For a Markov decision process with countably infinite states, the optimal value may not be achievable in the set of stationary policies. In this paper, we study the existence conditions of an optimal stationary policy in a countable-state Markov decision process under the long-run average criterion. With a properly defined metric on the policy space of ergodic MDPs, the existence of an optimal stationary policy can be guaranteed by the compactness of the space and the continuity of the long-run average cost with respect to the metric. We further extend this condition by some assumptions which can be easily verified in control problems of specific systems, such as queueing systems. Our results make a complementary contribution to the literature in the sense that our method is capable to handle the cost function unbounded from both below and above, only at the condition of continuity and ergodicity. Several examples are provided to illustrate the application of our main results.
For a countable-state Markov decision process we introduce an embedding which produces a finite-state Markov decision process. The finite-state embedded process has the same optimal cost, and moreover, it has the same dynamics as the original process when restricting to the approximating set. The embedded process can be used as an approximation which, being finite, is more convenient for computation and implementation.
On Linear Programming for Constrained and Unconstrained Average-Cost Markov Decision Processes with Countable Action Spaces and Strictly Unbounded Costs
We consider killed Markov decision processes for countable models on a finite time-interval. Existence of a uniform $\varepsilon$-optimal policy is proven. We show the correctness of the fundamental equation. The optimal control problem is reduced to a similar problem for the derived model. We receive an optimality equation and a method for the construction of simple optimal policies. The sufficiency of simple policies for countable models is proven. We show the correctness of the Markovian property. Additionally, a dynamic programming principle is considered.
The infinite horizon risk-sensitive discounted-cost and ergodic-cost nonzero-sum stochastic games for controlled Markov chains with countably many states are analyzed. For the discounted-cost game, we prove the existence of Nash equilibrium strategies in the class of Markov strategies under fairly general conditions. Under an additional geometric ergodicity condition and a small cost criterion, the existence of Nash equilibrium strategies in the class of stationary Markov strategies is proved for the ergodic-cost game.
This paper is devoted to solving a time-inconsistent risk-sensitive control problem with parameter $\e$ and its limit case ($\e\rightarrow0^+$) for countable-stated Markov decision processes (MDPs for short). Since the cost functional is time-inconsistent, it is impossible to find a global optimal strategy for both cases. Instead, for each case, we will prove the existence of time-inconstant equilibrium strategies which verify the so-called step-optimality. Moreover, we prove the convergence of $\e$-equilibriums and the corresponding value functions as $\e\rightarrow0^+$.
We consider average-cost Markov decision processes (MDPs) with Borel state spaces, countable, discrete action spaces, and strictly unbounded one-stage costs. For the minimum pair approach, we introduce a new majorization condition on the state transition stochastic kernel, in place of the commonly required continuity conditions on the MDP model. We combine this majorization condition with Lusin's theorem to prove the existence of a stationary minimum pair, i.e., a stationary policy paired with an invariant probability measure induced on the state space, with the property that the pair attains …
We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.
We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $\varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $\varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev…
We consider discrete-time Markov Decision Processes with Borel state and action spaces and universally measurable policies. For several long-run average cost criteria, we establish the following optimality results: the optimal average cost functions are lower semianalytic, there exist universally measurable semi-Markov or history-dependent $ε$-optimal policies, and similar results hold for the minimum average costs achievable by Markov or stationary policies. We then analyze the structure of the optimal average cost functions, proving sufficient conditions for them to be constant almost everyw…
This paper extends to Continuous-Time Jump Markov Decision Processes (CTJMDP) the classic result for Markov Decision Processes stating that, for a given initial state distribution, for every policy there is a (randomized) Markov policy, which can be defined in a natural way, such that at each time instance the marginal distributions of state-action pairs for these two policies coincide. It is shown in this paper that this equality takes place for a CTJMDP if the corresponding Markov policy defines a nonexplosive jump Markov process. If this Markov process is explosive, then at each time instan…
This paper presents a new condition for the existence of optimal stationary policies in average-cost continuous-time Markov decision processes with unbounded cost and transition rates, arising from controlled queueing systems. This condition is closely related to the stability of queueing systems. It suggests that the proof of the stability can be exploited to verify the existence of an optimal stationary policy. This new condition is easier to verify than existing conditions. Moreover, several conditions are provided which suffice for the average-cost optimality equality to hold.
