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
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 × 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 × 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 ( λw=10−3 )
0.3260
4.66%
3.21%
0.60%
Tied init (constr. Dec)
L1 ( λw=10−3 )
0.5486
51.62%
14.07%
14.10%
Untied init
L2 ( λw=10−5 )
0.5061
7.38%
4.91%
2.90%
Tied init (constr. Dec)
L2 ( λw=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 ( cdec ) across TopK sparsity levels (mean across 3 seeds; across-seed SDs are ≤2×10−4 and barely visible at this scale). Left: computed over all features; L2 reduces cdec , 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 ≥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).
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 C2R (Cross-sample Consistency Regularization). C2R 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 C2R 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.
Haoran Jin, Xiting Wang, Shijie Ren +2
University of Science and Technology of China · State Key Laboratory of Cognitive Intelligence · Gaoling School of Artificial Intelligence, Renmin University of China
Sparse autoencoders (SAEs) are one of the main methods to interpret the inner workings of deep neural networks (DNNs), decomposing activations into higher-dimensional features. However, they exhibit critical shortcomings where a large fraction of features are never activated and are unstable. Despite variants of SAEs that attempt to mitigate these issues, they require additional data, resampling, or training. We propose the aligned training, a parameter-free reparameterization of SAEs that simultaneously improves reconstruction quality, eliminates dead features, and significantly enhances stability across training seeds. Our approach is motivated by an overlooked observation that SAE feature quality, measured by the inner product between encoder and decoder directions (which we call the alignment score), follows a bimodal distribution across all modern architectures. The proposed aligned training enforces a geometric constraint between the encoder and decoder such that their inner product equals one for every feature, which removes a source of degeneracy in the SAE training without adding any hyperparameters. Across multiple models, dictionary sizes, and sparsity levels, the aligned training shows Pareto improvements on the SAEBench benchmarks. Beyond improving dead features, stability and reconstruction, our method readily integrates with techniques in mechanical interpretability such as Top/BatchTop-K architectures and p-Annealing. Overall, the aligned training substantially improves feature quality and stability of SAE without computational complexity or cost.
Michał Brzozowski, Neo Christopher Chung
Samsung AI Center, Warsaw, Poland · University of Warsaw, Poland
Sparse autoencoders (SAEs) are widely used in mechanistic interpretability to project LLM activations onto sparse latent spaces. However, sparsity alone is an imperfect proxy for interpretability, and current training objectives often result in brittle latent representations. SAEs are known to be prone to feature absorption, where general features are subsumed by more specific ones due to co-occurrence, degrading interpretability despite high reconstruction fidelity. Recent negative results on Out-of-Distribution (OOD) performance further underscore broader robustness related failures tied to under-specified training objectives. We address this by proposing a masking-based regularization that randomly replaces tokens during training to disrupt co-occurrence patterns. This improves robustness across SAE architectures and sparsity levels reducing absorption, enhancing probing performance, and narrowing the OOD gap. Our results point toward a practical path for more reliable interpretability tools.