$\kappa$-Explorer: A Unified Framework for Active Model Estimation in MDPs

In tabular Markov decision processes (MDPs) with perfect state observability, each trajectory provides active samples from the transition distributions conditioned on state-action pairs. Consequently, accurate model estimation depends on how the exploration policy allocates visitation frequencies in accordance with the intrinsic complexity of each transition distribution. Building on recent work on coverage-based exploration, we introduce a parameterized family of decomposable and concave objective functions $U_\kappa$ that explicitly incorporate both intrinsic estimation complexity and extrinsic visitation frequency. Moreover, the curvature $\kappa$ provides a unified treatment of various global objectives, such as the average-case and worst-case estimation error objectives. Using the closed-form characterization of the gradient of $U_\kappa$, we propose $\kappa$-Explorer, an active exploration algorithm that performs Frank-Wolfe-style optimization over state-action occupancy measures. The diminishing-returns structure of $U_\kappa$ naturally prioritizes underexplored and high-variance transitions, while preserving smoothness properties that enable efficient optimization. We establish tight regret guarantees for $\kappa$-Explorer and further introduce a fully online and computationally efficient surrogate algorithm for practical use. Experiments on benchmark MDPs demonstrate that $\kappa$-Explorer provides superior performance compared to existing exploration strategies.