Good question! The short answer is that we tried the simplest thing and it happened to work well. (The additional computational cost of higher order kernels is negligible.) We did look at other kernels but my intuition, borne out by the results (to be included in a v2 shortly, along with some typos and bugfixes to the pseudo-code), was that this would not make much difference. This is because we're using KDE in a very high variance regime already, and yet the SAEs seem to train fine: given a batch size of 4096 and bandwidth of 0.001, hardly any activations (even for the most frequent features) end up having kernels that capture the threshold (i.e. that are included in the gradient estimate). So it's not obvious that improving the bias-variance trade-off slightly by switching to a different kernel is going to make that much difference. In this sense, the way we use KDE here is very different from e.g. using KDE to visualise empirical distributions, and so we need to be careful about how we transfer intuitions between these domains.

You can't do JumpReLU -> ReLU in the current setup, as there would be no threshold parameters to train with the STE (i.e. it would basically train a ReLU SAE with no sparsity penalty). In principle you should be able to train the other SAE parameters by adding more pseudo derivatives, but we found this didn't work as well as just training the threshold (so there was then no point in trying this ablation). L0 -> L1 leads to worse Pareto curves (I can't remember off the top of my head which side of Gated, but definitely not significantly better than Gated) - it's a good question whether this resolves the high frequency features; my guess is it would (as I think Gated SAEs basically approximate JumpReLU SAEs with a L1 loss) but we didn't check this.

Thanks! I was worried about how well KDE would work because of sparsity, but in practice it seems to work quite well. My intuition is that the penalty only matters for more frequent features (i.e. features that typically do fire, perhaps multiple times, within a 4096 batch), plus the threshold is only updated a small amount at each step (so the noise averages out reliably over many steps). In any case, we tried increasing batch size and turning on momentum (with the idea this would reduce noise), but neither seemed to particularly help performance (keeping the number of tokens constant). One thing I suspect would further improve things is to have a per-feature bandwidth parameter, as I imagine the constant bandwidth we use is too narrow for some features and too wide for others; but to do this we also need to find a reliable way to adapt the bandwidth during training.

We found that exactly that form of sparsity penalty did improve shrinkage with standard (ungated) SAEs, and provide a decent boost to loss recovered at low L0. (We didn't evaluate interpretability though.) But then we hit upon Gated SAEs which looked even better, and for which modifying the sparsity penalty in this way feels less necessary, so we haven't experimented with combining the two.

Good question - we're planning to post an update on this point about combining the new sparsity penalty from Anthropic with Gated SAEs. The TL;DR is that you can replace the L1 term in the Gated SAE loss with the analogous (gated feature magnitudes dotted with decoder magnitudes) sparsity term introduced by Anthropic and thereby do away with the decoder norms constraint and resampling. If you're going to do this, you also need to either unfreeze the decoder in the auxiliary task, or freeze the decoder weights where they appear in the sparsity penalty; both attain reasonably similar performance, and are definitely better than having the decoder weights frozen in one place but not the other. Put together, this seems to a marginal hit (versus the original Gated loss with L1 penalty and resampling) when comparing Pareto curves, but may be worth to the extent it simplifies training (with this loss function, the SAE training loop just becomes a vanilla neural network training loop).

PS With either the original (L1-based) loss or the modified loss of the previous paragraph, some of the other improvements suggested in the Anthropic post -- in particular, initializing the encoder weights to the transpose of the decoder weights (only at initialisation, not tying them thereafter), and warming up lambda. My point about the new loss not being Pareto better than L1 applies only if you compare like with like -- i.e. apply these other improvements in both cases.

Thanks for sharing your findings - this was an interesting idea to test out! I played around with the notebook you linked to on this and noticed that the logistic regression training accuracy is also pretty low for earlier layers when using the encoded hidden representations. This was initially surprising (surely it should be easy to overfit with such a high dimensional input space and only ~1000 examples?) until I noticed that the number of 'on' features is pretty low, especially for early layer SAEs. 

For example, the layer 2 SAE only has (the same) 2 features on over all examples in the dataset, so effectively you're training a classifier after doing a dimensionality reduction down to 2 dimensions. This may be a tall order even if you used (say) PCA to choose those 2 dimensions, but in the case of the pretrained SAE those two dimensions were chosen to optimise reconstruction on the full data distribution (of which this dataset is rather unrepresentative). The upshot is that unless you're lucky (and the SAE happened to pick features that correspond to sentiment), it makes sense you lose a lot of classification performance.

In contrast, the final SAEs have hundreds of features that are 'on' over the dataset, so even if none of those features directly relate to sentiment, the chances are good that you have preserved enough of the structure in the original hidden state to be able to recover sentiment. On the other hand, even at this end of the spectrum, note you haven't really projected to a higher dimensional space - you've gone from ~1000 dimensions to a similar or fewer number of effective dimensions - so it's not so surprising performance still doesn't match training a classifier on the hidden states directly.

All in all, I think this gave me a couple of useful insights:

• It's important to have really, really high fidelity with SAEs if you want to keep L0 (number of on features) low while at the same time be able to use the SAE for very narrow distribution analysis. (E.g. in this case, if the layer 2 SAE really had encoded the concept of sentiment, then it wouldn't have mattered that only 2 features were on on average across the dataset.)
• I originally shared your initial hypothesis (about projecting to a higher dimensional space making concepts more separable), but have updated to thinking that I shouldn't think of sparse "high dimensional" projections in the same way as dense projections. My new mental model for sparse projections is that you're actually projecting down to a lower dimensional space, but where the projection is task dependent (i.e. the SAE's relu chooses which projections it thinks are relevant). (Think of it a bit like a mixture of experts dimensionality reduction algorithm.) So the act of projection will only help with classification performance if the dimensions chosen by the filter are actually relevant to the problem (which requires a really good SAE), otherwise you're likely to get worse performance than if you hadn't projected at all.