Eluder-based Regret for Stochastic Contextual MDPs
This provides an efficient and rate-optimal solution for regret minimization in CMDPs under general offline function approximation, addressing a key problem in reinforcement learning theory.
The paper tackles regret minimization in Stochastic Contextual MDPs by proposing the E-UC³RL algorithm, which achieves a regret guarantee of Õ(H³√(T|S||A|d_E(P) log(|F||P|/δ))) under minimal assumptions of realizable function classes and offline regression oracles.
We present the E-UC$^3$RL algorithm for regret minimization in Stochastic Contextual Markov Decision Processes (CMDPs). The algorithm operates under the minimal assumptions of realizable function class and access to \emph{offline} least squares and log loss regression oracles. Our algorithm is efficient (assuming efficient offline regression oracles) and enjoys a regret guarantee of $ \widetilde{O}(H^3 \sqrt{T |S| |A|d_{\mathrm{E}}(\mathcal{P}) \log (|\mathcal{F}| |\mathcal{P}|/ δ) )}) , $ with $T$ being the number of episodes, $S$ the state space, $A$ the action space, $H$ the horizon, $\mathcal{P}$ and $\mathcal{F}$ are finite function classes used to approximate the context-dependent dynamics and rewards, respectively, and $d_{\mathrm{E}}(\mathcal{P})$ is the Eluder dimension of $\mathcal{P}$ w.r.t the Hellinger distance. To the best of our knowledge, our algorithm is the first efficient and rate-optimal regret minimization algorithm for CMDPs that operates under the general offline function approximation setting. In addition, we extend the Eluder dimension to general bounded metrics which may be of separate interest.