Dimension-free Regret for Learning Asymmetric Linear Dynamical Systems

• URL: http://arxiv.org/abs/2502.06545v1
• Date: Mon, 10 Feb 2025 15:10:06 GMT
• Title: Dimension-free Regret for Learning Asymmetric Linear Dynamical Systems
• Authors: Annie Marsden, Elad Hazan,
• Abstract summary: We introduce a novel method that overcomes the trade-off, dimension-free regret despite the presence of matrices.<n>Our method combines spectral filtering with linear predictors and employs Chebyshevs in the complex plane to construct a novel spectral filtering basis.<n>We prove that as long as the transition matrix has eigenvalues with complex component bounded by $1/mathrmpoly log$, then our method achieves regret $tildeO(9/10)$ when compared to the best linear predictor in hindsight.
• Score: 19.415741153449265
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\tilde{O}(T^{9/10})$ when compared to the best linear dynamical predictor in hindsight.

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