Li-Yu Chen

Liyu Chen, Haipeng Luo

We initiate the study of dynamic regret minimization for goal-oriented reinforcement learning modeled by a non-stationary stochastic shortest path problem with changing cost and transition functions. We start by establishing a lower bound $Ω((B_{\star} SAT_{\star}(Δ_c + B_{\star}^2Δ_P))^{1/3}K^{2/3})$, where $B_{\star}$ is the maximum expected cost of the optimal policy of any episode starting from any state, $T_{\star}$ is the maximum hitting time of the optimal policy of any episode starting from the initial state, $SA$ is the number of state-action pairs, $Δ_c$ and $Δ_P$ are the amount of changes of the cost and transition functions respectively, and $K$ is the number of episodes. The different roles of $Δ_c$ and $Δ_P$ in this lower bound inspire us to design algorithms that estimate costs and transitions separately. Specifically, assuming the knowledge of $Δ_c$ and $Δ_P$, we develop a simple but sub-optimal algorithm and another more involved minimax optimal algorithm (up to logarithmic terms). These algorithms combine the ideas of finite-horizon approximation [Chen et al., 2022a], special Bernstein-style bonuses of the MVP algorithm [Zhang et al., 2020], adaptive confidence widening [Wei and Luo, 2021], as well as some new techniques such as properly penalizing long-horizon policies. Finally, when $Δ_c$ and $Δ_P$ are unknown, we develop a variant of the MASTER algorithm [Wei and Luo, 2021] and integrate the aforementioned ideas into it to achieve $\widetilde{O}(\min\{B_{\star} S\sqrt{ALK}, (B_{\star}^2S^2AT_{\star}(Δ_c+B_{\star}Δ_P))^{1/3}K^{2/3}\})$ regret, where $L$ is the unknown number of changes of the environment.

Liyu Chen, Andrea Tirinzoni, Matteo Pirotta et al.

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 result with lower bounds when prior knowledge of the hitting time of the optimal policy is available and when we restrict optimality by competing against policies with bounded hitting time. Finally, we design an algorithm with matching upper bounds in these cases. This settles the sample complexity of learning $ε$-optimal polices in SSP with generative models. We also initiate the study of learning $ε$-optimal policies without access to a generative model (i.e., the so-called best-policy identification problem), and show that sample-efficient learning is impossible in general. On the other hand, efficient learning can be made possible if we assume the agent can directly reach the goal state from any state by paying a fixed cost. We then establish the first upper and lower bounds under this assumption. Finally, using similar analytic tools, we prove that horizon-free regret is impossible in SSPs under general costs, resolving an open problem in (Tarbouriech et al., 2021c).

Liyu Chen, Andrea Tirinzoni, Alessandro Lazaric et al.

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of $ε$-optimal policies reaching a set $\mathcal{S}_L^{\rightarrow}$ of incrementally $L$-controllable states. We introduce a novel layered decomposition of the set of incrementally $L$-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of $\tilde{\mathcal{O}}(LS^{\rightarrow}_{L(1+ε)}Γ_{L(1+ε)} A \ln^{12}(S^{\rightarrow}_{L(1+ε)})/ε^2)$, where $S^{\rightarrow}_{L(1+ε)}$ is the number of states that are incrementally $L(1+ε)$-controllable, $A$ is the number of actions, and $Γ_{L(1+ε)}$ is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of $L^2$ and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of $\tilde{\mathcal{O}}(LS^{\rightarrow}_{L}A\ln^{12}(S^{\rightarrow}_{L})/ε^2)$, outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

Liyu Chen, Rahul Jain, Haipeng Luo

We study regret minimization for infinite-horizon average-reward Markov Decision Processes (MDPs) under cost constraints. We start by designing a policy optimization algorithm with carefully designed action-value estimator and bonus term, and show that for ergodic MDPs, our algorithm ensures $\widetilde{O}(\sqrt{T})$ regret and constant constraint violation, where $T$ is the total number of time steps. This strictly improves over the algorithm of (Singh et al., 2020), whose regret and constraint violation are both $\widetilde{O}(T^{2/3})$. Next, we consider the most general class of weakly communicating MDPs. Through a finite-horizon approximation, we develop another algorithm with $\widetilde{O}(T^{2/3})$ regret and constraint violation, which can be further improved to $\widetilde{O}(\sqrt{T})$ via a simple modification, albeit making the algorithm computationally inefficient. As far as we know, these are the first set of provable algorithms for weakly communicating MDPs with cost constraints.