We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

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.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

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} .

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 discounted optimal policies with a discount factor close to $1$, but they only work under strong or hard-to-verify assumptions such as ergodicity or weakly communicating MDPs. In this paper, we show that when the discount factor is larger than the Blackwell discount factor $\gamma_{\mathrm{bw}}$, all discounted optimal policies become Blackwell- and average-optimal, and we derive a general upper bound on $\gamma_{\mathrm{bw}}$. The upper bound on $\gamma_{\mathrm{bw}}$ provides the first reduction from average and Blackwell optimality to discounted optimality, without any assumptions, and new polynomial-time algorithms for average- and Blackwell-optimal policies. Our work brings new ideas from the study of 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.

Although in recent years reinforcement learning has become very popular the number of successful applications to different kinds of operations research problems is rather scarce. Reinforcement learning is based on the well-studied dynamic programming technique and thus also aims at finding the best stationary policy for a given Markov Decision Process, but in contrast does not require any model knowledge. The policy is assessed solely on consecutive states (or state-action pairs), which are observed while an agent explores the solution space. The contributions of this paper are manifold. First we provide deep theoretical insights to the widely applied standard discounted reinforcement learning framework, which give rise to the understanding of why these algorithms are inappropriate when permanently provided with non-zero rewards, such as costs or profit. Second, we establish a novel near-Blackwell-optimal reinforcement learning algorithm. In contrary to former method it assesses the average reward per step separately and thus prevents the incautious combination of different types of state values. Thereby, the Laurent Series expansion of the discounted state values forms the foundation for this development and also provides the connection between the two approaches. Finally, we prove the viability of our algorithm on a challenging problem set, which includes a well-studied M/M/1 admission control queuing system. In contrast to standard discounted reinforcement learning our algorithm infers the optimal policy on all tested problems. The insights are that in the operations research domain machine learning techniques have to be adapted and advanced to successfully apply these methods in our settings.