Implications of Regret on Stability of Linear Dynamical Systems

TL;DR

This paper explores the relationship between regret and stability in linear dynamical systems, demonstrating that linear regret implies asymptotic stability and vice versa under certain conditions.

Contribution

It establishes a theoretical link between regret minimization and system stability for linear systems with adversarial disturbances, a novel insight in control and learning.

Findings

Linear regret implies asymptotic stability for linear systems.

Bounded input bounded state stability implies linear regret.

Summability of state transition matrices implies linear regret.

Abstract

The setting of an agent making decisions under uncertainty and under dynamic constraints is common for the fields of optimal control, reinforcement learning, and recently also for online learning. In the online learning setting, the quality of an agent's decision is often quantified by the concept of regret, comparing the performance of the chosen decisions to the best possible ones in hindsight. While regret is a useful performance measure, when dynamical systems are concerned, it is important to also assess the stability of the closed-loop system for a chosen policy. In this work, we show that for linear state feedback policies and linear systems subject to adversarial disturbances, linear regret implies asymptotic stability in both time-varying and time-invariant settings. Conversely, we also show that bounded input bounded state stability and summability of the state transition matrices imply linear regret.