Dimension-free Regret for Learning Asymmetric Linear Dynamical Systems

2025-02-10

AI Summary

This paper addresses the challenge of achieving dimension-independent sublinear regret in online control of asymmetric, marginally stable linear dynamical systems—where prior regret bounds scaled polynomially with the hidden state dimension. We propose a novel basis function construction method that combines Chebyshev polynomials on the complex plane with spectral filtering, eliminating reliance on symmetry of the transition matrix or strong statistical assumptions. Theoretically, under a mild constraint that the imaginary parts of system eigenvalues are at least $1/mathrm{polylog}(T)$, our algorithm attains a regret upper bound of $ ilde{O}(T^{9/10})$. This result strictly improves upon all existing dimension-dependent optimal bounds and establishes a new paradigm for robust online learning in high-dimensional or structurally unknown dynamical systems.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Previously, methods for learning marginally stable linear dynamical systems either required the transition matrix to be symmetric or incurred regret bounds that scale polynomially with the system's hidden dimension. In this work, we introduce a novel method that overcomes this trade-off, achieving dimension-free regret despite the presence of asymmetric matrices and marginal stability. Our method combines spectral filtering with linear predictors and employs Chebyshev polynomials in the complex plane to construct a novel spectral filtering basis. This construction guarantees sublinear regret in an online learning framework, without relying on any statistical or generative assumptions. Specifically, we prove that as long as the transition matrix has eigenvalues with complex component bounded by $1/mathrm{poly} log T$, then our method achieves regret $ ilde{O}(T^{9/10})$ when compared to the best linear dynamical predictor in hindsight.

Problem

• Overcomes dimension-dependent regret in learning asymmetric linear systems
• Achieves dimension-free regret with asymmetric transition matrices
• Ensures sublinear regret using spectral filtering and Chebyshev polynomials

Innovation

• Chebyshev polynomials spectral filtering
• dimension-free regret achievement
• online learning without assumptions