In this paper we develop a method to compute the solution to a countable (finite or infinite) set of equations that occurs in many different fields including Markov processes that model queueing systems, birth-and-death processes and inventory systems. The method provides a fast and exact computation of the inverse of the matrix of the coefficients of the system. In contrast, alternative inverse techniques perform much slower and work only for finite size matrices. Furthermore, we provide a procedure to construct the eigenvalues of the matrix under consideration.
This paper, based on the compactness-continuity and finite value conditions, establishes the sufficiency of the class of stationary policies out of the general class of history-dependent ones for a constrained continuous-time Markov decision process in Borel state and action spaces with total nonnegative cost criteria. The controlled process is not necessarily absorbing, the discount factor can be identically equal to zero, and the transition rates can also be equal to zero, which account for the major technical difficulties. Models with these features seemingly had not been handled in the pre…
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.
We study countably infinite Markov decision processes with Büchi objectives, which ask to visit a given subset of states infinitely often. A question left open by T.P. Hill in 1979 is whether there always exist $\varepsilon$-optimal Markov strategies, i.e., strategies that base decisions only on the current state and the number of steps taken so far. We provide a negative answer to this question by constructing a non-trivial counterexample. On the other hand, we show that Markov strategies with only 1 bit of extra memory are sufficient.
In a variety of applications, an agent's success depends on the knowledge that an adversarial observer has or can gather about the agent's decisions. It is therefore desirable for the agent to achieve a task while reducing the ability of an observer to infer the agent's policy. We consider the task of the agent as a reachability problem in a Markov decision process and study the synthesis of policies that minimize the observer's ability to infer the transition probabilities of the agent between the states of the Markov decision process. We introduce a metric that is based on the Fisher informa…
This paper studies the approximation of optimal control policies by quantized (discretized) policies for a very general class of Markov decision processes (MDPs). The problem is motivated by applications in networked control systems, computational methods for MDPs, and learning algorithms for MDPs. We consider the finite-action approximation of stationary policies for a discrete-time Markov decision process with discounted and average costs under a weak continuity assumption on the transition probability, which is a significant relaxation of conditions required in earlier literature. The discr…
We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.
In this paper, we consider the problem of optimization and learning for constrained and multi-objective Markov decision processes, for both discounted rewards and expected average rewards. We formulate the problems as zero-sum games where one player (the agent) solves a Markov decision problem and its opponent solves a bandit optimization problem, which we here call Markov-Bandit games. We extend Q-learning to solve Markov-Bandit games and show that our new Q-learning algorithms converge to the optimal solutions of the zero-sum Markov-Bandit games, and hence converge to the optimal solutions o…
We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded. Under a certain Lyapunov stability assumption on the dynamics, we establish the existence of a value and saddle point equilibrium. Further we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Finally, we present and analyze an illustrative example.
