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Efficient Spectral Control of Partially Observed Linear Dynamical Systems

Anand Brahmbhatt, Gon Buzaglo, Sofiia Druchyna, Elad Hazan

TL;DR

This work addresses online control of partially observed linear dynamical systems with adversarial disturbances by introducing Double Spectral Control (DSC), a two-level spectral learning framework that reframes control as online convex optimization over spectral features. DSC builds a universal Hankel-based spectral basis to convexify the disturbance-response map and delivers a provable regret bound of $Regret_T(DSC) = \tilde{O}(\sqrt{T}/\gamma^{11})$ with per-step polylogarithmic time, achieving an exponential improvement in the dependence on the stability margin $\gamma$ over prior methods. Theoretical analysis connects spectral controllers to a class of diagonalizable LDCs, showing approximation guarantees and providing a complete regret proof, while empirical results in the appendix corroborate scalability and practical performance. The approach offers a principled, scalable route for online control under partial information and adversarial loss, with potential extensions to unknown dynamics and bandit feedback.

Abstract

We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

Efficient Spectral Control of Partially Observed Linear Dynamical Systems

TL;DR

This work addresses online control of partially observed linear dynamical systems with adversarial disturbances by introducing Double Spectral Control (DSC), a two-level spectral learning framework that reframes control as online convex optimization over spectral features. DSC builds a universal Hankel-based spectral basis to convexify the disturbance-response map and delivers a provable regret bound of with per-step polylogarithmic time, achieving an exponential improvement in the dependence on the stability margin over prior methods. Theoretical analysis connects spectral controllers to a class of diagonalizable LDCs, showing approximation guarantees and providing a complete regret proof, while empirical results in the appendix corroborate scalability and practical performance. The approach offers a principled, scalable route for online control under partial information and adversarial loss, with potential extensions to unknown dynamics and bandit feedback.

Abstract

We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

Paper Structure

This paper contains 32 sections, 15 theorems, 102 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Regret Minimization with Partial Observation.
  3. Marginal Stability and Spectral Filters.
  4. Our Contributions
  5. New algorithm: Double Spectral Control (DSC).
  6. Exponential runtime improvement in terms of condition number.
  7. Empirical Performance.
  8. Related Work
  9. Control of Linear Dynamical Systems.
  10. Online Control and Adversarial Disturbances.
  11. Partial Observation.
  12. Spectral Filtering.
  13. Convex Relaxations in Control.
  14. Broader Landscape.
  15. Online Convex Optimization.
  16. ...and 17 more sections

Key Result

Theorem 4.1

Let $c_t$ be any sequence of convex Lipschitz cost functions satisfying Assumption assm:lipschitz, and let the LDS be controllable (Definition assm:controllable) and satisfy Assumption assm:bounded-system. Then, Algorithm alg:mainA achieves the following regret bound: where $\tilde{O}$ hides poly-logarithmic factors in $\frac{T}{\gamma}$ and constants, and $\mathcal{S}$ is the class of LDCs defin

Figures (3)

  • Figure 1: Entries of the first six eigenvectors of $H_{500}$, plotted coordinate-wise.
  • Figure 2: Illustration of the Double Spectral Control (DSC) method. The learner receives the observed signal $\mathbf{y}_t$, from which it computes the natural observation $\mathbf{y}^{\text{nat}}$ in an online manner by maintaining a fictitious internal state. This signal is processed through two levels of spectral operations: a spectral lifting stage using filters $\boldsymbol{\phi}$, followed by a spectral filtering stage using filters $\boldsymbol{\varphi}$. The resulting features are linearly combined to generate the control action $\mathbf{u}$, which is applied to the linear dynamical system (LDS).
  • Figure 3: Comparison of Controllers: LQG, GRC, and DSC with 95% Confidence Intervals over 100 trials under Different Input Signal and Perturbation Settings.

Theorems & Definitions (38)

  • Definition 3.1
  • Definition 3.4
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Theorem 4.1: Main Theorem
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 4.4
  • ...and 28 more