Sample Complexity Bounds for Stochastic Shortest Path with a Generative Model
For reinforcement learning theorists, this work provides fundamental hardness results for SSP, distinguishing it from other settings and guiding algorithm design.
The paper establishes the first sample complexity lower bound for learning an ε-optimal policy in Stochastic Shortest Path (SSP) with a generative model, showing Ω(SAB₊³/(c_min ε²)) samples are needed, and provides matching algorithms up to log factors. It reveals that SSP is strictly harder than finite-horizon and discounted settings, and is unlearnable when c_min=0 unless the optimal policy has bounded hitting time.
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 $c_{\min}$, and maximum expected cost of the optimal policy over all states $B_{\star}$, where any algorithm requires at least $Ω(SAB_{\star}^3/(c_{\min}ε^2))$ samples to return an $ε$-optimal policy with high probability. Surprisingly, this implies that whenever $c_{\min} = 0$ 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 $c_{\min} = 0$, but only under the condition that the optimal policy has a bounded hitting time to the goal state.