Very concise summary of the procedure described in this paper:
1. Run the model once across a dataset to estimate loss curvature per MLP weight matrix via K-FAC (activation/gradient covariances).
2. Decompose each weight matrix into curvature-ordered components; low-curvature directions correspond most to verbatim memorization, higher curvature to shared/general mechanisms.
3. Edit by dropping the low-curvature subspace and keep only the top directions.
> In extending from studying per-example to bulk memorization, we propose a novel inversion of the
previous interpretation of loss curvature: while individual memorized points are associated with high curvature, the direction of curvature varies across examples, meaning that, averaged across multiple
examples, memorization directions are actually flatter than generalizing directions, which maintain a consistent moderate curvature across points
Ah! I figured I should be very circumspect in the question since I hadn't read in full and there could be some crazy reason it's actually the opposite.
This sounds more correct to me. I've read previously somewhere that better generalization is usually associated with wider, smoother minima, and this is why regularization is important, because it has a smoothing function on the loss landscape.
Yes. This is also not hard to see intuitively from scratch.
Say you have a smooth but highly flexible model y = f(x) and some data points you are fitting with a machine learning algorithm. For whatever reason, the algorithm decides it wants to reduce training error by interpolating some specific point, (x0,y0), without negatively affecting training error on nearby points. The direct, guaranteed successful way to do this is to adjust the model to y0 = f(x0) exactly on x0 by adding a Dirac delta there, leaving the rest of f exactly as-is. But this cannot be done on a differentiable model, as it would create a discontinuity. The next best thing that such a model can actually do is replace the Dirac delta with a smooth but very narrow bump (e.g. Gaussian). But this narrow bump will inevitably have extremely high curvature at x0, since the bump is flat at x0 and it has to merge with the neighborhood around x0 in a very short distance.
Think of driving: if you have to change lanes in a very short distance, you're going to have to steer hard. Steering is curvature.
The decomposition they use "averages out the points of high curvature" therefore those components of the decomposition which correspond to "higher curvature" are those components which are used across multiple data points. Therefore they are the "general reasoning"
Now, about the paper-that’s super interesting. I imagine the dream here is to distil down into a “reasoning” core. Or maybe reclaim space for more generalization. Lots of interesting use cases.
> Our work enhances the understanding of memorization in neural networks with practical applications towards removing it
Cool stuff. In a recent podcast Karpathy was also talking about this. He sees this as the next "target": models that don't memorise, because you can look it up in an oracle, but still keep the "reasoning" qualities.
How can you generalize without facts? They are the foundation on which generalization is built. Like programming without memorizing the keywords. Unless you make a distinction between facts that let you generalize, and facts that do not, like random ID numbers.
We want the LLM to learn the multiplication algorithm not an incomplete set of tables. The algorithm might be smaller and will be more complete.
Honestly, our technology has outpaced our epistemology. So we don't really know what a fact is or isn't. Are facts what we call our supervised learning experiences? You think the sun rises, no the earth spins. Your belief that the sun rises helps you predict sunset and sunrise. Your belief would be quaint to someone born and raised on a space station. Apollos chariot moves the sun across the sky doesn't it?
There is a related line of work that suggests spikes in the ESD are related to the generalization vs memorization too; e.g.,
From Spikes to Heavy Tails: Unveiling the Spectral Evolution of Neural Networks (https://openreview.net/pdf?id=DJHB8eBUnt)
Very concise summary of the procedure described in this paper:
1. Run the model once across a dataset to estimate loss curvature per MLP weight matrix via K-FAC (activation/gradient covariances).
2. Decompose each weight matrix into curvature-ordered components; low-curvature directions correspond most to verbatim memorization, higher curvature to shared/general mechanisms.
3. Edit by dropping the low-curvature subspace and keep only the top directions.
Thank you!
I think you may have accidentally switched low and high in #2, no? The abstract speaks of high curvature as associated with memorization:
> curvature for memorized training points is much sharper than non memorized
Actually, no! Look at this in the paper
> In extending from studying per-example to bulk memorization, we propose a novel inversion of the previous interpretation of loss curvature: while individual memorized points are associated with high curvature, the direction of curvature varies across examples, meaning that, averaged across multiple examples, memorization directions are actually flatter than generalizing directions, which maintain a consistent moderate curvature across points
Ah! I figured I should be very circumspect in the question since I hadn't read in full and there could be some crazy reason it's actually the opposite.
This sounds more correct to me. I've read previously somewhere that better generalization is usually associated with wider, smoother minima, and this is why regularization is important, because it has a smoothing function on the loss landscape.
Yes. This is also not hard to see intuitively from scratch.
Say you have a smooth but highly flexible model y = f(x) and some data points you are fitting with a machine learning algorithm. For whatever reason, the algorithm decides it wants to reduce training error by interpolating some specific point, (x0,y0), without negatively affecting training error on nearby points. The direct, guaranteed successful way to do this is to adjust the model to y0 = f(x0) exactly on x0 by adding a Dirac delta there, leaving the rest of f exactly as-is. But this cannot be done on a differentiable model, as it would create a discontinuity. The next best thing that such a model can actually do is replace the Dirac delta with a smooth but very narrow bump (e.g. Gaussian). But this narrow bump will inevitably have extremely high curvature at x0, since the bump is flat at x0 and it has to merge with the neighborhood around x0 in a very short distance.
Think of driving: if you have to change lanes in a very short distance, you're going to have to steer hard. Steering is curvature.
That's very reminiscent of the idea behind the SAM (Sharpness Aware Minimization) family of optimizers.
The decomposition they use "averages out the points of high curvature" therefore those components of the decomposition which correspond to "higher curvature" are those components which are used across multiple data points. Therefore they are the "general reasoning"
Thank you for this huge time saver.
Now, about the paper-that’s super interesting. I imagine the dream here is to distil down into a “reasoning” core. Or maybe reclaim space for more generalization. Lots of interesting use cases.
> Our work enhances the understanding of memorization in neural networks with practical applications towards removing it
Cool stuff. In a recent podcast Karpathy was also talking about this. He sees this as the next "target": models that don't memorise, because you can look it up in an oracle, but still keep the "reasoning" qualities.
How can you generalize without facts? They are the foundation on which generalization is built. Like programming without memorizing the keywords. Unless you make a distinction between facts that let you generalize, and facts that do not, like random ID numbers.
We want the LLM to learn the multiplication algorithm not an incomplete set of tables. The algorithm might be smaller and will be more complete.
Honestly, our technology has outpaced our epistemology. So we don't really know what a fact is or isn't. Are facts what we call our supervised learning experiences? You think the sun rises, no the earth spins. Your belief that the sun rises helps you predict sunset and sunrise. Your belief would be quaint to someone born and raised on a space station. Apollos chariot moves the sun across the sky doesn't it?
A very similar idea is presented here in the first 5 minutes of this recent talk. But more from observing a kink in loss curves.
https://youtu.be/UyK3DgWY7yw?si=NN3f9Erik8o_Nfbs