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Compute Optimal Inference and Provable Amortisation Gap in Sparse Autoencoders
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Proves that SAE linear-nonlinear encoders have a non-zero amortisation gap for sparse inference, then empirically compares SAE, MLP, sparse coding and SAE+ITO encoders on synthetic data and GPT-2 Layer 9 activations, finding MLP features score higher on automated interpretability.
Problem
Sparse autoencoder encoders are linear-nonlinear maps that may be insufficient for accurate sparse inference, creating an amortisation gap relative to optimisation-based sparse coding.
Approach
Decouple encoding from decoding to compare amortised encoders (SAE, MLP) with iterative sparse inference (sparse coding, SAE+ITO), and prove via a rank argument that a single-pass linear-nonlinear encoder cannot recover all K-sparse codes when M<N.
Key results
- In the known-dictionary setting, MLPs reach an MCC nearly 10% higher than the standard SAE at convergence.
- On GPT-2 Layer 9, MLP features have median automated-interpretability F1 of 0.83 versus about 0.6 for SAE and LCA features (Kruskal-Wallis H=1856.33, p<0.001).
- With unknown codes and dictionary, sparse coding achieves latent MCC over 10% higher than other methods on synthetic data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the single-layer SAE encoder with an MLP encoder | SAE on known sparse codes · Synthetic sparse inference, known sparse codes | MCC: approximately 0.1 higher | Helped |
| replace the single-layer SAE encoder with an MLP encoder | SAE with known dictionary · Synthetic sparse inference, known dictionary | MCC: nearly $10\%$ higher | Helped |
| optimise sparse coefficients at inference time using the learned SAE dictionary | SAE · Synthetic sparse inference, known dictionary | MCC | Helped |
| use sparse coding instead of an SAE | SAE · Synthetic sparse inference, unknown sparse codes and dictionary | latent MCC: over $10\%$ higher | Helped |
| use sparse coding instead of an SAE | SAE · Synthetic dictionary learning, unknown sparse codes and dictionary | dictionary MCC: approximately $10\%$ | Helped |
| use sparse coding instead of an SAE | SAE · Synthetic sparse inference across varying $N$, $M$, and $K$ | latent MCC | Helped |
| use sparse coding instead of other methods | Sparse inference methods · Synthetic sparse inference at matched L0 | MCC | Helped |
| replace the SAE encoder with an MLP encoder | SAE on GPT-2 Small Layer 9 activations · GPT-2 Small Layer 9 feature interpretability | median F1-score: around $0.6$ -> $0.83$ | Helped |
| replace the SAE encoder with an MLP encoder | SAE on GPT-2 Small Layer 9 activations · GPT-2 Small Layer 9 residual stream | normalised MSE: 0.061 -> 0.055 | Helped |
| increase MLP hidden width (ablation) | MLP autoencoder · Synthetic dictionary learning across three $N, M, K$ regimes | latent and dictionary MCC | Helped |
| include a bias parameter (ablation) | SAE, MLP, and LCA models · Synthetic dictionary learning | reconstruction loss | No clear effect |
| include a bias parameter (ablation) | MLP · Synthetic dictionary learning | latent MCC and L0 | Mixed |
Surprising findings
- MLP encoders yield more interpretable GPT-2 features than SAEs, despite MLP's higher FLOPs and more dead neurons (22% vs 11%).
- Sparse coding shows an initial decrease in performance before surpassing other methods in the unknown-codes setting.
Limitations
- LCA implementation not optimised for scale, so its interpretability may be underestimated.
- Analysis limited to constant sparsity and uncorrelated channels, which may not capture real-world complexity.
- SAE+ITO did not use advanced techniques such as matching pursuit.
Experimental setup
Datasets: OpenWebText synthetic sparse sources
Benchmarks: synthetic sparse recovery (MCC) OpenWebText GPT-2 Layer 9 interpretability
Metrics: MCC normalised MSE L0 F1-score for interpretability FLOPs
Baselines: SAE MLP encoder sparse coding SAE+ITO LCA
Builds on: sparse coding k-sparse autoencoders Gated SAE JumpReLU SAE LISTA
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