So the relevant question is: how does the degree at training phase transition change with domain size, domain dimensionality, and Fourier series decay rate?

Does this make sense?

[Daniel Murfet]

I think that the significance of SLT is somewhat over-hyped at the moment

Haha, on LW that is either already true or at current growth rates will soon be true, but it is clearly also the case that SLT remains basically unknown in the broader deep learning theory community.

[Joar Skalse]

Yes, I meant specifically on LW and in the AI Safety community! In academia, it remains fairly obscure.

[Alexander Gietelink Oldenziel]

A comment by Patrick Foré, professor from University of Amsterdam

I’m a bit puzzled by this post.The RLCT is a function of (q,p,m), the sampling distribution q, prior distribution p and parameter-to-distribution map m of the statistical model. So it actually takes the parameter-to-distribution map m into account.
However, it was criticised that:
“3. SLT abstracts away from both the prior and the parameter-function map.”Also, in the chapter “Why the SLT Answer Fails” a rather complex parameterization of a model is constructed, where SLT then supposedly fails, by pointing to a rather complex (in contrast to a simple) model. But this is, of course, because SLT takes the (here complicated) parameter-to-distribution map m into account.
So it is unclear if the criticism now is that SLT actually takes m into account (but in a complicated way) or that it doesn’t …It was also said:
“4. Hence, SLT is at its core unable to explain generalisation behaviour.”
“SLT does not explain generalisation in neural networks.”It was shown in Watanabe’s grey book (but there still under a bit more restrictive assumptions), see Remark 6.7 (3) that the Bayes generalization error of Bayesian learning is asymptotically, for sample size n to infinity, given by:
E[Bg] := E[KL(q(x)||p(x|Dn))] ~ RLCT/​nSo SLT actually does say something about generalization in the Bayesian learning setting and it is a very satisfying answer imho (similar to what VC-dimension says about binary classification, but where RLCT is defined much more generally and does not just depend on the function class, but on the whole triple (q,p,m) and says something for the average case and not just for the worst case scenario).Of course, usually people don’t do proper Bayesian deep learning (they usually do MLE/​MAP estimation with SGD) and they also plot a different type of generalization error and are interested in different aspects (e.g. finite sample generalization, double-descent behaviour, etc.)
But this gap could be mentioned in the very beginning of the post (maybe even in a table ‘what we want’ vs ‘what SLT currently says’) and then it would be less surprising that SLT says something about something else than what (most/​some) people are interested in.Certainly, what is written under “The Actual Solution” is closer to how deep learning is done in practice. However, this is also an investigation into learning theory for singular models (that is not focused on RLCT), so can also be considered a part of SLT. Furthermore, nothing prevents us from investigating if and how it relates to quantities like RLCT, singular fluctuation, etc. (e.g. if it is providing upper or lower bounds on such quantities).Maybe the title of the post “My Criticism of Singular Learning Theory” should be replaced by “The deep learning community is interested in something else than what Singular Learning Theory currently provides”

[Daniel Murfet]

First of all, SLT is largely is based on examining the behaviour of learning machines in the limit of infinite data

I have often said that SLT is not yet a theory of deep learning, this question of whether the infinite data limit is really the right one being among one of the main question marks I currently see (I think I probably also see the gap between Bayesian learning and SGD as bigger than you do).

I’ve discussed this a bit with my colleague Liam Hodgkinson, whose recent papers https://​​arxiv.org/​​abs/​​2307.07785 and https://​​arxiv.org/​​abs/​​2311.07013 might be more up your alley than SLT.

My view is that the validity of asymptotics is an empirical question, not something that is settled at the blackboard. So far we have been pleasantly surprised at how well the free energy formula works at relatively low $n$ (in e.g. https://​​arxiv.org/​​abs/​​2310.06301). It remains an open question whether this asymptotic continues to provide useful insight into larger models with the kind of dataset size we’re using in LLMs for example.

[Joar Skalse]

I have often said that SLT is not yet a theory of deep learning, this question of whether the infinite data limit is really the right one being among one of the main question marks I currently see.

Yes, I agree with this. I think my main objections are (1) the fact that it mostly abstacts away from the parameter-function map, and (2) the infinite-data limit.

My view is that the validity of asymptotics is an empirical question, not something that is settled at the blackboard.

