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Efficient Spectral Control of Partially Observed Linear Dynamical Systems
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Proposes Double Spectral Control (DSC), a convex improper-learning controller for partially observed LDS under adversarial disturbances that uses two levels of Hankel-eigenvector spectral filters, achieving O(sqrt(T)/gamma^11) regret with polylog(T/gamma) per-step runtime.
Problem
Online control of partially observed linear dynamical systems with adversarial disturbances and adversarially chosen convex losses, competing with the best stable linear dynamical controller while keeping runtime efficient in the stability margin.
Approach
A two-level spectral approximation: LDCs are approximated by a long-memory open-loop spectral-projection controller over the natural observation sequence, which is then expressed as a convolution over double-filtered (lifted) observations using universal Hankel-eigenvector filters; the resulting policy is linear in learnable parameters and trained by projected online gradient descent.
Key results
- DSC achieves regret O(sqrt(T)/gamma^11) against the best diagonalizably stable LDC, matching the best known regret rate for this setting.
- Per-round runtime is polylog(T/gamma), exponentially improving over the polynomial dependence on the stability margin of GRC.
- Paper reports only preliminary empirical evaluations supporting the theory (appendix; details not in provided text).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DSC's two-level spectral approximation instead of GRC | Partially observed LDS with adversarial disturbances · Partially observed LDS with adversarial disturbances | runtime: \(\tilde{O}\left(\text{poly}\left(\gamma^{-1}\right)\log T\right)\) -> \( \mathrm{polylog}\left(T/\gamma\right)\) | Helped |
| Use DSC instead of GRC | Partially observed LDS with adversarial disturbances · Partially observed LDS with adversarial disturbances | regret: \(\tilde{O}\left(\text{poly}\left(\gamma^{-1}\right)\sqrt{T}\right)\) -> \(\tilde{O}\left(\text{poly}\left(\gamma^{-1}\r | No clear effect |
Limitations
- Assumes known system matrices (A, B, C) and full-information loss feedback; no bandit extension.
- Competitor class limited to diagonalizably stable LDCs; complex eigenstructure not covered.
- Primarily theoretical; evaluation on real-world control tasks absent.
Experimental setup
Metrics: regret per-step runtime
Baselines: GRC GPC AdaptOn LQG
Builds on: spectral filtering gradient response controller online gradient descent gradient perturbation controller
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