Sample Complexity Bounds for Stochastic Shortest Path with a Generative Model

We study the sample complexity of learning an ε-optimal policy in the Stochastic Shortest Path (SSP) problem. We first derive sample complexity bounds when the learner has access to a generative model. We show that there exists a worst-case SSP instance with S states, A actions, minimum cost cmin, and maximum expected cost of the optimal policy over all states Bstar, where any algorithm requires at least Omega(SABstar3/(cminε2)) samples to return an ε-optimal policy with high probability. Surprisingly, this implies that whenever cmin = 0 an SSP problem may not be learnable, thus revealing that learning in SSPs is strictly harder than in the finite-horizon and discounted settings. We complement this lower bound with an algorithm that matches it, up to logarithmic factors, in the general case, and an algorithm that matches it up to logarithmic factors even when cmin = 0, but only under the condition that the optimal policy has a bounded hitting time to the goal state.