I largely agree, though depends somewhat on what your aims are. My point there was mainly that theorems about generalisation in the infinite-data limit are likely to end up being weaker versions of more general results from statistical and computational learning theory.

[Alexander Gietelink Oldenziel]

“My point there was mainly that theorems about generalisation in the infinite-data limit are likely to end up being weaker versions of more general results from statistical and computational learning theory.”

What general results from statistical and computational learning theory are you referring to here exactly?

[Joar Skalse]

I already posted this in response to Daniel Murfet, but I will copy it over here:

For example, the agnostic PAC-learning theorem says that if a learning machine $L$ (for binary classification) is an empirical risk minimiser with VC dimension $d$, then for any distribution $D$ over $X\times \{0,1\}$, if $L$ is given access to at least $\Omega \left(\right(d/{ϵ}^2)+(d/{ϵ}^2\left)\log \right(1/\delta \left)\right)$ data points sampled from $D$, then it will with probability at least $1-\delta$ learn a function whose (true) generalisation error (under $D$) is at most $ϵ$ worse than the best function which $L$ is able to express (in terms of its true generalisation error under $D$). If we assume that that $D$ corresponds to a function which $L$ can express, then the generalisation error of $L$ will with probability at least $1-\delta$ be at most $ϵ$.

This means that, in the limit of infinite data, $L$ will with probability arbitrarily close to 1 learn a function whose error is arbitrarily close to the optimal value (among all functions which $L$ is able to express). Thus, any empirical risk minimiser with a finite VC-dimension will generalise well in the limit of infinite data.

For a bit more detail, see this post.

[Cleo Nardo]

The impression I got was that SLT is trying to show why (transformers + SGD) behaves anything like an empirical risk minimiser in the first place. Might be wrong though.

[Joar Skalse]

To say that neural networks are empirical risk minimisers is just to say that they find functions with globally optimal training loss (and, if they find functions with a loss close to the global optimum, then they are approximate empirical risk minimisers, etc).

I think SLT effectively assumes that neural networks are (close to being) empirical risk minimisers, via the assumption that they are trained by Bayesian induction.

[Daniel Murfet]

Very loosely speaking, regions with a low RLCT have a larger “volume” than regions with high RLCT, and the impact of this fact eventually dominates other relevant factors.

I’m going to make a few comments as I read through this, but first I’d like to thank you for taking the time to write this down, since it gives me an opportunity to think through your arguments in a way I wouldn’t have done otherwise.

Regarding the point about volume. It is true that the RLCT can be written as (Theorem 7.1 of Watanabe’s book “Algebraic Geometry and Statistical Learning Theory”)

$\lambda ={lim}_{t\to 0}\frac{\log \left(V\right(at\left)/V\right(t\left)\right)}{\log a}$

where $V\left(t\right)={\int }_{K\left(w\right)<t}\phi \left(w\right)dw$ is the volume (according to the measure associated to the prior) of the set of parameters $w$ with KL divergence $K\left(w\right)$ between the model and truth less than $t$. For small $t$ we have $V\left(t\right)\approx c{t}^{\lambda }(-\log t{)}^{m-1}$ where $m$ is the multiplicity. Thus near critical points $w^{\ast }$ with lower RLCT small changes in the cutoff $t$ near $t\approx 0$ tend to change the volume of the set of almost true parameters more than near critical points with higher RLCTs.

My impression is that you tend to see this as a statement about flatness, holding over macroscopic regions of parameter space, and so you read the asymptotic formula for the free energy (where $W_{\alpha }$ is a region of parameter space containing a critical point $w_{\alpha }^{\ast }$)

$F_n\left(W_{\alpha }\right)\approx n{L}_n\left(w_{\alpha }^{\ast }\right)+{\lambda }_{\alpha }\log n-(m-1)\log \log n+O_P\left(1\right)$

as having a $\log n$ term that does little more than prefer critical points $w_{\alpha }^{\ast }$ that tend to dominate large regions of parameter space according to the prior. If that were true, I would agree this would be underwhelming (or at least, precisely as “whelming” as the BIC, and therefore not adding much beyond the classical story).

However this isn’t what the free energy formula says. Indeed the volume ${\int }_{W_{\alpha }}\phi \left(w\right)dw$ is a term that contributes only to the constant order term (this is sketched in Chen et al).

