Online Markov Decision Processes with Aggregate Bandit Feedback

TL;DR

This paper introduces a new efficient algorithm for online Markov Decision Processes with aggregate bandit feedback, achieving sublinear regret by reducing the problem to a novel bandit setting called Distorted Linear Bandits.

Contribution

It presents a computationally efficient algorithm with $O( oot{K})$ regret for a new MDP variant and introduces the Distorted Linear Bandits framework with an online mirror descent approach.

Findings

Achieves $O( oot{K})$ regret in the new MDP setting.

Develops a novel reduction to Distorted Linear Bandits.

Provides an efficient online algorithm with $O( oot{T})$ regret for DLB.

Abstract

We study a novel variant of online finite-horizon Markov Decision Processes with adversarially changing loss functions and initially unknown dynamics. In each episode, the learner suffers the loss accumulated along the trajectory realized by the policy chosen for the episode, and observes aggregate bandit feedback: the trajectory is revealed along with the cumulative loss suffered, rather than the individual losses encountered along the trajectory. Our main result is a computationally efficient algorithm with $O(\sqrt{K})$ regret for this setting, where $K$ is the number of episodes. We establish this result via an efficient reduction to a novel bandit learning setting we call Distorted Linear Bandits (DLB), which is a variant of bandit linear optimization where actions chosen by the learner are adversarially distorted before they are committed. We then develop a computationally-efficient online algorithm for DLB for which we prove an $O(\sqrt{T})$ regret bound, where $T$ is the number of time steps. Our algorithm is based on online mirror descent with a self-concordant barrier regularization that employs a novel increasing learning rate schedule.