Information Theoretic Regret Bounds for Online Nonlinear Control

TL;DR

This paper introduces the LC^3 algorithm for online nonlinear control, achieving near-optimal regret bounds in unknown systems modeled in RKHS, with empirical validation on various control tasks.

Contribution

It proposes a novel control algorithm with dimension-independent regret bounds for unknown nonlinear systems in RKHS, advancing theoretical and practical control methods.

Findings

LC^3 achieves near-optimal O(√T) regret bounds.

The regret bound depends on information-theoretic quantities, not system dimension.

Empirical results demonstrate effective learning and exploration in nonlinear control tasks.

Abstract

This work studies the problem of sequential control in an unknown, nonlinear dynamical system, where we model the underlying system dynamics as an unknown function in a known Reproducing Kernel Hilbert Space. This framework yields a general setting that permits discrete and continuous control inputs as well as non-smooth, non-differentiable dynamics. Our main result, the Lower Confidence-based Continuous Control ($LC^3$) algorithm, enjoys a near-optimal $O(\sqrt{T})$ regret bound against the optimal controller in episodic settings, where $T$ is the number of episodes. The bound has no explicit dependence on dimension of the system dynamics, which could be infinite, but instead only depends on information theoretic quantities. We empirically show its application to a number of nonlinear control tasks and demonstrate the benefit of exploration for learning model dynamics.