Implications of Regret on Stability of Linear Dynamical Systems
This work addresses the problem of ensuring stability in online learning and control for researchers and practitioners, but it is incremental as it builds on existing concepts of regret and stability.
The paper tackles the relationship between regret and stability in linear dynamical systems with adversarial disturbances, showing that linear regret implies asymptotic stability and that bounded input bounded state stability with summable transition matrices implies linear regret.
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.