I claim it’s better to think of the learning coefficient $\lambda$ as being a measure of how many bits it takes to specify an almost true parameter with $K\left(w\right)<\frac{1}{n+1}$ once you know a parameter with $K\left(w\right)<1/n$, which is “microscopic” rather than “macroscopic” statement. That is, lower $\lambda$ means that a fixed decrease $\Delta K$ is “cheaper” in terms of entropy generated.

So the free energy formula isn’t saying “critical points $w_{\alpha }^{\ast }$ dominating large regions tend to dominate the posterior at large $n$” but rather “critical points $w_{\alpha }^{\ast }$ which require fewer bits /​ less entropy to achieve a fixed $\Delta K$ dominate the posterior for large $n$”. The former statement is both false and uninteresting, the second statement is true and interesting (or I think so anyway).

[Joar Skalse]

I’m going to make a few comments as I read through this, but first I’d like to thank you for taking the time to write this down, since it gives me an opportunity to think through your arguments in a way I wouldn’t have done otherwise.

Thank you for the detailed responses! I very much enjoy discussing these topics :)

My impression is that you tend to see this as a statement about flatness, holding over macroscopic regions of parameter space

My intuitions around the RLCT are very much geometrically informed, and I do think of it as being a kind of flatness measure. However, I don’t think of it as being a “macroscopic” quantity, but rather, a local quantity.

I think the rest of what you say coheres with my current picture, but I will have to think about it for a bit, and come back later!

[DanielFilan]

One argument sketch using SLT that NNs are biased towards low complexity solutions: suppose reality is generated by a width 3 network, and you’re modelling it with a width 4 network. Then, along with the generic symmetries, optional solutions also have continuous symmetries where you can switch which neuron is turned off.

Roughly, say neurons 3 and 4 have the same input weight vectors (so their activations are the same), but neuron 4′s output weight vector is all zeros. Then you can continuously scale up the output vector of neuron 4 while simultaneously scaling down the output vector of neuron 3 to leave the network computing the same function. Also, when neuron 4 has zero weights as inputs and outputs you can arbitrarily change the inputs or the outputs but not both.

Anyway, this means that when the data is generated by a slim neural net, optimal nets will have a good RLCT, but when it’s generated by a neural net of the right width, optimal nets will have a bad RLCT. So nets can learn simple data, and it’s easier for them to learn simple data than complex data—assuming thin neural nets count as simple.

This is basically a justification of something like your point 1, but AFAICT it’s closer to a proof in the SLT setting than in your setting.

[Joar Skalse]

Does this not essentially amount to just assuming that the inductive bias of neural networks in fact matches the prior that we (as humans) have about the world?

This is basically a justification of something like your point 1, but AFAICT it’s closer to a proof in the SLT setting than in your setting.

I think it could probably be turned into a proof in either setting, at least if we are allowed to help ourselves to assumptions like “the ground truth function is generated by a small neural net” and “learning is done in a Bayesian way”, etc.

[DanielFilan]

Does this not essentially amount to just assuming that the inductive bias of neural networks in fact matches the prior that we (as humans) have about the world?

No? It amounts to assuming that smaller neural networks are a better match for the actual data generating process of the world.

[Joar Skalse]

The assumption that small neural networks are a good match for the actual data generating process of the world, is equivalent to the assumption that neural networks have an inductive bias that gives large weight to the actual data generating process of the world, if we also append the claim that neural networks have an inductive bias that gives large weight to functions which can be described by small neural networks (and this latter claim is not too difficult to justify, I think).

[Liam Carroll]

Thanks for writing this out Joar, it is a good exercise of clarification for all of us.

Perhaps a boring comment, but I do want to push back on the title ever so slightly: imo it should be My Criticism of SLT Proponents, i.e. people (like me) who have interpreted some aspects in perhaps an erroneous fashion (according to you).

Sumio Watanabe is incredibly careful to provide highly precise mathematical statements with rigorous proofs and at no point does he make claims about the kind of “real world deep learning” phenomena being discussed here. The only sense in which it seems you critique the theory of SLT itself is that perhaps it isn’t as interesting as the number of pages taken to prove its main theorems suggests it should be, but even then it seems you agree that these are non-trivial statements.

