# Improved Sample Complexity for Incremental Autonomous Exploration in MDPs

We investigate the exploration of an unknown environment when no reward function is provided. Building on the incremental exploration setting introduced by Lim and Auer [1], we define the objective of learning the set of ϵ-optimal goal-conditioned policies attaining all states that are incrementally reachable within L steps (in expectation) from a reference state s_0. In this paper, we introduce a novel model-based approach that interleaves discovering new states from s_0 and improving the accuracy of a model estimate that is used to compute goal-conditioned policies to reach newly discovered states. The resulting algorithm, DisCo, achieves a sample complexity scaling as Õ(L^5 S_L+ϵΓ_L+ϵ A ϵ^-2), where A is the number of actions, S_L+ϵ is the number of states that are incrementally reachable from s_0 in L+ϵ steps, and Γ_L+ϵ is the branching factor of the dynamics over such states. This improves over the algorithm proposed in [1] in both ϵ and L at the cost of an extra Γ_L+ϵ factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an ϵ/c_min-optimal policy for any cost-sensitive shortest-path problem defined on the L-reachable states with minimum cost c_min. Finally, we report preliminary empirical results confirming our theoretical findings.

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