# Regret Bounds for Adaptive Nonlinear Control

arXiv: https://arxiv.org/abs/2011.13101 · HTML: https://mlsanity.com/paper/2011.13101/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2020.

## Built on by (most cited first, 25 of 25)

- [On the Sample Complexity of Stability Constrained Imitation Learning](https://mlsanity.com/paper/2102.09161/) (2021)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Meta-Adaptive Nonlinear Control: Theory and Algorithms](https://mlsanity.com/paper/2106.06098/) (2021)
- [TaSIL: Taylor Series Imitation Learning](https://mlsanity.com/paper/2205.14812/) (2022)
- [Optimal Exploration for Model-Based RL in Nonlinear Systems](https://mlsanity.com/paper/2306.09210/) (2023)
- [Online Control of Unknown Time-Varying Dynamical Systems](https://mlsanity.com/paper/2202.07890/) (2022)
- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
- [How are policy gradient methods affected by the limits of control?](https://mlsanity.com/paper/2206.06863/) (2022)
- [Nonparametric adaptive control and prediction: theory and randomized algorithms](https://mlsanity.com/paper/2106.03589/) (2021)
- [Generalization in Supervised Learning Through Riemannian Contraction](https://mlsanity.com/paper/2201.06656/) (2022)
- [Active Learning for Control-Oriented Identification of Nonlinear Systems](https://mlsanity.com/paper/2404.09030/) (2024)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [The Sample Complexity of Online Reinforcement Learning: A Multi-model Perspective](https://mlsanity.com/paper/2501.15910/) (2025)
- [Regret Analysis of Learning-Based MPC with Partially-Unknown Cost Function](https://mlsanity.com/paper/2108.02307/) (2021)
- [Robust Online Control with Model Misspecification](https://mlsanity.com/paper/2107.07732/) (2021)
- [Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems](https://mlsanity.com/paper/2111.03772/) (2021)
- [Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control](https://mlsanity.com/paper/2407.05781/) (2024)
- [Adversarially Robust Stability Certificates can be Sample-Efficient](https://mlsanity.com/paper/2112.10690/) (2021)
- [Optimal Competitive-Ratio Control](https://mlsanity.com/paper/2206.01782/) (2022)
- [Adaptive Robust Model Predictive Control via Uncertainty Cancellation](https://mlsanity.com/paper/2212.01371/) (2022)
- [Data-Guided Regulator for Adaptive Nonlinear Control](https://mlsanity.com/paper/2311.12230/) (2023)
- [Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem](https://mlsanity.com/paper/2411.15972/) (2024)
- [Logarithmic Regret for Nonlinear Control](https://mlsanity.com/paper/2501.10261/) (2025)
- [A Framework for Adaptive Stabilisation of Nonlinear Stochastic Systems](https://mlsanity.com/paper/2511.17436/) (2025)
- [Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guarantees](https://mlsanity.com/paper/2609.32289/) (2026)

## Abstract (arXiv)

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 widetildeO(√T) in expectation. Furthermore, we show that when the input is subject to a k timestep delay, the regret degrades to widetildeO(k √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.