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Gradient surfing: the hidden role of regularization

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In particular, can we use noise to make a model grok even in the absence of regularization (which is currently a requirement to make models grok with SGD)?

Worth noting that you can get grokking in some cases without explicit regularization with *full batch gradient descent, *if you use an adaptive optimizer, due to the *slingshot mechanism*: https://arxiv.org/abs/2206.04817

Unfortunately, reproducing slingshots reliably was pretty challenging for me; I could consistently get it to happen with 2+ layer transformers but not reliably on 1 layer transformers (and not at all on 1-layer MLPs).

(As an aside, I also think grokking is not very interesting to study -- if you want a generalization phenomena to study, I'd just study a task without grokking, and where you can get immediately generalization or memorization depending on hyperparameters.)

As for other forms of noise inducing grokking: we do see grokking with dropout! So there's some reason to think noise -> grokking.

(Source: Figure 28 from https://arxiv.org/abs/2301.05217)

Also worth noting that grokking is pretty hyperparameter sensitive -- it's possible you just haven't found the right size/form of noise yet!

Unfortunately, reproducing slingshots reliably was pretty challenging for me; I could consistently get it to happen with 2+ layer transformers but not reliably on 1 layer transformers (and not at all on 1-layer MLPs).

Shallow/wide NNs seem to be bad in a lot of ways. Have you tried instead 'skinny' NNs with a bias towards depth, which ought to have inductive biases towards more algorithmic, less memorization-heavy solutions? (Particularly for MLPs, which are notorious for overfitting due to their power.)

Have you tried instead 'skinny' NNs with a bias towards depth,

I haven't -- the problem with skinny NNs is stacking MLP layers quickly makes things uninterpretable, and my attempts to reproduce slingshot -> grokking were done with the hope of interpreting the model before/after the slingshots.

That being said, you're probably correct that having more layers does seem related to slingshots.

(Particularly for MLPs, which are notorious for overfitting due to their power.)

What do you mean by power here?

What do you mean by power here?

Just a handwavy term for VC dimension, expressivity, number of unique models, or whatever your favorite technical reification of "can be real smart and learn complicated stuff" is.

The negative result tells us that the strong form of the claim "regularization = navigability" is probably wrong. Having a smaller weight norm actually is good for generalization (just as the learning theorists would have you believe). You'll have better luck moving along the set of minimum loss weights in the way that minimizes the norm than in any other way.

Have you seen the Omnigrok work? It directly argues that weight norm is directly related to grokking:

Similarly, Figure 7 from https://arxiv.org/abs/2301.05217 also makes this point, but less strongly:

That being said, it's possible that both group composition tasks (like the mod add stuff) and MNIST are pretty special datasets, in that generalizing solutions have small weight norm and memorization solutions have large weight norm. It might be worth constructing tasks where generalizing solutions have large weight norm, and seeing what happens.

In a previous post, I demonstrated that Brownian motion near singularities defies our expectations from "regular" physics. Singularities trap random motion and take up more of the equilibrium distribution than you'd expect from the Gibbs measure.

In the computational probability community, this is a well-known

pathology. Sampling techniques like Hamiltonian Monte Carlo get stuck in corners, and this is something to avoid. You typically don't want biased estimates of the distribution you're trying to sample.In deep learning, I argued, this behavior might be less a bug than a feature.

The claim of singular learning theory is that models near singularities have lower effective dimensionality. From Occam's razor, we know that simpler models generalize better, so if the dynamics of SGD get stuck at singularities, it would suggest an explanation (at least in part) for why SGD

works: the geometry of the loss landscape biases your optimizer towards good solutions.This is not a particularly novel claim. Similar versions of the claim been made before by Mingard et al. and Valle Pérez et al.. But from what I can tell, the proposed mechanism, of singularity "stickiness", is quite different.

Moreover, it offers a new possible explanation for the role of regularization. If exploring the set of points with minimum training loss is enough to get to generalization, then perhaps the role of regularizer is not just to privilege "simpler" functions but also to make exploration

possible.In the absence of regularization, SGD can't easily move between points of equal loss. When it reaches the bottom of a valley, it's pretty much stuck. Adding a term like weight decay breaks this invariance. It frees the neural network to surf the loss basin, so it can

accidentallystumble across better generalizing solutions.So could we improve generalization by exploring the bottom of the loss basin in other ways — without regularization or even without SGD? Could we, for example, get a model to grok through random drift?

…No. We can't.

That is to say I haven't succeeded yet. Still, in the spirit of "null results are results", let me share the toy model that motivated this hypothesis and the experiments that have (as of yet) failed to confirm it.

## The inspiration: a toy model

First, let's take a look at the model that inspired the hypothesis.

Let's begin by modifying the example of the previous post to include an optional regularization term controlled by λ:

U(x)=a(min(x02,x12))+λ2(x−c)2.We deliberately center the regularization away from the origin at c=(−1,−1) so it doesn't already privilege the singularity at the origin.