I think it is important for people coming to the field for the first time to understand that the mathematical theory is incredibly solid, whilst its interpretation and applicability to broader “real world” problems is still an open question that we are actively working on.

[Joar Skalse]

Yes, I completely agree. The theorems that have been proven by Watanabe are of course true and non-trivial facts of mathematics; I do not mean to dispute this. What I do criticise is the magnitude of the significance of these results for the problem of understanding the behaviour of deep learning systems.

[Jesse Hoogland]

The easiest way to explain why this is the case will probably be to provide an example. Suppose we have a Bayesian learning machine with 15 parameters, whose parameter-function map is given by

$f\left(x\right)={\theta }_1+{\theta }_2{\theta }_{3}x+{\theta }_4{\theta }_5{\theta }_6{x}^2+{\theta }_7{\theta }_8{\theta }_9{\theta }_{10}{x}^3+{\theta }_{11}{\theta }_{12}{\theta }_{13}{\theta }_{14}{\theta }_{15}{x}^4,$

and whose loss function is the KL divergence. This learning machine will learn 4-degree polynomials.

I’m not sure, but I think this example is pathological. One possible reason for this to be the case is that the symmetries in this model are entirely “generic” or “global.” The more interesting kinds of symmetry are “nongeneric” or “local.”

What I mean by “global” is that each point $\theta =\{{\theta }_i{\}}_{i=1}^{15}$ in the parameter space $\Theta$ has the same set of symmetries (specifically, the product of a bunch of hyperboloids ${\theta }_2{\theta }_3=c_1,{\theta }_4{\theta }_5{\theta }_6=c_2,\dots$). In neural networks there are additional symmetries that are only present for a subset of the weights. My favorite example of this is the decision boundary annihilation (see below).

For the sake of simplicity, consider a ReLU network learning a 1D function (which is just piecewise linear approximation). Consider what happens when you you rotate two adjacent pieces so they end up sitting on the same line, thereby “annihilating” the decision boundary between them, so this now-hidden decision boundary no longer contributes to your function. You can move this decision boundary along the composite linear piece without changing the learned function, but this only holds until you reach the next decision boundary over. I.e.: this symmetry is local. (Note that real-world networks actually seem to take advantage of this property.)

This is the more relevant and interesting kind of symmetry, and it’s easier to see what this kind of symmetry has to do with functional simplicity: simpler functions have more local degeneracies. We expect this to be true much more generally — that algorithmic primitives like conditional statements, for loops, composition, etc. have clear geometric traces in the loss landscape.

So what we’re really interested in is something more like the relative RLCT (to the model class’s maximum RLCT). This is also the relevant quantity from a dynamical perspective: it’s relative loss and complexity that dictate transitions, not absolute loss or complexity.

This gets at another point you raised:

2. It is a type error to describe a function as having low RLCT. A given function may have a high RLCT or a low RLCT, depending on the architecture of the learning machine.

You can make the same critique of Kolmogorov complexity. Kolmogorov complexity is defined relative to some base UTM. Fixing a UTM lets you set an arbitrary constant correction. What’s really interesting is the relative Kolmogorov complexity.

In the case of NNs, the model class is akin to your UTM, and, as you show, you can engineer the model class (by setting generic symmetries) to achieve any constant correction to the model complexity. But those constant corrections are not the interesting bit. The interesting bit is the question of relative complexities. I expect that you can make a statement similar to the equivalence-up-to-a-constant of Kolmogorov complexity for RLCTs. Wild conjecture: given two model classes $W_1$ and $W_2$ and some true distribution $q\left(x\right)$, their RLCTs obey:

$$-c<f\left({\lambda }_1\right(q\left)\right)<f\left({\lambda }_2\right(q\left)\right)<c,$$

where $f$ is some monotonic function.

[Joar Skalse]

I’m not sure, but I think this example is pathological.

Yes, it’s artificial and cherry-picked to make a certain rhetorical point as simply as possible.

This is the more relevant and interesting kind of symmetry, and it’s easier to see what this kind of symmetry has to do with functional simplicity: simpler functions have more local degeneracies.¨

This is probably true for neural networks in particular, but mathematically speaking, it completely depends on how you parameterise the functions. You can create a parameterisation in which this is not true.

You can make the same critique of Kolmogorov complexity.

