Near-optimal Regret Bounds for Stochastic Shortest Path

Stochastic shortest path (SSP) is a well-known problem in planning and control, in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent is unaware of the environment dynamics (i.e., the transition function) and has to repeatedly play for a given number of episodes, while learning the problem's optimal solution. Unlike other well-studied models in reinforcement learning (RL), the length of an episode is not predetermined (or bounded) and is influenced by the agent's actions. Recently, \cite{tarbouriech2019noregret} studied this problem in the context of regret minimization, and provided an algorithm whose regret bound is inversely proportional to the square root of the minimum instantaneous cost. In this work we remove this dependence on the minimum cost---we give an algorithm that guarantees a regret bound of $\widetilde{O}(B^{3/2} S \sqrt{A K})$, where $B$ is an upper bound on the expected cost of the optimal policy, $S$ is the number of states, $A$ is the number of actions and $K$ is the total number of episodes. We additionally show that any learning algorithm must have at least $\Omega(B \sqrt{S A K})$ regret in the worst case.