Stable and Steerable Sparse Autoencoders with Weight Regularization

Piotr Jedryszek; Oliver M. Crook

stat.MLMar 4, 2026

Stable and Steerable Sparse Autoencoders with Weight Regularization

Authors: Piotr Jedryszek, Oliver M. Crook

Organizations: Department of Biology, University of Oxford, Oxford, UK · Kavli Institute for Nanoscience Discovery, University of Oxford, Oxford, UK · Department of Chemistry, University of Oxford, Oxford, UK

Abstract

Sparse autoencoders (SAEs) are widely used to extract human-interpretable features from neural network activations, but their learned features can vary substantially across random seeds and training choices. To improve stability, we studied weight regularization by adding L1 or L2 penalties on encoder and decoder weights, and evaluate how regularization interacts with common SAE training defaults. On MNIST, we observe that L2 weight regularization produces a core of highly aligned features and, when combined with tied initialization and unit-norm decoder constraints, it dramatically increases cross-seed feature consistency. For TopK SAEs trained on language model activations (Pythia-70M-deduped), adding a small L2 weight penalty increased the fraction of features shared across three random seeds and roughly doubles steering success rates, while leaving the mean of automated interpretability scores essentially unchanged. Finally, in the regularized setting, activation steering success becomes better predicted by auto-interpretability scores, suggesting that regularization can align text-based feature explanations with functional controllability.

Figures & tables

Figure 1: MNIST cosine similarity and feature visualizations. Left column (A–C): Histograms of encoder–decoder cosine similarity for base, L1, and L2 SAEs (1,568 latents). L2 creates a bimodal distribution with a small high-alignment core. Right panels (D–I): Feature pairs showing encoder and decoder as 28 $\times$ 28 heatmaps (blue = negative, red = positive). Left subcolumns show random samples; right subcolumns show high cosine-similarity features. L2’s high-alignment features capture clean strokes and curves. The plotted features come from SAEs without decoder norm constraints.

Figure 2: MNIST shared vs. random feature visualization. Encoder weights as 28 $\times$ 28 heatmaps (blue = negative, red = positive). Top two rows: random features; bottom two rows: features shared between SAEs with seed 0 and 2. Shared features capture clean strokes and curves; random features appear noisy. Features shown are from the SAE with tied weights and constrained decoder but no weight penalty.

Setting	Weight penalty	Mean max cos	Frac paired ( $>0.7$ )	Frac shared (All)	Frac shared (Alive)
Untied init	none	0.2400	3.57%	0.00%	0.00%
Tied init (constr. Dec)	none	0.4783	48.28%	1.74%	1.74%
Untied init	L1 ( $\lambda_{w}=10^{-3}$ )	0.3260	4.66%	3.21%	0.60%
Tied init (constr. Dec)	L1 ( $\lambda_{w}=10^{-3}$ )	0.5486	51.62%	14.07%	14.10%
Untied init	L2 ( $\lambda_{w}=10^{-5}$ )	0.5061	7.38%	4.91%	2.90%
Tied init (constr. Dec)	L2 ( $\lambda_{w}=10^{-5}$ )	0.3858	42.88%	12.54%	22.50%

Table 1: MNIST cross-seed feature consistency (3 seeds) with unit-norm decoder columns. Weight regularization interacts strongly with decoder constraints and tied initialization; L2 regularization increases the fraction of strictly shared features by an order of magnitude.

Figure 3: Encoder–decoder cosine similarity distributions for regularized (“reg”) vs. unregularized (“no reg”) SAEs across architectures.Features with zero encoder or decoder norm are excluded; for TopK-L2, the majority of features have collapsed to zero norm during training (see Appendix C ). Distributions are of the Pareto-best SAE for each architecture and the unregularized SAE with corresponding

k

Figure 4: Pythia-70M TopK: cross-seed feature consistency metrics across sparsity levels (

k

) for unregularized (blue) and L2-regularized (orange) SAEs. L2 weight regularization substantially increases sharedness, particularly among alive features.

Figure 5: Decoder pairwise cosine similarity ( $c_{\text{dec}}$ ) across TopK sparsity levels (mean across 3 seeds; across-seed SDs are $\leq 2\times 10^{-4}$ and barely visible at this scale). Left: computed over all features; L2 reduces $c_{\text{dec}}$ , partly driven by dead features. Right: computed over alive features only (encoder–decoder alignment $>0.1$ ). At $k{=}40$ , alive-feature orthogonality is nearly identical; at higher $k$ , surviving L2 features are more orthogonal than the full unregularized dictionary.

Figure 6: Pythia-70M TopK

k{=}40

: (A) Steering success rate (LLM judge score

\geq 4

); L2 regularization roughly doubles the rate. (B) Auto-interpretability score distributions remain similar across conditions. (C) Spearman correlation between auto-interpretability and steering success; regularization strengthens the link.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: MNIST reconstruction MSE. Boxplots measured on the first 250 MNIST examples for three SAE variants (base, L1, L2) under four latent selection strategies: full SAE (all latents active), top-250 high-cosine-similarity latents, 250 random latents, and all latents except the top-250. A small number of high-cosine similarity latents accounts for most reconstruction capacity in L2-regularized SAEs.

Figure 8: TopK cosine similarity including dead features. Encoder–decoder cosine similarity distributions for regularized vs. unregularized TopK SAEs, retaining features where the encoder or decoder norm is zero. The majority of L2-regularized features collapse to zero norm during training.

Figure 9: SAEBench Pareto analysis. Aggregated top-3 Pareto configurations per architecture. Regularized models frequently appear on the Pareto frontier. Radar plots showing normalized SAEBench metrics. Red = Pareto-optimal; gray = dominated. Axes: TPP@10, CE score, SCR@10, SP Top-1 (higher = better).

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Sparse autoencoders decompose neural network activations into sparse features to study, interpret, and control learned representations.

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Sparse Autoencoders (SAEs) are widely used to interpret large language models by decomposing activations into sparse, human-understandable features, but scaling to large dictionaries exposes fundamental challenges. Systematic studies reveal pervasive feature splitting that fragments coherent concepts into non-atomic latents and widespread feature absorption that creates arbitrary exceptions in general features, severely compromising latent reliability. These issues stem from inconsistent latent assignment across samples: without cross-sample constraints, per-sample optimization often allows a single underlying concept to be inconsistently distributed across multiple redundant or interfering latents. To address this, we introduce C

^2

R (Cross-sample Consistency Regularization). C

^2

R explicitly encourages that each semantic feature is consistently represented by a unified latent across the batch by penalizing the co-activation of directionally similar latents. Comprehensive evaluation demonstrates that C

^2

R effectively mitigates both splitting and absorption while, crucially, preserving reconstruction fidelity, providing a principled solution that enhances latent interpretability without degrading model performance. Source code is available at https://github.com/hr-jin/Cross-sample-Consistency-Regularization.

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