Discounted Markov Decision Processes Research Articles

Managing Resources for Shared Micromobility: Approximate Optimality in Large-Scale Systems

We consider the problem of managing resources in shared micromobility systems (bike sharing and scooter sharing). An important task in managing such systems is periodic repositioning/recharging/sourcing of units to avoid stockouts or excess inventory at nodes with unbalanced flows. We consider a discrete-time model; each period begins with an initial inventory at each node in the network, and then, customers (demand) materialize at the nodes. Each customer picks up a unit at the origin node and drops it off at a randomly sampled destination node with an origin-specific probability distribution. We model the above network inventory management problem as an infinite horizon discrete-time discounted Markov decision process (MDP) and prove the asymptotic optimality of a novel mean-field approximation to the original MDP as the number of stations becomes large. To compute an approximately optimal policy for the mean-field dynamics, we provide an algorithm with a running time that is logarithmic in the desired optimality gap. Lastly, we compare the performance of our mean field-based policy with state-of-the-art heuristics via numerical experiments, including experiments using Austin scooter-sharing data. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02023 .

Preferences, risk neutrality and risk-sensitive MDPs

Preferences, risk neutrality and risk-sensitive MDPs

Fast Rates for the Regret of Offline Reinforcement Learning

We study the regret of offline reinforcement learning in an infinite-horizon discounted Markov decision process (MDP). While existing analyses of common approaches, such as fitted Q-iteration (FQI), suggest root-n convergence for regret, empirical behavior exhibits much faster convergence. In this paper, we present a finer regret analysis that exactly characterizes this phenomenon by providing fast rates for the regret convergence. First, we show that given any estimate for the optimal quality function, the regret of the policy it defines converges at a rate given by the exponentiation of the estimate’s pointwise convergence rate, thus speeding up the rate. The level of exponentiation depends on the level of noise in the decision-making problem, rather than the estimation problem. We establish such noise levels for linear and tabular MDPs as examples. Second, we provide new analyses of FQI and Bellman residual minimization to establish the correct pointwise convergence guarantees. As specific cases, our results imply one-over-n rates in linear cases and exponential-in-n rates in tabular cases. We extend our findings to general function approximation by extending our results to regret guarantees based on Lp-convergence rates for estimating the optimal quality function rather than pointwise rates, where L2 guarantees for nonparametric estimation can be ensured under mild conditions. Funding: This work was supported by the Division of Information and Intelligent Systems, National Science Foundation [Grant 1846210].

Risk-sensitive discounted Markov decision processes with unbounded reward functions and Borel spaces

This paper attempts to study the risk-sensitive discounted discrete-time Markov decision processes in Borel spaces, in which the reward functions are allowed to be unbounded from above and from below. We find mild conditions imposed on the primitive data of the decision processes, which not only ensure the existence of a solution to the optimality equation (OE in short), but also are the generalization of the bounded reward case. Furthermore, using the OE and a novel technique, we prove the existence of an optimal policy out of the class of randomized history-dependent policies. Finally, we illustrate our results with an inventory system.

On Supervised Online Rolling-Horizon Control for Infinite-Horizon Discounted Markov Decision Processes

This note re-visits the rolling-horizon control approach to the problem of Markov decision process (MDP) with infinite-horizon discounted expected reward criterion. Distinguished from the classical value-iteration approaches, we develop an asynchronous on-line algorithm based on policy iteration integrated with a multi-policy improvement method of policy switching. A sequence of monotonically improving solutions to the forecast-horizon sub-MDP is generated by updating the current solution only at the currently visited state, building in effect a rolling-horizon control policy for the MDP over infinite horizon. Feedbacks from “supervisors,” if available, can be also incorporated while updating. We focus on the convergence issue with a relation to the transition structure of the MDP. Either a global convergence to an optimal forecast-horizon policy or a local convergence to a “locally-optimal” fixed-policy in a finite time is achieved by the algorithm depending on the structure.

Deterministic Discounted Markov Decision Processes with Fuzzy Rewards/Costs

Deterministic Discounted Markov Decision Processes with Fuzzy Rewards/Costs

Optimal discharge of patients from intensive care via a data-driven policy learning framework

Clinical decision support tools rooted in machine learning and optimization can provide significant value to healthcare providers through better management of intensive care units. In particular, it is important that intensive care unit patient discharge decisions account for the nuanced trade-off between decreasing the length of stay and the risk of readmission or death after discharge of a patient. This work introduces a comprehensive framework (i.e., not geared towards any particular disease or condition) for capturing this trade-off and to recommend optimal discharge timing decisions given the electronic health records of a patient. A data-driven approach is used to derive a parsimonious, discrete state space representation to represent the physiological condition of a given patient. Based on this model and a given cost function, an infinite-horizon discounted Markov decision process is formulated and solved numerically to compute an optimal discharge policy, whose performance is assessed using off-policy evaluation strategies. Extensive numerical experiments are performed to validate the proposed framework using real-life intensive care unit patient data.