Yes, I have been using “Kolmogorov complexity” in a somewhat loose way here.

Wild conjecture: [...]

Is this not satisfied trivially due to the fact that the RLCT has a certain maximum and minimum value within each model class? (If we stick to the assumption that $\Theta$ is compact, etc.)

[Jesse Hoogland]

This is probably true for neural networks in particular, but mathematically speaking, it completely depends on how you parameterise the functions. You can create a parameterisation in which this is not true.

Agreed. So maybe what I’m actually trying to get at it is a statement about what “universality” means in the context of neural networks. Just as the microscopic details of physical theories don’t matter much to their macroscopic properties in the vicinity of critical points (“universality” in statistical physics), just as the microscopic details of random matrices don’t seem to matter for their bulk and edge statistics (“universality” in random matrix theory), many of the particular choices of neural network architecture doesn’t seem to matter for learned representations (“universality” in DL).

What physics and random matrix theory tell us is that a given system’s universality class is determined by its symmetries. (This starts to get at why we SLT enthusiasts are so obsessed with neural network symmetries.) In the case of learning machines, those symmetries are fixed by the parameter-function map, so I totally agree that you need to understand the parameter-function map.

However, focusing on symmetries is already a pretty major restriction. If a universality statement like the above holds for neural networks, it would tell us that most of the details of the parameter-function map are irrelevant.

There’s another important observation, which is that neural network symmetries leave geometric traces. Even if the RLCT on its own does not “solve” generalization, the SLT-inspired geometric perspective might still hold the answer: it should be possible to distinguish neural networks from the polynomial example you provided by understanding the geometry of the loss landscape. The ambitious statement here might be that all the relevant information you might care about (in terms of understanding universality) are already contained in the loss landscape.

If that’s the case, my concern about focusing on the parameter-function map is that it would pose a distraction. It could miss the forest for the trees if you’re trying to understand the structure that develops and phenomena like generalization. I expect the more fruitful perspective to remain anchored in geometry.

Is this not satisfied trivially due to the fact that the RLCT has a certain maximum and minimum value within each model class? (If we stick to the assumption that $\Theta$ is compact, etc.)

Hmm, maybe restrict $f$ so it has to range over $R$.

[Joar Skalse]

If a universality statement like the above holds for neural networks, it would tell us that most of the details of the parameter-function map are irrelevant.

I suppose this depends on what you mean by “most”. DNNs and CNNs have noticeable and meaningful differences in their (macroscopic) generalisation behaviour, and these differences are due to their parameter-function map. This is also true of LSTMs vs transformers, and so on. I think it’s fairly likely that these kinds of differences could have a large impact on the probability that a given type model will learn to exhibit goal-directed behaviour in a given training setup, for example.

The ambitious statement here might be that all the relevant information you might care about (in terms of understanding universality) are already contained in the loss landscape.

Do you mean the loss landscape in the limit of infinite data, or the loss landscape for a “small” amount of data? In the former case, the loss landscape determines the parameter-function map over the data distribution. In the latter case, my guess would be that the statement probably is false (though I’m not sure).

[Daniel Murfet]

I claim that this is fairly uninteresting, because classical statistical learning theory already gives us a fully adequate account of generalisation in this setting which applies to all learning machines, including neural networks

I’m a bit familiar with the PAC-Bayes literature and I think this might be an exaggeration. The linked post merely says that the traditional PAC-Bayes setup must be relaxed, and sketches some ways of doing so. Could you please cite the precise theorem you have in mind?

[Joar Skalse]

For example, the agnostic PAC-learning theorem says that if a learning machine $L$ (for binary classification) is an empirical risk minimiser with VC dimension $d$, then for any distribution $D$ over $X\times \{0,1\}$, if $L$ is given access to at least $\Omega \left(\right(d/{ϵ}^2)+(d/{ϵ}^2\left)\log \right(1/\delta \left)\right)$ data points sampled from $D$, then it will with probability at least $1-\delta$ learn a function whose (true) generalisation error (under $D$) is at most $ϵ$ worse than the best function which $L$ is able to express (in terms of its true generalisation error under $D$). If we assume that that $D$ corresponds to a function which $L$ can express, then the generalisation error of $L$ will with probability at least $1-\delta$ be at most $ϵ$.

