Regret Bounds for Adaptive Nonlinear Control
This work addresses adaptive control for nonlinear systems, providing theoretical guarantees for practitioners in robotics or aerospace, though it is incremental in extending regret analysis to nonlinear settings.
The paper tackles the problem of adaptive control for nonlinear systems with unmodeled disturbances, proving the first finite-time regret bounds of O(√T) for certainty equivalence adaptive control compared to an oracle, which degrades to O(k√T) with input delays.
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.