#### Authors

Jean Tarbouriech, Matteo Pirotta, Michal Valko, Alessandro Lazaric

#### Abstract

We study the problem of exploring an unknown environment when no reward function is provided to the agent. Building on the incremental exploration setting introduced by Lim and Auer (2012), we define the objective of learning the set of $\epsilon$-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. The resulting algorithm, DisCo, achieves a sample complexity scaling as $\widetilde{O}_{\epsilon}(L^5 S_{L+\epsilon} \Gamma_{L+\epsilon} A \epsilon^{-2})$, where $A$ is the number of actions, $S_{L+\epsilon}$ is the number of states that are incrementally reachable from $s_0$ in $L+\epsilon$ steps, and $\Gamma_{L+\epsilon}$ is the branching factor of the dynamics over such states. This improves over the algorithm proposed in (Lim and Auer, 2012) in both $\epsilon$ and $L$ at the cost of an extra $\Gamma_{L+\epsilon}$ factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an $\epsilon/c_{\min}$-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.