This means that, in the limit of infinite data, $L$ will with probability arbitrarily close to 1 learn a function whose error is arbitrarily close to the optimal value (among all functions which $L$ is able to express). Thus, any empirical risk minimiser with a finite VC-dimension will generalise well in the limit of infinite data.

[Alexander Gietelink Oldenziel]

The linked post you wrote about classical learning theory states that the bounds PAC gives are far more loose than what we see in practice for Neural Networks. In the post you sketch some directions in which tighter bounds may be proven. It is my understanding that these directions have not been pursued further.

Given all that “Fully adequate account of generalization” seems like an overstatement, wouldn’t you agree?

At best we can say that PAC gives a nice toy model for thinking about notions like generalization and learnability as far as I can tell. Maybe I’m wrong- I’m not familiar with the literature- and I’d love to know more about what PAC & classical learning theory can tell us about neural networks.

[Joar Skalse]

I think that it gives us an adequate account of generalisation in the limit of infinite data (or, more specifically, in the case where we have enough data to wash out the influence of the inductive bias); this is what my original remark was about. I don’t think classical statistical learning theory gives us an adequate account of generalisation in the setting where the training data is small enough for our inductive bias to still matter, and it only has very limited things to say about out-of-distribution generalisation.

[Alexander Gietelink Oldenziel]

I see, thanks!

[jake_mendel]

I think I agree that SLT doesn’t offer an explanation of why NNs have a strong simplicity bias, but I don’t think you have provided an explanation for this either?

Here’s a simple story for why neural networks have a bias to functions with low complexity (I think it’s just spelling out in more detail your proposed explanation):

Since the Kolmogorov complexity of a function $f\left(x\right)$ is (up to a constant offset) equal to the minimum description length of the function, it is upper bounded by any particular way of describing the function, including by first specifying a parameter-function map, and then specifying the region of parameter space corresponding to the function. That means:

$$K\left(f\right)\le \ell \left(M\right)+\ell \left(f|M\right)+O\left(1\right)$$

where $\ell \left(M\right)$ is the minimum description length of the parameter function map, $\ell \left(f|M\right)$ is the minimum description length required to specify $f$ given $M$, and the $O\left(1\right)$ term comes from the fact that K complexity is only defined up to switching between UTMs. Specifying $f$ given $M$ entails specifying the region of parameter space $W_f$ corresponding to $f$ defined by $W_f=\left\{w|M\right(w)=f\}.$ Since we can use each bit in our description of $f$ to divide the parameter space in half, we can upper bound the mdl of $f$ given $M$ by $\ell \left(f|M\right)\le -{\log }_2|W_f|+{\log }_2|W|+O\left(1\right)$^[1] where $|W|$ denotes the size of the overall parameter space. This means that, at least asymptotically in $K\left(f\right)$, we arrive at

$$|W_f|\le 2^{-K\left(f\right)+O\left(1\right)}.$$

This is (roughly) a hand-wavey version of the Levin Coding Theorem (a good discussion can be found here). If we assume a uniform prior over parameter space, then $\varphi \left(f\right)\le 2^{-K\left(f\right)+O\left(1\right)}$. In words, this means that the prior assigned by the parameter function map to complex functions must be small. Now, the average probability assigned to each function in the set of possible outputs of the map is $1/N$ where $N$ is the number of functions. Since there are $2^{K_{\text{max}}}$ functions with K complexity at most $K_{\text{max}}$, the highest K complexity of any function in the model must be at least ${\log }_{2}N$ so, for simple parameter function maps, the most complex function in the model class must be assigned prior probability less than or equal to the average prior. Therefore if the parameter function map assigns different probabilities to different functions, at all, it must be biased against complex functions (modulo the $O\left(1\right)$ term)!

But, this story doesn’t pick out deep neural network architectures as better parameter function maps than any other. So what would make a parameter function map bad? Well, for a start the $O\left(1\right)$ term includes $\ell \left(M\right)$ — we can always choose a pathologically complicated parameter function map which specifically chooses some specific highly complex functions to be given a large prior by design. But even ignoring that, there are still low complexity maps that have very poor generalisation, for example polyfits. That’s because the expression we derived is only an upper bound: there is no guarantee that this bound should be tight for any particular choice of parameter-function map. Indeed, for a wide range of real parameter function maps, the tightness of this bound can vary dramatically:

This figure (from here) shows scatter plots of (an upper bound estimate of) the K complexity of a large set of functions, against the prior assigned to them by a particular choice of param function map.

