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From Generalization Analysis to Optimization Designs for State Space Models
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Derives a data-dependent generalization bound for state space models that couples the SSM kernel with the mean and covariance of the input process, then uses it to propose a scale-based initialization rule and a regularizer for SSM training.
Problem
Existing generalization bounds for SSMs and linear RNNs use only input and parameter norms and ignore the temporal dependencies of sequence data, which leaves SSM initialization and regularization without a principled basis.
Approach
A data-dependent bound whose dominant term is the SSM kernel weighted by the input's pointwise standard deviation and mean. This yields (1) a closed-form rescaling of the output matrix C at initialization so output scale is O(1) regardless of temporal structure, and (2) a regularizer that penalizes this term instead of parameter norms or weight decay.
Key results
- On LRA, S4-Legs average test accuracy rises from 86.89 (baseline) to 87.18 with the initialization rescaling alone and 86.97 with regularization alone; S4D-Legs improves from 86.38 to 86.82 with rescaling and 86.60 with regularization alone.
- On small S4D-Legs models, combining rescaling and regularization gives average accuracy 82.25 vs 81.03 baseline; small S4-Legs reaches 81.61 vs 81.01.
- On synthetic Gaussian noise (b=0.01), the regularizer lowers test MSE from 4.70 to 0.60 when combined with rescaling, and the rescaling reduces training MSE from 2.50 to 0.20 at b=0.01.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add initialization rescaling based on the generalization measure (ablation) | S4-Legs on synthetic Gaussian white noise · Gaussian white noise, b=1, 0.1, 0.01 | training loss (MSE): 0.15 -> 0.11; 0.67 -> 0.27; 2.50 -> 0.20 | Helped |
| Add generalization-bound regularization (ablation) | S4-Legs on synthetic Gaussian white noise · Gaussian white noise, b=1, 0.1, 0.01 | test loss (MSE): 0.25 -> 0.20; 1.01 -> 0.75; 4.70 -> 1.06 | Helped |
| Add initialization rescaling and regularization (ablation) | S4-Legs on synthetic Gaussian white noise · Gaussian white noise, b=1, 0.1, 0.01 | test loss (MSE): 0.25 -> 0.18; 1.01 -> 0.59; 4.70 -> 0.60 | Helped |
| Add generalization-bound regularization (ablation) | S4-Legs · LRA tasks overall | test accuracy | Mixed |
| Add generalization-bound regularization (ablation) | S4D-Legs · LRA tasks overall | test accuracy | Helped |
| Add initialization rescaling (ablation) | S4-Legs and S4D-Legs · LRA benchmark | test accuracy | Mixed |
| Combine initialization rescaling and generalization-bound regularization (ablation) | S4-Legs · LRA Image | accuracy: 87.41 -> 88.63 | Helped |
| Combine initialization rescaling and generalization-bound regularization (ablation) | S4D-Legs · LRA Image | accuracy: 86.69 -> 88.12 | Helped |
| Use regularization coefficient λ=10^{-3} instead of λ=0 (ablation) | S4-Legs · LRA ListOps | accuracy: 61.16 -> 61.63 | Helped |
| Use regularization coefficient λ=10^{-3} instead of λ=0 (ablation) | S4-Legs · LRA Retrieval | accuracy: 91.21 -> 88.25 | Hurt |
| Use regularization coefficient λ=10^{-3} instead of λ=0 (ablation) | S4-Legs · LRA Pathfinder | accuracy: 95.89 -> 89.06 | Mixed |
| Use regularization coefficient λ=10^{-4} on PathX (ablation) | S4-Legs and S4D-Legs · LRA PathX | test accuracy | Hurt |
Surprising findings
- Adding the rescaling alone barely changes average LRA accuracy for S4-Legs (86.89 to 87.18 at most), with the authors attributing this to batch/layer norm reducing the temporal dependence, so the method's gain is smaller on real data than on synthetic data.
- Weight decay on the hidden matrix A dramatically hurts ListOps for S4-Legs (49.90 vs 61.16 baseline).
Limitations
- Theory covers only single-layer linear SSMs; the multi-layer and nonlinear extension is left open and the method uses a heuristic per-layer rescaling.
- Feature dependencies are ignored by treating all features as independent and averaging over the feature dimension.
- The regularizer is highly sensitive to lambda on Pathfinder and PathX, where large values cause training to fail (e.g., 50.54% accuracy).
Experimental setup
Datasets: Long Range Arena (ListOps, Text, Retrieval, Image, Pathfinder, PathX) synthetic Gaussian white noise sequences
Benchmarks: Long Range Arena synthetic Gaussian white noise regression
Metrics: test accuracy test MSE training MSE generalization measure psi^2/sqrt(n) running time per epoch
Baselines: S4-Legs without proposed methods S4D-Legs without proposed methods weight decay on A filter norm regularization
Builds on: S4 HiPPO Rademacher complexity
Closest papers by meaning
- Autocorrelation Matters: Understanding the Role of Initialization Schemes for State Space Models (ICLR 2025 · similarity 0.91)
- Generalization Error Analysis for Selective State-Space Models Through the Lens of Attention (NeurIPS 2025 · similarity 0.89)
- A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments (ICML 2025 · similarity 0.87)
- HOPE for a Robust Parameterization of Long-memory State Space Models (ICLR 2025 · similarity 0.87)
- Provable Benefits of Complex Parameterizations for Structured State Space Models (NeurIPS 2024 · similarity 0.87)
- State Space Models are Provably Comparable to Transformers in Dynamic Token Selection (ICLR 2025 · similarity 0.86)
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