Because I don't think this is realistically useful, I don't think this at all reduces my probability that your techniques are fake and your models of interpretability are wrong.

Maybe the groundedness you're talking about comes from the fact that you're doing interp on a domain of practical importance?

??? Come on, there's clearly a difference between "we can find an Arabic feature when we go looking for anything interpretable" vs "we chose from the relatively small set of practically important things and succeeded in doing something interesting in that domain". I definitely agree this isn't yet close to "doing something useful, beyond what well-tuned baselines can do". But this should presumably rule out some hypotheses that current interpretability results are due to an extreme streetlight effect?

(I suppose you could have already been 100% confident that results so far weren't the result of extreme streetlight effect and so you didn't update, but imo that would just make you overconfident in how good current mech interp is.)

(I'm basically saying similar things as Lawrence.)

Sounds plausible, but why does this differentially impact the generalizing algorithm over the memorizing algorithm?

Perhaps under normal circumstances both are learned so fast that you just don't notice that one is slower than the other, and this slows both of them down enough that you can see the difference?

Daniel Filan: But I would’ve guessed that there wouldn’t be a significant complexity difference between the frequencies. I guess there’s a complexity difference in how many frequencies you use.

Vikrant Varma: Yes. That’s one of the differences: how many you use and their relative strength and so on. Yeah, I’m not really sure. I think this is a question we pick out as a thing we would like to see future work on.

My pet hypothesis here is that (a) by default, the network uses whichever frequencies were highest at initialization (for which there is significant circumstantial evidence) and (b) the amount of interference differs significantly based on which frequencies you use (which in turn changes the quality of the logits holding parameter norm fixed, and thus changes efficiency).

In principle this can be tested by randomly sampling frequency sets, simulating the level of interference you'd get, using that to estimate the efficiency + critical dataset size for that grokking circuit. This gives you a predicted distribution over critical dataset sizes, which you could compare against the actual distribution.

Tbc there are other hypotheses too, e.g. perhaps different frequency sets are easier / harder to implement by the neural network architecture.

This suggestion seems less expressive than (but similar in spirit to) the "rescale & shift" baseline we compare to in Figure 9. The rescale & shift baseline is sufficient to resolve shrinkage, but it doesn't capture all the benefits of Gated SAEs.

The core point is that L1 regularization adds lots of biases, of which shrinkage is just one example, so you want to localize the effect of L1 as much as possible. In our setup L1 applies to $\text{ReLU}\left({\pi }_{\text{gate}}\right(x\left)\right)$, so you might think of ${\pi }_{\text{gate}}$ as "tainted", and want to use it as little as possible. The only thing you really need L1 for is to deter the model from setting too many features active, i.e. you need it to apply to one bit per feature (whether that feature is on / off). The Heaviside step function makes sure we are extracting just that one bit, and relying on ${\text{f}}_{\text{mag}}$ for everything else.

Thinking on this a bit more, this might actually reflect a general issue with the way we think about feature shrinkage; namely, that whenever there is a nonzero angle between two vectors of the same length, the best way to make either vector close to the other will be by shrinking it.

This was actually the key motivation for building this metric in the first place, instead of just looking at the ratio $\frac{E\left[||\hat{x}|{|}^2\right]}{E\left[||x|{|}^2\right]}$. Looking at the $\gamma$ that would optimize the reconstruction loss ensures that we're capturing only bias from the L1 regularization, and not capturing the "inherent" need to shrink the vector given these nonzero angles. (In particular, if we computed $\frac{E\left[||\hat{x}|{|}^2\right]}{E\left[||x|{|}^2\right]}$ for Gated SAEs, I expect that would be below 1.)

I think the main thing we got wrong is that we accidentally treated $E[||\hat{x}-x|{|}^2]$ as though it were $E[||\hat{x}-\gamma x|{|}^2]$. To the extent that was the main mistake, I think it explains why our results still look how we expected them to -- usually $\gamma$ is going to be close to 1 (and should be almost exactly 1 if shrinkage is solved), so in practice the error introduced from this mistake is going to be extremely small.

We're going to take a closer look at this tomorrow, check everything more carefully, and post an update after doing that. I think it's probably worth waiting for that -- I expect we'll provide much more detailed derivations that make everything a lot clearer.

Possibly I'm missing something, but if you don't have $L_{\text{aux}}$, then the only gradients to $W_{\text{gate}}$ and $b_{\text{gate}}$ come from $L_{\text{sparsity}}$ (the binarizing Heaviside activation function kills gradients from $L_{\text{reconstruct}}$), and so ${\pi }_{\text{gate}}$ would be always non-positive to get perfect zero sparsity loss. (That is, if you only optimize for L1 sparsity, the obvious solution is "none of the features are active".)

(You could use a smooth activation function as the gate, e.g. an element-wise sigmoid, and then you could just stick with $L_{\text{incorrect}}$ from the beginning of Section 3.2.2.)

Is it accurate to summarize the headline result as follows?

• Train a Transformer to predict next tokens on a distribution generated from an HMM.
• One optimal predictor for this data would be to maintain a belief over which of the three HMM states we are in, and perform Bayesian updating on each new token. That is, it maintains $p(\text{hidden state}=H_i)$.
• Key result: A linear probe on the residual stream is able to reconstruct $p(\text{hidden state}=H_i)$.

(I don't know what Computational Mechanics or MSPs are so this could be totally off.)

EDIT: Looks like yes. From this post:

Part of what this all illustrates is that the fractal shape is kinda… baked into any Bayesian-ish system tracking the hidden state of the Markov model. So in some sense, it’s not very surprising to find it linearly embedded in activations of a residual stream; all that really means is that the probabilities for each hidden state are linearly represented in the residual stream.