# Regret-optimal Estimation and Control

arXiv: https://arxiv.org/abs/2106.12097 · HTML: https://mlsanity.com/paper/2106.12097/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [Online Optimization : Competing with Dynamic Comparators](https://mlsanity.com/paper/1501.06225/) (2015)
- [Online Control with Adversarial Disturbances](https://mlsanity.com/paper/1902.08721/) (2019)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [Thinking Fast and Slow: Optimization Decomposition Across Timescales](https://mlsanity.com/paper/1704.07785/) (2017)
- [The Nonstochastic Control Problem](https://mlsanity.com/paper/1911.12178/) (2019)
- [Logarithmic Regret for Adversarial Online Control](https://mlsanity.com/paper/2003.00189/) (2020)
- [Online Optimal Control with Linear Dynamics and Predictions: Algorithms and Regret Analysis](https://mlsanity.com/paper/1906.11378/) (2019)
- [Adaptive Regret for Control of Time-Varying Dynamics](https://mlsanity.com/paper/2007.04393/) (2020)
- [Non-stationary Online Learning with Memory and Non-stochastic Control](https://mlsanity.com/paper/2102.03758/) (2021)
- [Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization](https://mlsanity.com/paper/1905.12776/) (2019)
- [Smoothed Online Optimization for Regression and Control](https://mlsanity.com/paper/1810.10132/) (2018)
- [Making Non-Stochastic Control (Almost) as Easy as Stochastic](https://mlsanity.com/paper/2006.05910/) (2020)
- [The Power of Linear Controllers in LQR Control](https://mlsanity.com/paper/2002.02574/) (2020)
- [Regret-optimal measurement-feedback control](https://mlsanity.com/paper/2011.12785/) (2020)
- [Adaptive Control and Regret Minimization in Linear Quadratic Gaussian (LQG) Setting](https://mlsanity.com/paper/2003.05999/) (2020)

## Built on by (most cited first, 10 of 10)

- [Competitive Control](https://mlsanity.com/paper/2107.13657/) (2021)
- [Optimal Dynamic Regret in LQR Control](https://mlsanity.com/paper/2206.09257/) (2022)
- [Predictive Linear Online Tracking for Unknown Targets](https://mlsanity.com/paper/2402.10036/) (2024)
- [A Distributionally Robust Approach to Regret Optimal Control using the Wasserstein Distance](https://mlsanity.com/paper/2304.06783/) (2023)
- [Optimal Competitive-Ratio Control](https://mlsanity.com/paper/2206.01782/) (2022)
- [Online estimation and control with optimal pathlength regret](https://mlsanity.com/paper/2110.12544/) (2021)
- [Follow the Clairvoyant: an Imitation Learning Approach to Optimal Control](https://mlsanity.com/paper/2211.07389/) (2022)
- [Online Control of Linear Systems under Unbounded Noise](https://mlsanity.com/paper/2402.10252/) (2024)
- [Efficient Spectral Control of Partially Observed Linear Dynamical Systems](https://mlsanity.com/paper/2505.20943/) (2025)
- [Distributionally Robust Regret Optimal Control Under Moment-Based Ambiguity Sets](https://mlsanity.com/paper/2512.10906/) (2025)

## Abstract (arXiv)

We consider estimation and control in linear time-varying dynamical systems from the perspective of regret minimization. Unlike most prior work in this area, we focus on the problem of designing causal estimators and controllers which compete against a clairvoyant noncausal policy, instead of the best policy selected in hindsight from some fixed parametric class. We show that the regret-optimal estimator and regret-optimal controller can be derived in state-space form using operator-theoretic techniques from robust control and present tight,data-dependent bounds on the regret incurred by our algorithms in terms of the energy of the disturbances. Our results can be viewed as extending traditional robust estimation and control, which focuses on minimizing worst-case cost, to minimizing worst-case regret. We propose regret-optimal analogs of Model-Predictive Control (MPC) and the Extended KalmanFilter (EKF) for systems with nonlinear dynamics and present numerical experiments which show that our regret-optimal algorithms can significantly outperform standard approaches to estimation and control.