AlphaSnake: Policy Iteration on a Nondeterministic NP-Hard Markov Decision Process (Student Abstract)

Reinforcement learning has been used to approach well-known NP-hard combinatorial problems in graph theory. Among these, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP), where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning -- or win rate -- with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over 0.5 (a uniform random policy achieves a win rate < 2.57 x 10^{-15}), demonstrating the versatility of AlphaZero in tackling NP-hard problems.

A unified algorithm framework for mean-variance optimization in discounted Markov decision processes

This paper studies the risk-averse mean-variance optimization in infinite-horizon discounted Markov decision processes (MDPs). The involved variance metric concerns reward variability during the whole process, and future deviations are discounted to their present values. This discounted mean-variance optimization yields a reward function dependent on a discounted mean, and this dependency renders traditional dynamic programming methods inapplicable since it suppresses a crucial property—time-consistency. To deal with this unorthodox problem, we introduce a pseudo mean to transform the untreatable MDP to a standard one with a redefined reward function in standard form and derive a discounted mean-variance performance difference formula. With the pseudo mean, we propose a unified algorithm framework with a bilevel optimization structure for the discounted mean-variance optimization. The framework unifies a variety of algorithms for several variance-related problems, including, but not limited to, risk-averse variance and mean-variance optimizations in discounted and average MDPs. Furthermore, the convergence analyses missing from the literature can be complemented with the proposed framework as well. Taking the value iteration as an example, we develop a discounted mean-variance value iteration algorithm and prove its convergence to a local optimum with the aid of a Bellman local-optimality equation. Finally, we conduct a numerical experiment on portfolio management to validate the proposed algorithm.

Geometry and convergence of natural policy gradient methods

We study the convergence of several natural policy gradient (NPG) methods in infinite-horizon discounted Markov decision processes with regular policy parametrizations. For a variety of NPGs and reward functions we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows with the metrics proposed by Kakade and Morimura and co-authors by observing that these arise from the Hessian geometries of conditional entropy and entropy respectively. Further, we obtain sublinear convergence rates for Hessian geometries arising from other convex functions like log-barriers. Finally, we interpret the discrete-time NPG methods with regularized rewards as inexact Newton methods if the NPG is defined with respect to the Hessian geometry of the regularizer. This yields local quadratic convergence rates of these methods for step size equal to the inverse penalization strength.

Turnpikes in Finite Markov Decision Processes and Random Walk

In this paper we revise the theory of turnpikes in discounted Markov decision processes, prove the turnpike theorem for the undiscounted model, and apply the results to the specific random walk.KeywordsturnpikeMarkov decision processdiscounted rewardaverage rewardrandom walkstochastic knapsack problem

Policy Gradient for Continuing Tasks in Discounted Markov Decision Processes

Reinforcement learning aims to find policies that maximize an expected cumulative reward in Markov decision processes with unknown transition probabilities. Policy gradient (PG)-algorithms use stochastic gradients of the value function to update the policy. A major drawback of PG-algorithms is that they are limited to episodic tasks (multiple finite-horizon trajectories) unless stringent stationarity assumptions are imposed on the trajectories. Hence, they need restarts and cannot be fully implemented online, which is critical for systems need to adapt to new tasks and/or environments in deployment. Moreover, the standard stationary formulation ignores transient behaviors. This motivates our study of discounted MDPs of infinite horizon without restarts. However, it is unknown if in this case following stochastic PG-type estimates would improve the policy. The main result of this work is to establish that when policies belong to a reproducing kernel Hilbert space (RKHS), and the kernel is selected properly, then these PG-estimates are ascent directions for the value function conditioned to any arbitrary initial point. This allows us to prove convergence of our online algorithm to the local optima. A numerical example shows that an agent running our online algorithm learns to navigate and succeeds in a surveillance task that requires looping between two goal locations. This example corroborates our theoretical findings about the ascent directions of subsequent stochastic gradients. It also shows how our online algorithm guides the agent through a continuing cyclic trajectory that does not comply with the standard stationarity assumptions in the literature for non-episodic training.

An optimistic value iteration for mean–variance optimization in discounted Markov decision processes

This paper proposes an optimistic value iteration for steady-state mean–variance optimization in infinite-horizon discounted Markov decision processes (MDPs). The involved variance metric concerns reward variability in the long run, and future deviations are discounted to their present values. This mean–variance optimality criterion is time-inconsistent since its reward function depends on the mean, which renders traditional dynamic programming methods inapplicable. A family of policy/value iteration algorithms can be developed under a bilevel optimization algorithm framework, but there are still several problems to solve for a reinforcement learning (RL) extension. One problem is that in the value iteration, the inner optimization should achieve a near-optimal solution to ensure convergence before the outer update, which is required in every outer iteration. However, in an RL scenario, it is impractical to determine when the inner optimization should be stopped. To deal with this problem, we propose an optimistic value iteration, where the outer updates can be merged into the inner optimization with a learning rate. We prove the algorithm convergence and conduct a numerical experiment on portfolio management to validate the proposed algorithm.

