Regret Bounds for Adaptive Nonlinear Control

TL;DR

This paper establishes the first finite-time regret bounds for adaptive nonlinear control with unmodeled disturbances, demonstrating near-optimal performance guarantees in stochastic settings and analyzing effects of input delays.

Contribution

It introduces novel finite-time regret bounds for adaptive nonlinear control with matched uncertainty, connecting stability theory with online optimization analysis.

Findings

Regret bounded by (\u007f\, T) in stochastic setting

Regret degrades to (k  T) with input delay

Links classical stability notions with modern regret analysis

Abstract

We study the problem of adaptively controlling a known discrete-time nonlinear system subject to unmodeled disturbances. We prove the first finite-time regret bounds for adaptive nonlinear control with matched uncertainty in the stochastic setting, showing that the regret suffered by certainty equivalence adaptive control, compared to an oracle controller with perfect knowledge of the unmodeled disturbances, is upper bounded by $\widetilde{O}(\sqrt{T})$ in expectation. Furthermore, we show that when the input is subject to a $k$ timestep delay, the regret degrades to $\widetilde{O}(k \sqrt{T})$. Our analysis draws connections between classical stability notions in nonlinear control theory (Lyapunov stability and contraction theory) and modern regret analysis from online convex optimization. The use of stability theory allows us to analyze the challenging infinite-horizon single trajectory setting.