I originally ran some of these experiments on 7B and got very different results, that PCA plot of 7B looks familiar (and bizarre).

I found that the PCA plot for 7B for larger_than and smaller_than individually looked similar to that for 13B, but that the PCA plot for larger_than + smaller_than looked degenerate in the way I screenshotted. Are you saying that your larger_than + smaller_than PCA looked familiar for 7B?

I suppose there are two things we want to separate: "truth" from likely statements, and "truth" from what humans think (under some kind of simulacra framing).  I think this approach would allow you to do the former, but not the latter.  And to be honest, I'm not confident on TruthfulQA's ability to do the latter either.

Agreed on both points.

We differ slightly from the original GoT paper in naming, and use got_cities to refer to both the cities and neg_cities datasets. The same is true for sp_en_trans and larger_than. We don't do this for cities_cities_{conj,disj} and leave them unpaired.

Thanks for clarifying! I'm guessing this is what's making the GoT datasets much worse for generalization (from and to) in your experiments. For 13B, it mostly seemed to me that training on negated statements helped for generalization to other negated statements, and that pairing negated statements with unnegated statements in training data usually (but not always) made generalization to unnegated datasets a bit worse. (E.g. the cities -> sp_en_trans generalization is better than cities + neg_cities -> sp_en_trans generalization.)

Very cool! Always nice to see results replicated and extended on, and I appreciated how clear you were in describing your experiments.

Do smaller models also have a generalised notion of truth?

In my most recent revision of GoT^[1] we did some experiments to see how truth probe generalization changes with model scale, working with LLaMA-2-7B, -13B, and -70B. Result: truth probes seems to generalize better for larger models. Here are the relevant figures.

Some other related evidence from our visualizations:

We summed things up like so, which I'll just quote in its entirety:

Overall, these visualizations suggest a picture like the following: as LLMs scale (and perhaps, also as a fixed LLM progresses through its forward pass), they hierarchically develop and linearly represent increasingly general abstractions. Small models represent surface-level characteristics of their inputs; these surface-level characteristics may be sufficient for linear probes to be accurate on narrow training distributions, but such probes are unlikely to generalize out-of-distribution. Large models linearly represent more abstract concepts, potentially including abstract notions like “truth” which capture shared properties of topically and structurally diverse inputs. In middle regimes, we may find linearly represented concepts of intermediate levels of abstraction, for example, “accurate factual recall” or “close association” (in the sense that “Beijing” and “China” are closely associated). These concepts may suffice to distinguish true/false statements on individual datasets, but will only generalize to test data for which the same concepts
suffice.

How do we know we’re detecting truth, and not just likely statements?

One approach here is to use a dataset in which the truth and likelihood of inputs are uncorrelated (or negatively correlated), as you kinda did with TruthfulQA. For that, I like to use the "neg_" versions of the datasets from GoT, containing negated statements like "The city of Beijing is not in China." For these datasets, the correlation between truth value and likelihood (operationalized as LLaMA-2-70B's log probability of the full statement) is strong and negative (-0.63 for neg_cities and -.89 for neg_sp_en_trans). But truth probes still often generalize well to these negated datsets. Here are results for LLaMA-2-70B (the horizontal axis shows the train set, and the vertical axis shows the test set).

We also find that the probe performs better than LDA in-distribution, but worse out-of-distribution:

Yep, we found the same thing -- LDA improves things in-distribution, but generalizes work than simple DIM probes.

Why does got_cities_cities_conj generalise well?

I found this result surprising, thanks! I don't really have great guesses for what's going on. One thing I'll say is that it's worth tracking differences between various sorts of factual statements. For example, for LLaMA-2-13B it generally seemed to me that there was better probe transfer between factual recall datasets (e.g. cities and sp_en_trans, but not larger_than). I'm not really sure why the conjunctions are making things so much better, beyond possibly helping to narrow down on "truth" beyond just "correct statement of factual recall." 

I'm not surprised that cities_cities_conj and cities_cities_disj are so qualitatively different -- cities_cities_disj has never empirically played well with the other datasets (in the sense of good probe transfer) and I don't really know why. 

1. ^{^}

   This is currently under review, but not yet on arxiv, sorry about that! Code in the nnsight branch here. I'll try to come back to add a link to the paper once I post it or it becomes publicly available on OpenReview, whichever happens first.

This comment is about why we were getting different MSE numbers. The answer is (mostly) benign -- a matter of different scale factors. My parallel comment, which discusses why we were getting different CE diff numbers is the more important one.

When you compute MSE loss between some activations $x$ and their reconstruction $\hat{x}$, you divide by variance of $x$, as estimated from the data in a batch. I'll note that this doesn't seem like a great choice to me. Looking at the resulting training loss:

$\parallel x-\hat{x}{\parallel }_2^2/Var\left(x\right)+\lambda \parallel f{\parallel }_1$

where $f$ is the encoding of $x$ by the autoencoder and $\lambda$ is the L1 regularization constant, we see that if you scale $x$ by some constant $\alpha$, this will have no effect on the first term, but will scale the second term by $\alpha$. So if activations generically become larger in later layers, this will mean that the sparsity term becomes automatically more strongly weighted.

