Online learning with dynamics: A minimax perspective

TL;DR

This paper develops a minimax framework for online learning in dynamic, stateful environments, providing bounds on regret that account for policy complexity and environment stability, applicable to various complex settings.

Contribution

It introduces a unifying analysis framework for online learning with dynamics, deriving regret bounds that incorporate policy class complexity and environment stability, including new bounds for non-linear dynamics.

Findings

Derived upper bounds on minimax regret for dynamic environments.

Established necessary conditions for online learnability in this setting.

Extended regret analysis to non-linear dynamics with non-convex losses.

Abstract

We study the problem of online learning with dynamics, where a learner interacts with a stateful environment over multiple rounds. In each round of the interaction, the learner selects a policy to deploy and incurs a cost that depends on both the chosen policy and current state of the world. The state-evolution dynamics and the costs are allowed to be time-varying, in a possibly adversarial way. In this setting, we study the problem of minimizing policy regret and provide non-constructive upper bounds on the minimax rate for the problem. Our main results provide sufficient conditions for online learnability for this setup with corresponding rates. The rates are characterized by 1) a complexity term capturing the expressiveness of the underlying policy class under the dynamics of state change, and 2) a dynamics stability term measuring the deviation of the instantaneous loss from a certain counterfactual loss. Further, we provide matching lower bounds which show that both the complexity terms are indeed necessary. Our approach provides a unifying analysis that recovers regret bounds for several well studied problems including online learning with memory, online control of linear quadratic regulators, online Markov decision processes, and tracking adversarial targets. In addition, we show how our tools help obtain tight regret bounds for a new problems (with non-linear dynamics and non-convex losses) for which such bounds were not known prior to our work.