Efficient Bias-Span-Constrained Exploration-Exploitation in Reinforcement Learning
This work addresses the challenge of reinforcement learning in MDPs with large diameters but small bias spans, providing a computationally efficient solution for scenarios where prior algorithms were impractical or less effective.
The paper tackles the problem of efficient exploration-exploitation in weakly-communicating Markov decision processes (MDPs) by introducing SCAL, an algorithm that achieves a regret bound of $\widetilde{O}(c\sqrt{\Gamma SAT})$, significantly improving over existing methods like UCRL and PSRL, which scale linearly with the MDP diameter $D$.
We introduce SCAL, an algorithm designed to perform efficient exploration-exploitation in any unknown weakly-communicating Markov decision process (MDP) for which an upper bound $c$ on the span of the optimal bias function is known. For an MDP with $S$ states, $A$ actions and $Γ\leq S$ possible next states, we prove a regret bound of $\widetilde{O}(c\sqrt{ΓSAT})$, which significantly improves over existing algorithms (e.g., UCRL and PSRL), whose regret scales linearly with the MDP diameter $D$. In fact, the optimal bias span is finite and often much smaller than $D$ (e.g., $D=\infty$ in non-communicating MDPs). A similar result was originally derived by Bartlett and Tewari (2009) for REGAL.C, for which no tractable algorithm is available. In this paper, we relax the optimization problem at the core of REGAL.C, we carefully analyze its properties, and we provide the first computationally efficient algorithm to solve it. Finally, we report numerical simulations supporting our theoretical findings and showing how SCAL significantly outperforms UCRL in MDPs with large diameter and small span.