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 $\tilde{O}(L^5 S_{L+ε} Γ_{L+ε} A ε^{-2})$, 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.