Stability-constrained Markov Decision Processes using MPC

In this paper, we consider solving discounted Markov Decision Processes (MDPs) under the constraint that the resulting policy is stabilizing. In practice MDPs are solved based on some form of policy approximation. We will leverage recent results proposing to use Model Predictive Control (MPC) as a structured approximator in the context of Reinforcement Learning, which makes it possible to introduce stability requirements directly inside the MPC-based policy. This will restrict the solution of the MDP to stabilizing policies by construction. Because the stability theory for MPC is most mature for the undiscounted MPC case, we will first show in this paper that stable discounted MDPs can be reformulated as undiscounted ones. This observation will entail that the undiscounted MPC-based policy with stability guarantees will produce the optimal policy for the discounted MDP if it is stable, and the best stabilizing policy otherwise.

Constrained Discounted Stochastic Games

In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state–action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in a locally convex topological vector space. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on an approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using the so-called weighted norm approach.

Nonasymptotic Analysis of Monte Carlo Tree Search

In “Nonasymptotic Analysis of Monte Carlo Tree Search,” D. Shah, Q. Xie, and Z. Xu consider the popular tree-based search strategy, the Monte Carlo Tree Search (MCTS), in the context of the infinite-horizon discounted Markov decision process. They show that MCTS with an appropriate polynomial rather than logarithmic bonus term indeed leads to the desired convergence property. The authors derive the results by establishing a polynomial concentration property of regret for a class of nonstationary multiarm bandits. Furthermore, using this as a building block, they demonstrate that MCTS, combined with nearest neighbor supervised learning, acts as a “policy improvement” operator that can iteratively improve value function approximation.

A new parallelized of hierarchical value iteration algorithm for discounted Markov decision processes

Markov Decision Process (MDP) is a popular mathematical framework for modeling stochastic sequential problems under uncertainty. These models appear in many applications, such as computer science, engineering, telecommunications, and finance, among others. One of the most challenging goals is to deal with complexity reduction in the case of large MDP. In this paper; we propose an optimal strategy deals with large MDP under discount reward. The proposed approach is based on an intelligent combination of a decomposition technique and an efficient parallel strategy. The global MDP is splitting into several 'sub-MDPs', subsequently, these MDPs are classified by level following the strongly connected components principle. A master-slave strategy base on Message Passing Interface (MPI) is proposed to solve the obtained problem. The performance of the proposed approach is shown in terms of scalability, cost, and execution speed.

Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization

Preconditioning and Regularization Enable Faster Reinforcement Learning Natural policy gradient (NPG) methods, in conjunction with entropy regularization to encourage exploration, are among the most popular policy optimization algorithms in contemporary reinforcement learning. Despite the empirical success, the theoretical underpinnings for NPG methods remain severely limited. In “Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization”, Cen, Cheng, Chen, Wei, and Chi develop nonasymptotic convergence guarantees for entropy-regularized NPG methods under softmax parameterization, focusing on tabular discounted Markov decision processes. Assuming access to exact policy evaluation, the authors demonstrate that the algorithm converges linearly at an astonishing rate that is independent of the dimension of the state-action space. Moreover, the algorithm is provably stable vis-à-vis inexactness of policy evaluation. Accommodating a wide range of learning rates, this convergence result highlights the role of preconditioning and regularization in enabling fast convergence.

Variance reduced value iteration and faster algorithms for solving Markov decision processes

AbstractIn this paper we provide faster algorithms for approximately solving discounted Markov decision processes in multiple parameter regimes. Given a discounted Markov decision process (DMDP) with |S| states, |A| actions, discount factor γ ∈ (0, 1), and rewards in the range [−M, M], we show how to compute an ϵ‐optimal policy, with probability 1 − δ in time (Note: We use to hide polylogarithmic factors in the input parameters, that is, .) This contribution reflects the first nearly linear time, nearly linearly convergent algorithm for solving DMDPs for intermediate values of γ. We also show how to obtain improved sublinear time algorithms provided we can sample from the transition function in O(1) time. Under this assumption we provide an algorithm which computes an ϵ‐optimal policy for with probability 1 − δ in time Furthermore, we extend both these algorithms to solve finite horizon MDPs. Our algorithms improve upon the previous best for approximately computing optimal policies for fixed‐horizon MDPs in multiple parameter regimes. Interestingly, we obtain our results by a careful modification of approximate value iteration. We show how to combine classic approximate value iteration analysis with new techniques in variance reduction. Our fastest algorithms leverage further insights to ensure that our algorithms make monotonic progress towards the optimal value. This paper is one of few instances in using sampling to obtain a linearly convergent linear programming algorithm and we hope that the analysis may be useful more broadly.

Feasible Bases for a Polytope Related to the Hamilton Cycle Problem

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.