This paper deals with control of partially observable discrete-time stochastic systems. It introduces and studies Markov Decision Processes with Incomplete Information and with semi-uniform Feller transition probabilities. The important feature of these models is that their classic reduction to Completely Observable Markov Decision Processes with belief states preserves semi-uniform Feller continuity of transition probabilities. Under mild assumptions on cost functions, optimal policies exist, optimality equations hold, and value iterations converge to optimal values for these models. In parti…
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 …
This paper proves continuity of value functions in discounted periodic-review single-commodity total-cost inventory control problems with \revision{continuous inventory levels,} fixed ordering costs, possibly bounded inventory storage capacity, and possibly bounded order sizes for finite and infinite horizons. In each of these constrained models, the finite and infinite-horizon value functions are continuous, there exist deterministic Markov optimal finite-horizon policies, and there exist stationary deterministic Markov optimal infinite-horizon policies. For models with bounded inventory stor…
We show that combinations of optimal (stationary) policies in unichain Markov decision processes are optimal. That is, let M be a unichain Markov decision process with state space S, action space A and policies π_j^*: S -> A (1\leq j\leq n) with optimal average infinite horizon reward. Then any combination πof these policies, where for each state i in S there is a j such that π(i)=π_j^*(i), is optimal as well. Furthermore, we prove that any mixture of optimal policies, where at each visit in a state i an arbitrary action π_j^*(i) of an optimal policy is chosen, yields optimal average reward, t…
Markov decision process models and algorithms can be used to identify optimal policies for dispatching ambulances to spatially distributed customers, where the optimal policies indicate the ambulance to dispatch to each customer type in each state. Since the optimal solutions are dependent on Markov state variables, they may not always correspond to a simple set of rules when implementing the policies in practice. Restricted policies that conform to a priority list for each type of customer may be desirable for use in practice, since such policies are transparent, explainable, and easy to impl…
In hierarchical planning for Markov decision processes (MDPs), temporal abstraction allows planning with macro-actions that take place at different time scale in form of sequential composition. In this paper, we propose a novel approach to compositional reasoning and hierarchical planning for MDPs under temporal logic constraints. In addition to sequential composition, we introduce a composition of policies based on generalized logic composition: Given sub-policies for sub-tasks and a new task expressed as logic compositions of subtasks, a semi-optimal policy, which is optimal in planning with…
We consider an auto-scaling technique in a cloud system where virtual machines hosted on a physical node are turned on and off depending on the queue's occupation (or thresholds), in order to minimise a global cost integrating both energy consumption and performance. We propose several efficient optimisation methods to find threshold values minimising this global cost: local search heuristics coupled with aggregation of Markov chain and with queues approximation techniques to reduce the execution time and improve the accuracy. The second approach tackles the problem with a Markov Decision Proc…
Peking University（北京大学）；University of Groningen（格罗宁根大学）
In stochastic dynamic environments, team Markov games have emerged as a versatile paradigm for studying sequential decision-making problems of fully cooperative multi-agent systems. However, the optimality of the derived policies is usually sensitive to model parameters, which are typically unknown and required to be estimated from noisy data in practice. To mitigate the sensitivity of optimal policies to these uncertain parameters, we propose a robust model of team Markov games in this paper, where agents utilize robust optimization approaches to update strategies. This model extends team Mar…
We consider a new form of reinforcement learning (RL) that is based on opportunities to directly learn the optimal control policy and a general Markov decision process (MDP) framework devised to support these opportunities. Derivations of general classes of our control-based RL methods are presented, together with forms of exploration and exploitation in learning and applying the optimal control policy over time. Our general MDP framework extends the classical Bellman operator and optimality criteria by generalizing the definition and scope of a policy for any given state. We establish the con…
Boston University（波士顿大学）；IIT-Bombay（印度理工学院孟买分校）；Zhejiang University（浙江大学）
We consider the problem of learning a policy for a Markov decision process consistent with data captured on the state-actions pairs followed by the policy. We assume that the policy belongs to a class of parameterized policies which are defined using features associated with the state-action pairs. The features are known a priori, however, only an unknown subset of them could be relevant. The policy parameters that correspond to an observed target policy are recovered using $\ell_1$-regularized logistic regression that best fits the observed state-action samples. We establish bounds on the dif…
This paper first describes a class of uncertain stochastic control systems with Markovian switching, and derives an Itô-Liu formula for Markov-modulated processes. And we characterize an optimal control law, which satisfies the generalized Hamilton-Jacobi-Bellman (HJB) equation with Markovian switching. Then, by using the generalized HJB equation, we deduce the optimal consumption and portfolio policies under uncertain stochastic financial markets with Markovian switching. Finally, for constant relative risk-aversion (CRRA) felicity functions, we explicitly obtain the optimal consumption and p…
