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Measuring Progress in Dictionary Learning for Language Model Interpretability with Board Game Models
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Proposes supervised metrics (board reconstruction and coverage) for sparse autoencoder quality using chess and Othello GPT models with ground-truth board state properties, plus a p-annealing SAE training method that matches Gated SAEs.
Problem
Evaluating sparse autoencoder quality for language model interpretability is hard because there is no ground-truth set of interpretable features; existing proxies (L0, loss recovered, autointerpretability) do not directly measure whether features capture meaningful concepts.
Approach
Use LMs trained on chess and Othello transcripts, where board state properties give formal ground-truth interpretable features, to define supervised coverage and board reconstruction metrics; and introduce p-annealing, which anneals an L_p^p sparsity penalty from p=1 toward p=0.2 with coefficient annealing, to train standard SAEs competitive with Gated SAEs.
Key results
- Standard SAEs trained with p-annealing reach coverage comparable to Gated SAEs trained without p-annealing on chess board-state properties.
- Board reconstruction for trained-GPT SAEs reaches 0.85 (chess) and 0.95 (Othello), versus 0.98 and 0.99 for linear probes.
- Coverage and reconstruction distinguish SAEs with 4096 vs 8192 hidden dimensions, which unsupervised proxies (loss recovered, L0) do not.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add p-annealing to Standard SAE training instead of using a constant L1 penalty (ablation) | Standard SAE · Chess and Othello board-game models | coverage | Helped |
| Add p-annealing to Standard SAE training instead of using a constant L1 penalty (ablation) | Standard SAE · Chess and Othello board-game models | board reconstruction | Helped |
| Use Gated SAE architecture instead of Standard SAE architecture (ablation) | SAE · Chess board-game model | loss recovered | Helped |
| Increase SAE hidden dimension from 4096 to 8192 (ablation) | Standard SAE · Chess and Othello board-game models | coverage | Helped |
| Increase SAE hidden dimension from 4096 to 8192 (ablation) | SAE · Chess and Othello board-game models | loss recovered | No clear effect |
Surprising findings
- Unsupervised proxy metrics (loss recovered, L0) rate SAEs with 4096 and 8192 hidden dimensions equally, while the supervised metrics show clear gains from larger dictionaries.
- Linear probes fall below F1 0.95 on 6 of 15 chess strategy properties, suggesting chess-GPT does not represent these linearly.
Limitations
- Metrics depend on researcher-chosen board state properties, which may miss relevant features.
- Applicable only to chess and Othello domains, with unclear generalization to natural language.
- Evaluates how well SAEs capture model representations, not downstream interpretability utility.
Experimental setup
Datasets: synthetic Othello game transcripts Lichess chess games
Benchmarks: chess board state properties Othello board state properties
Metrics: coverage board reconstruction F1 L0 loss recovered relative reconstruction bias
Baselines: Standard SAE (L1) Gated SAE linear probe random GPT SAE
Builds on: sparse autoencoders Gated SAE dictionary_learning linear probes p-continuation
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