Now, instead of viewing U(x) as a potential and exploring it with Brownian motion, we'll treat it as a loss function and use stochastic gradient descent to optimize for x.

We'll start our optimizer at a uniformly sampled random point in this region and take T=100 steps down the gradient (with optional momentum controlled by β). After each gradient step, we'll inject a bit of Gaussian noise to simulate the "stochasticity." Altogether, the update rule for x is as follows:

x(t+1)←x(t)+η v(t+1)+δ(t),with momentum updated according to:

v(t+1)←β⋅v(t)+(1−β)⋅(−∇xU),and noise given by,

δ(t)i∼N(0,ϵ2).If we sample the final obtained position, x(T) over independent initializations, then, in the absence of regularization and in the presence of a small noise term, we'll get a distribution that looks like the figure on the left.

Unlike the case of random motion, the singularity at the origin is now

repulsive. Good luck finding those simple solutions now.However, as soon as we turn on the regularization (middle figure) or increase the noise term (figure on the right), the singularity once again becomes favored. This is true

even though the origin no longer minimizes the overall loss.So noise and regularization

^{[1]}bias SGD towards singularities. If you buy that singularities correspond to simpler solutions, this might mean a novel, unexplored^{[2]}inductive bias towards generalization.## Grokking through random motion

Let's test this hypothesis in a more realistic setting.

Here's one application: can we use random drift at the bottom of the loss basin to induce grokking?

^{[3]}In other words, given a model with low training error but high test error, can we improve test performance just by increasing the amount of "lateral" motion (that preserves rather than lowers loss).In particular, can we use noise to make a model grok even in the absence of regularization (which is currently a requirement to make models grok with SGD)?

If this were true, it wouldn't give us enough evidence to distinguish the contribution of singularities from other factors like simple functions taking up more weight-space volume, but it'd be a start.

Unfortunately, it seems pretty difficult to grok without regularization.

^{[3]}Some of the techniques I've used to explore the loss basin have had minor success (up to a ∼5% sustained bump in test accuracy over Adam without weight decay). But nothing approaches the sustained grokking of Adam with weight decay.

## Weird regularizations and additional noise

One way to encourage lateral motion is to abuse the regularization term. I tried both a repulsive variant (where I reward the model for displacement from its SGD-primed starting point) and a variant that selected for large weight norms (by punishing distance from a larger weight norm). I also tried a variant that simply adds extra noise after each gradient step.

All of these improve test performance somewhat. What seems to happen is that these variants dislodge the model from a very fragile initial equilibrium to find a new slightly better and more stable solution. It outperforms Adam without weight decay but quickly plateaus (and, in the case of "antiregularization", slowly declines after reaching a peak).

Of the three, antiregularization is able to reach the highest improvements in accuracy (up to 40%), but the improvements are unstable, and final performance is worse than it started out. Still, the fact that improving generalization is at all possible while increasing weight norm is intriguing. And the fact that the variance in performance is so high could be evidence that favors an explanation in terms of singularities over flat basins, but high-dimensional spaces are weird, and the results are anything but conclusive.

Note: these variants only work for small-to-zero momentum, and they work best near the inflection point of 50% test accuracy.

## Hamiltonian Monte Carlo

Another idea to explore the set of minimum loss points is to steal from physics: the trick behind Hamiltonian Monte Carlo is to view the loss as a potential energy and then to simulate a physical particle moving through that potential energy landscape.

From the equipartition theorem, we can associate the starting loss to an inverse temperature, which describes a physical system with that loss as its average.

The system will still accept changes that decrease the loss, but it will also occasionally accept changes that increase the loss. Over the long run, these two forces will balance out, so the expected loss remains constant.

Though this would be the "cleanest" form of exploration (it is precisely the Brownian motion discussed previously), it doesn't seem to work (regardless of the choice of hyperparameters and of where I start HMC along the grokking curve).

Then again, I also can't get this to work even in the presence of regularization. In this case, by lowering the temperature, HMC should limit to a slow version of gradient descent. The code seems right, so I'm inclined to write it off to high-dimensional spaces being hard and my compute budget being limited.

## What does this mean for singularities?

The negative result tells us that the strong form of the claim "regularization = navigability" is probably wrong. Having a smaller weight norm actually is good for generalization (just as the learning theorists would have you believe). You'll have better luck moving along the set of minimum loss weights in the way that minimizes the norm than in any other way.

But the observation that you can, for a time, increase generalization performance by selecting for much larger norms suggests we can't outright reject the weaker version of the claim. Simply exploring the minimum loss set may —

still— privilege simpler solutions.^{^}Momentum doesn't have much of an effect on this example.

^{^}From what I've seen so far.

^{^}On the standard modular addition task.

^{^}Or maybe fortunately. Otherwise, this would seem a whole lot more capabilities-risky.