Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: SAE reconstruction errors are (empirically) pathological, published by wesg on March 29, 2024 on LessWrong.
Summary
Sparse Autoencoder (SAE) errors are empirically pathological: when a reconstructed activation vector is distance ϵ from the original activation vector, substituting a randomly chosen point at the same distance changes the next token prediction probabilities significantly less than substituting the SAE reconstruction[1] (measured by both KL and loss). This is true for all layers of the model (~2x to ~4.5x increase in KL and loss over baseline) and is not caused by feature suppression/shrinkage.
Assuming others replicate, these results suggest the proxy reconstruction objective is behaving pathologically. I am not sure why these errors occur but expect understanding this gap will give us deeper insight into SAEs while also providing an additional metric to guide methodological progress.
Introduction
As the interpretability community allocates more resources and increases reliance on SAEs, it is important to understand the limitation and potential flaws of this method.
SAEs are designed to find a sparse overcomplete feature basis for a model's latent space. This is done by minimizing the joint reconstruction error of the input data and the L1 norm of the intermediate activations (to promote sparsity):
However, the true goal is to find a faithful feature decomposition that accurately captures the true causal variables in the model, and reconstruction error and sparsity are only easy-to-optimize proxy objectives. This begs the questions: how good of a proxy objective is this? Do the reconstructed representations faithfully preserve other model behavior? How much are we proxy gaming?
Naively, this training objective defines faithfulness as L2. But, another natural property of a "faithful" reconstruction is that substituting the original activation with the reconstruction should approximately preserve the next-token prediction probabilities. More formally, for a set of tokens T and a model M, let P=M(T) be the model's true next token probabilities.
Then let QSAE=M(T|do(xSAE(x))) be the next token probabilities after intervening on the model by replacing a particular activation x (e.g. a residual stream state or a layer of MLP activations) with the SAE reconstruction of x. The more faithful the reconstruction, the lower the KL divergence between P and Q (denoted as DKL(P||QSAE)) should be.
In this post, I study how DKL(P||QSAE) compares to several natural baselines based on random perturbations of the activation vectors x which preserve some error property of the SAE construction (e.g., having the same l2 reconstruction error or cosine similarity). I find that the KL divergence is significantly higher (2.2x - 4.5x) for the residual stream SAE reconstruction compared to the random perturbations and moderately higher (0.9x-1.7x) for attention out SAEs.
This suggests that the SAE reconstruction is not faithful by our definition, as it does not preserve the next token prediction probabilities.
This observation is important because it suggests that SAEs make systematic, rather than random, errors and that continuing to drive down reconstruction error may not actually increase SAE faithfulness. This potentially indicates that current SAEs are missing out on important parts of the learned representations of the model. The good news is that this KL-gap presents a clear target for methodological improvement and a new metric for evaluating SAEs. I intend to explore this in future work.
Intuition: how big a deal is this (KL) difference?
For some intuition, here are several real examples of the top-25 output token probabilities at the end of a prompt when patching in SAE and ϵ-random reconstructions compared to the original model's next-token distribution (note the use of ...

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