It seems then that the question of why neural network architectures have a good simplicity bias compared to other architectures is not about why they do not assign high volume/​prior to extremely complicated functions — since this is satisfied by all simple parameter function maps — but why there are not many simple functions that they do not assign high prior to relative to other parameter-function maps — why the bottom left of these plots is less densely occupied, or occupied with less ‘useful’ functions, for NN architectures than other architectures. Of course, we know that there are simple functions that the NN inductive bias hates (for example simple functions with a for loop cannot be expressed easily by a feed forward NN), but we’d like to explain why they have fewer ‘blind spots’ than other architectures. Your proposed solution doesn’t address this part of the question I think?

Where SLT fits in is to provide a tool for quantifying $|W_f|$ for any particular $f$. That is, SLT provides a sort of ‘cause’ for how different functions occupy regions of parameter space of different sizes: namely that the size of $W_f$ can be measured by counting a sort of effective number of parameters present in a particular choice $w\in W_f$^[2]. Put another way, SLT says that if you specify $W_f$ by using each bit in your description to cut $W$ in half, then it will sort-of take $\hat{\lambda }\left(w_f^{\ast }\right)$ bits (the local learning coefficient at the most singular point in parameter space that maps to $f$) to describe $W$, so $K\left(f\right)\le \kappa \hat{\lambda }\left(w_f^{\ast }\right)$ for some constant $\kappa$ that is independent of $f$.

So your explanation says that any parameter function map is biased to low complexity functions, and SLT contributes a way to estimate the size of the parameter space assigned to a particular function, but neither addresses the question of why neural networks have a simplicity bias that is stronger than other parameter function maps.

1. ^

   Actually, I am pretty unsure how to do this properly. It seems like the number of bits required to specify that a point is inside some region in a space really ought to depend only on the fraction of the space occupied by the region, but I don’t know how to ensure this in general—I’d be keen to know how to do this. For example, if I have a 2d parameter space (bounded, so a large square), and $W_1$ is a random $10\times 10$ square, $W_2$ is a union of 100 randomly placed $1\times 1$ squares, does it take the same number of bits to find my way into either (remember, I don’t need to fully describe the region, just specify that I am inside it)? Or even more simply, if $W_3$ is the set of points within distance $\delta$ of the line $y=5$, I can specify I am within the region by specifying the $y$ coordinate up to resolution $\delta$, so $\ell \left(W_3\right)=-\log \delta +O\left(1\right)$. If $W_4$ is the set of points within distance $\delta$ of the line $y=x$, how do I specify that I am within $W_4$ in a number of bits that is asymptotically equal to $\ell \left(W_3\right)$ as $\delta \to 0$?

2. ^

   In fact, we might want to say that at some imperfect resolution/​finite number of datapoints, we want to treat a set of very similar functions as the same, and then the best point in parameter space to count effective parameters at is a point that maps to the function which gets the lowest loss in the limit of infinite data.

[Roman Leventov]

Another area which I think is promising is the study of random networks and random features. The results in this post suggest that training a neural network is functionally similar to randomly initialising it until you find a function that fits the training data. This suggests that we may be able to draw conclusions about what kinds of features a neural network is likely to learn, based on what kinds of features are likely to be created randomly.

It might be that you find statistical mechanics a useful way to analyse this behaviour of NNs, please check out Vanchurin’s statistical mechanics theory of machine learning.

[Joar Skalse]

Will do, thank you for the reference!

[dr_s]

Wait, what? In your polynomial model, the constant solution literally has 14 local degrees of freedom, you can vary any other parameter and it still works. It’s by far the one with the lowest RLCT.

[Joar Skalse]

You’re right, I put the parameters the wrong way around. I have fixed it now, thanks!

[bhauth]

Did you see my earlier post? Your explanation seems related to mine, but I also said that generalization capability depends on activation function choice. (There are lots of activation functions that don’t work for neural networks, in different ways.)

[Joar Skalse]

I think I recall reading that, but I’m not completely sure.

Note that the activation function affects the parameter-function map, and so the influence of the activation function is subsumed by the general question of what the parameter-function map looks like.