🤖 AI Summary
This paper studies online control of linear systems under unbounded noise, unknown convex cost functions, and partial system knowledge—relaxing the prevalent bounded-noise assumption to merely requiring finite fourth-order moments. Methodologically, it integrates adaptive state feedback, online convex optimization, and robust estimation, employing moment-based constraints instead of boundedness conditions for analysis. Theoretically, it establishes the first high-probability regret bound of $ ilde{O}(sqrt{T})$ under unbounded noise. Furthermore, when the cost is strongly convex and the noise is sub-Gaussian, the regret improves to a poly-logarithmic rate of $mathrm{poly}(log T)$. These results significantly broaden the applicability and robustness limits of online control algorithms, providing rigorous theoretical guarantees for realistic stochastic environments with heavy-tailed disturbances.
📝 Abstract
This paper investigates the problem of controlling a linear system under possibly unbounded stochastic noise with unknown convex cost functions, known as an online control problem. In contrast to the existing work, which assumes the boundedness of noise, we show that an $ ilde{O}(sqrt{T}) $ high-probability regret can be achieved under unbounded noise, where $ T $ denotes the time horizon. Notably, the noise is only required to have a finite fourth moment. Moreover, when the costs are strongly convex and the noise is sub-Gaussian, we establish an $ O({
m poly} (log T)) $ regret bound.