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Reinforcement Learning/Average Cost Reward

Nuoya Xiong, Yihan Du, Longbo Huang

We investigate a novel safe reinforcement learning problem with step-wise violation constraints. Our problem differs from existing works in that we focus on stricter step-wise violation constraints and do not assume the existence of safe actions, making our formulation more suitable for safety-critical applications that need to ensure safety in all decision steps but may not always possess safe actions, e.g., robot control and autonomous driving.We propose an efficient algorithm SUCBVI, which guarantees $\widetilde{\mathcal{O}}(\sqrt{ST})$ or gap-dependent $\widetilde{\mathcal{O}}(S/\mathcal{C}_{\mathrm{gap}} + S^2AH^2)$ step-wise violation and $\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ regret. Lower bounds are provided to validate the optimality in both violation and regret performance with respect to the number of states $S$ and the total number of steps $T$. Moreover, we further study an innovative safe reward-free exploration problem with step-wise violation constraints. For this problem, we design algorithm SRF-UCRL to find a near-optimal safe policy, which achieves nearly state-of-the-art sample complexity $\widetilde{\mathcal{O}}((\frac{S^2AH^2}{\varepsilon}+\frac{H^4SA}{\varepsilon^2})(\log(\frac{1}{\delta})+S))$, and guarantees $\widetilde{\mathcal{O}}(\sqrt{ST})$ violation during exploration. Experimental results demonstrate the superiority of our algorithms in safety performance and corroborate our theoretical results.

Reinforcement Learning/Average Cost Reward

Julien Grand-Clément, Marek Petrik

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell optimality. Many existing approaches to computing average-optimal policies solve for discount-optimal policies with a discount factor close to $1$, but they only work under strong or hard-to-verify assumptions on the MDP structure such as unichain or ergodicity. We are the first to highlight the shortcomings of the classical definition of Blackwell optimality, which does not lead to simple algorithms for computing Blackwell-optimal policies and overlooks the pathological behaviors of optimal policies as regards the discount factors. To resolve this issue, in this paper, we show that when the discount factor is larger than the Blackwell discount factor $\gamma_{\sf bw}$, all discount-optimal policies become Blackwell- and average-optimal, and we derive a general upper bound on $\gamma_{\sf bw}$. Our upper bound on $\gamma_{\sf bw}$, parametrized by the bit-size of the rewards and transition probabilities of the MDP instance, provides the first reduction from average and Blackwell optimality to discounted optimality, without any assumptions, along with new polynomial-time algorithms. Our work brings new ideas from polynomials and algebraic numbers to the analysis of MDPs. Our results also apply to robust MDPs, enabling the first algorithms to compute robust Blackwell-optimal policies.

Reinforcement Learning/Average Cost Reward

Feiyang Wu, Jingyang Ke, Anqi Wu

We study the problem of Inverse Reinforcement Learning (IRL) with an average-reward criterion. The goal is to recover an unknown policy and a reward function when the agent only has samples of states and actions from an experienced agent. Previous IRL methods assume that the expert is trained in a discounted environment, and the discount factor is known. This work alleviates this assumption by proposing an average-reward framework with efficient learning algorithms. We develop novel stochastic first-order methods to solve the IRL problem under the average-reward setting, which requires solving an Average-reward Markov Decision Process (AMDP) as a subproblem. To solve the subproblem, we develop a Stochastic Policy Mirror Descent (SPMD) method under general state and action spaces that needs $\mathcal{O}(1/\varepsilon)$ steps of gradient computation. Equipped with SPMD, we propose the Inverse Policy Mirror Descent (IPMD) method for solving the IRL problem with a $\mathcal{O}(1/\varepsilon^2)$ complexity. To the best of our knowledge, the aforementioned complexity results are new in IRL with the average reward criterion. Finally, we corroborate our analysis with numerical experiments using the MuJoCo benchmark and additional control tasks.