# Optimal Competitive-Ratio Control

arXiv: https://arxiv.org/abs/2206.01782 · HTML: https://mlsanity.com/paper/2206.01782/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 3 Jun 2022.

## Builds on (most cited first, 17 of 17)

- [Online Control with Adversarial Disturbances](https://mlsanity.com/paper/1902.08721/) (2019)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems](https://mlsanity.com/paper/2003.11227/) (2020)
- [Logarithmic Regret for Adversarial Online Control](https://mlsanity.com/paper/2003.00189/) (2020)
- [Smoothed Online Optimization for Regression and Control](https://mlsanity.com/paper/1810.10132/) (2018)
- [Regret Bounds for Adaptive Nonlinear Control](https://mlsanity.com/paper/2011.13101/) (2020)
- [Making Non-Stochastic Control (Almost) as Easy as Stochastic](https://mlsanity.com/paper/2006.05910/) (2020)
- [Reinforcement Learning with Fast Stabilization in Linear Dynamical Systems](https://mlsanity.com/paper/2007.12291/) (2020)
- [Robustness and Consistency in Linear Quadratic Control with Untrusted Predictions](https://mlsanity.com/paper/2106.09659/) (2021)
- [Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds](https://mlsanity.com/paper/2111.07018/) (2021)
- [Regret-optimal Estimation and Control](https://mlsanity.com/paper/2106.12097/) (2021)
- [A System Level Approach to Controller Synthesis](https://mlsanity.com/paper/1610.04815/) (2016)
- [Competitive Control with Delayed Imperfect Information](https://mlsanity.com/paper/2010.11637/) (2020)
- [Competitive Control](https://mlsanity.com/paper/2107.13657/) (2021)
- [Safe Control with Minimal Regret](https://mlsanity.com/paper/2203.00358/) (2022)
- [On the Regret of H∞ Control](https://mlsanity.com/paper/2203.16237/) (2022)
- [A System Level Approach to Regret Optimal Control](https://mlsanity.com/paper/2202.13763/) (2022)

## Built on by (most cited first, 2 of 2)

- [Beyond Black-Box Advice: Learning-Augmented Algorithms for MDPs with Q-Value Predictions](https://mlsanity.com/paper/2307.10524/) (2023)
- [Follow the Clairvoyant: an Imitation Learning Approach to Optimal Control](https://mlsanity.com/paper/2211.07389/) (2022)

## Abstract (arXiv)

Inspired by competitive policy designs approaches in online learning, new control paradigms such as competitive-ratio and regret-optimal control have been recently proposed as alternatives to the classical H2 and H_∞ approaches. These competitive metrics compare the control cost of the designed controller against the cost of a clairvoyant controller, which has access to past, present, and future disturbances in terms of ratio and difference, respectively. While prior work provided the optimal solution for the regret-optimal control problem, in competitive-ratio control, the solution is only provided for the sub-optimal problem. In this work, we derive the optimal solution to the competitive-ratio control problem. We show that the optimal competitive ratio formula can be computed as the maximal eigenvalue of a simple matrix, and provide a state-space controller that achieves the optimal competitive ratio. We conduct an extensive numerical study to verify this analytical solution, and demonstrate that the optimal competitive-ratio controller outperforms other controllers on several large scale practical systems. The key techniques that underpin our explicit solution is a reduction of the control problem to a Nehari problem, along with a novel factorization of the clairvoyant controller's cost. We reveal an interesting relation between the explicit solutions that now exist for both competitive control paradigms by formulating a regret-optimal control framework with weight functions that can also be utilized for practical purposes.