I think a more principled choice would be something like

$\parallel x-\hat{x}{\parallel }_2+\lambda \parallel f{\parallel }_1$

where we're no longer normalizing by the variance, and are also using sqrt(MSE) instead of MSE. (This is what the dictionary_learning repo does.) When you scale $x$ by a constant $\alpha$, this entire expression scales by a factor of $\alpha$, so that the balance between reconstruction and sparsity remains the same. (On the other hand, this will mean that you might need to scale the learning rate by $1/\alpha$, so perhaps it would be reasonable to divide through this expression by $\parallel x{\parallel }_2$? I'm not sure.)

Also, one other thing I noticed: something which we both did was to compute MSE by taking the mean over the squared difference over the batch dimension and the activation dimension. But this isn't quite what MSE usually means; really we should be summing over the activation dimension and taking the mean over the batch dimension. That means that both of our MSEs are erroneously divided by a factor of the hidden dimension (768 for you and 512 for me).

This constant factor isn't a huge deal, but it does mean that:

1. The MSE losses that we're reporting are deceptively low, at least for the usual interpretation of "mean squared error"
2. If we decide to fix this, we'll need to both scale up our L1 regularization penalty by a factor of the hidden dimension (and maybe also scale down the learning rate).

This is a good lesson on how MSE isn't naturally easy to interpret and we should maybe just be reporting percent variance explained. But if we are going to report MSE (which I have been), I think we should probably report it according to the usual definition.

Yep, as you say, @Logan Riggs figured out what's going on here: you evaluated your reconstruction loss on contexts of length 128, whereas I evaluated on contexts of arbitrary length. When I restrict to context length 128, I'm able to replicate your results.

Here's Logan's plot for one of your dictionaries (not sure which)

and here's my replication of Logan's plot for your layer 1 dictionary

Interestingly, this does not happen for my dictionaries! Here's the same plot but for my layer 1 residual stream output dictionary for pythia-70m-deduped

(Note that all three plots have a different y-axis scale.)

Why the difference? I'm not really sure. Two guesses:

1. The model: GPT2-small uses learned positional embeddings whereas Pythia models use rotary embeddings
2. The training: I train my autoencoders on variable-length sequences up to length 128; left padding is used to pad shorter sequences up to length 128. Maybe this makes a difference somehow.

In terms of standardization of which metrics to report, I'm torn. On one hand, for the task your dictionaries were trained on (reconstruction activations taken from length 128 sequences), they're performing well and this should be reflected in the metrics. On the other hand, people should be aware that if they just plug your autoencoders into GPT2-small and start doing inference on inputs found in the wild, things will go off the rails pretty quickly. Maybe the answer is that CE diff should be reported both for sequences of the same length used in training and for arbitrary-length sequences?

My SAEs also have a tied decoder bias which is subtracted from the original activations. Here's the relevant code in dictionary.py

def encode(self, x):
        return nn.ReLU()(self.encoder(x - self.bias))
    
    def decode(self, f):
        return self.decoder(f) + self.bias
    
    def forward(self, x, output_features=False, ghost_mask=None):
            [...]
            f = self.encode(x)
            x_hat = self.decode(f)
            [...]
            return x_hat

Note that I checked that our SAEs have the same input-output behavior in my linked colab notebook. I think I'm a bit confused why subtracting off the decoder bias had to be done explicitly in your code -- maybe you used dictionary.encoder and dictionary.decoder instead of dictionary.encode and dictionary.decode? (Sorry, I know this is confusing.) ETA: Simple things I tried based on the hypothesis "one of us needs to shift our inputs by +/- the decoder bias" only made things worse, so I'm pretty sure that you had just initially converted my dictionaries into your infrastructure in a way that messed up the initial decoder bias, and therefore had to hand-correct it.

I note that the MSE Loss you reported for my dictionary actually is noticeably better than any of the MSE losses I reported for my residual stream dictionaries! Which layer was this? Seems like something to dig into.

At the time that I made this post, no, but this has been implemented in dictionary_learning since I saw your suggestion to do so in your linked post.

Another sanity check: when you compute CE loss using the same code that you use when computing CE loss when activations are reconstructed by the autoencoders, but instead of actually using the autoencoder you just plug the correct activations back in, do you get the same answer (~3.3) as when you evaluate CE loss normally?

In the notebook I link in my original comment, I check that the activations I get out of nnsight are the same as the activations that come from transformer_lens. Together with the fact that our sparsity statistics broadly align, I'm guessing that the issue isn't that I'm extracting different activations than you are.

Repeating my replication attempt with data from OpenWebText, I get this:

Broadly speaking, same story as above, except that the MSE losses look better (still not great), and that the CE reconstructed looks very bad for layer 1.

I don't much padding at all, that might be a big difference too.

Seems like there was a typo here -- what do you mean?

Logan Riggs reports that he tried to replicate your results and got something more similar to you. I think Logan is making decisions about padding and tokenization more like the decisions you make, so it's possible that the difference is down to something around padding and tokenization.

Possible next steps:

• Can you report your MSE Losses (instead of just variance explained)?
• Can you try to evaluate the residual stream dictionaries in the 5_32768 set released here? If you get CE reconstructed much better than mine, then it means that we're computing CE reconstructed in different ways, where your way consistently reports better numbers. If you get CE reconstructed much worse than mine, then it might mean that there's a translation error between our codebases (e.g. using different activations).