This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for inf-compact functions introduced in this paper.
This note describes sufficient conditions under which total-cost and average-cost Markov decision processes (MDPs) with general state and action spaces, and with weakly continuous transition probabilities, can be reduced to discounted MDPs. For undiscounted problems, these reductions imply the validity of optimality equations and the existence of stationary optimal policies. The reductions also provide methods for computing optimal policies. The results are applied to a capacitated inventory control problem with fixed costs and lost sales.
This paper studies average-cost Markov decision processes with semi-uniform Feller transition probabilities. This class of MDPs was recently introduced by the authors to study MDPs with incomplete information. This paper studies the validity of optimality inequalities, the existence of optimal policies, and the approximations of optimal policies by policies optimizing total discounted costs.
This paper provides conditions under which total-cost and average-cost Markov decision processes (MDPs) can be reduced to discounted ones. Results are given for transient total-cost MDPs with tran- sition rates whose values may be greater than one, as well as for average-cost MDPs with transition probabilities satisfying the condition that there is a state such that the expected time to reach it is uniformly bounded for all initial states and stationary policies. In particular, these reductions imply sufficient conditions for the validity of optimality equations and the existence of stationary…
This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total costs, and then it is extended under certain assumptions to MDPs w…
This paper describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs) with Borel state, observation, and action sets and with the expected total costs. Action sets may not be compact and one-step cost functions may be unbounded. The introduced conditions are also sufficient for the validity of optimality equations, semi-continuity of value functions, and convergence of value iterations to optimal values. Since POMDPs can be reduced to Completely Observable Markov Decision Processes (COMDPs), whose states are posterior state…
We propose a method of approximating multivariate Gaussian probabilities using dynamic programming. We show that solving the optimization problem associated with a class of discrete-time finite horizon Markov decision processes with non-Lipschitz cost functions is equivalent to integrating a Gaussian functions over polytopes. An approximation scheme for this class of MDPs is proposed and explicit error bounds under the supremum norm for the optimal cost to go functions are derived.
City University of Hong Kong（香港城市大学）；University of New Hampshire（新罕布什尔大学）；Imperial College London（帝国理工学院）
Robust Markov decision processes (MDPs) allow to compute reliable solutions for dynamic decision problems whose evolution is modeled by rewards and partially-known transition probabilities. Unfortunately, accounting for uncertainty in the transition probabilities significantly increases the computational complexity of solving robust MDPs, which severely limits their scalability. This paper describes new efficient algorithms for solving the common class of robust MDPs with s- and sa-rectangular ambiguity sets defined by weighted $L_1$ norms. We propose partial policy iteration, a new, efficient…
In classical Markov Decision Processes (MDPs), action costs and transition probabilities are assumed to be known, although an accurate estimation of these parameters is often not possible in practice. This study addresses MDPs under cost and transition probability uncertainty and aims to provide a mathematical framework to obtain policies minimizing the risk of high long-term losses due to not knowing the true system parameters. To this end, we utilize the risk measure value-at-risk associated with the expected performance of an MDP model with respect to parameter uncertainty. We provide mixed…
This paper studies convergence properties of optimal values and actions for discounted and average-cost Markov Decision Processes (MDPs) with weakly continuous transition probabilities and applies these properties to the stochastic periodic-review inventory control problem with backorders, positive setup costs, and convex holding/backordering costs. The following results are established for MDPs with possibly noncompact action sets and unbounded cost functions: (i) convergence of value iterations to optimal values for discounted problems with possibly non-zero terminal costs, (ii) convergence …
Efficient Algorithms for Robust Markov Decision Processes with $s$-Rectangular Ambiguity Sets
Department of Data Science, City University of Hong Kong（香港城市大学数据科学系）；Department of Computer Science, University of New Hampshire（新罕布什尔大学计算机科学系）；Imperial College Business School, Imperial College London（伦敦帝国理工学院商学院）
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
This paper proposes a formal approach to online learning and planning for agents operating in a priori unknown, time-varying environments. The proposed method computes the maximally likely model of the environment, given the observations about the environment made by an agent earlier in the system run and assuming knowledge of a bound on the maximal rate of change of system dynamics. Such an approach generalizes the estimation method commonly used in learning algorithms for unknown Markov decision processes with time-invariant transition probabilities, but is also able to quickly and correctly…
We introduce a framework to approximate a Markov Decision Process that stands on two pillars: state aggregation -- as the algorithmic infrastructure; and central-limit-theorem-type approximations -- as the mathematical underpinning of optimality guarantees. The theory is grounded in recent work Braverman et al (2020} that relates the solution of the Bellman equation to that of a PDE where, in the spirit of the central limit theorem, the transition matrix is reduced to its local first and second moments. Solving the PDE is $\textit{not}$ required by our method. Instead, we construct a "sister" …
Convex-Analytic View of Query Selection for Safety Verification of Markov Decision Processes
The Institute of Statistical Mathematics（统计数学研究所）；Tohoku University（东北大学）；RIKEN AIP（理化学研究所人工智能中心）；CyberAgent；National Institute of Informatics（信息学国立研究所）
