Online learning with dynamics: A minimax perspective
This work provides a unifying theoretical framework for online learning with dynamics, offering new insights and tight regret bounds for various problems, including those with non-linear dynamics and non-convex losses where such bounds were previously unknown.
This paper investigates online learning in dynamic environments where a learner interacts with a stateful system, choosing policies and incurring costs that depend on the policy and current state. The authors establish non-constructive upper bounds on the minimax rate for policy regret, providing sufficient conditions for online learnability and characterizing the rates by policy class expressiveness and dynamics stability. They also provide matching lower bounds, demonstrating the necessity of their complexity terms.
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.