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Exact finite approximations of average-cost countable Markov Decision Processes
Arie Leizarowitz, Adam Shwartz
TL;DR
Countable-state MDPs are difficult to analyze and approximate, especially when finite truncations provide only asymptotic guarantees. The paper constructs a finite-state embedding that preserves relevant dynamics and cost quantities, yielding an exact approximation with the same optimal cost under stated recurrence and non-drifting conditions. Approximate excursion calculations retain continuity but lose exact optimal-cost equality.
Problem
Countable-state MDPs are more difficult to study than finite-state MDPs, while truncation typically provides asymptotic approximations without error estimates.
Method
The paper constructs a finite-state MDP by embedding the dynamics and excursion costs of a countable-state MDP around a finite approximating set.
Results
The embedded MDP has the same optimal cost as the original MDP and an optimal policy agreeing with the original on corresponding states when a recurrent state and finite cycle cost are available.
Takeaways & Limitations
Finite embeddings provide exact finite approximations that are more convenient for computation and implementation while preserving the original model’s restricted dynamics and optimality.
Takeaways & Limitations
The exact result is restricted to non-drifting MDPs and requires a recurrent state with finite cycle cost; approximating excursion costs or times makes the embedding non-exact.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper addresses the analytical and numerical difficulty of countable-state MDPs under a long-run average-cost criterion. It proposes finite exact approximations that preserve optimal cost and restricted dynamics, while excluding drifting MDPs.
- Problem formulation: The model consists of a countable state space, compact state-dependent action sets, running costs, and probabilistic transitions, with long-run average cost minimized over admissible policies.
- Countable-state MDPs are harder to study analytically and numerically than finite-state MDPs.
- State clustering gives exact relations but requires equivalent states to share transitions and immediate costs under every action, a structure seldom found in applications.
- Truncation applies more generally, but its cost and policy guarantees are typically asymptotic and lack error estimates.
- The proposed finite embedding makes the approximating MDP’s optimal cost agree with the countable MDP’s and preserves optimal policies on corresponding approximating states.The approach is developed for general compact action spaces.
- Problem formulation: The analysis restricts attention to non-drifting MDPs because drifting MDPs cannot preserve cost structure, optimality, and dynamics in a finite approximation.The paper notes that coercive cost conditions can ensure non-drifting behavior.
2 The embedding
The embedding replaces excursions outside a finite subset with a finite-state construction that preserves return behavior, costs, and relevant policy performance. Its exactness follows from matching the induced dynamics and excursion quantities on corresponding states.
- The construction defines exit and return times η and τ, then uses excursion behavior outside Z to construct transitions and costs for auxiliary states ωi.If the initial state lies in Z, η is the first exit time and τ the first return time.
- The embedding associates finite subsets Z0 and Z through a one-to-one mapping e and imitates stationary-policy performance on corresponding states.The construction concerns finite subsets where the original policy has nontrivial dynamics.
- For states in Z, the embedded process matches the original process’s return-state distribution under corresponding stationary Markov policies.The embedding definition requires identical distributions at the first return times τ and τ0.
- The embedding accounts for excursion costs and times, whose exact matching is needed to preserve the original process’s cost flow and return behavior.The paper relates this calculation to cycle-time methods and Poisson-equation computations.
- The existence of embedding: Theorem 2.3 establishes that every finite set Z in a countable-state MDP admits a finite-state embedded MDP with the required embedding properties.The construction specifies the finite model’s state, action, cost, and transition components.
- Actions at auxiliary states encode entrance probabilities, excursion timing, and cost through α = (λq1, ..., λqn, c).The parameter λ preserves conditional entrance probabilities while adjusting expected entrance time to match the original process.
3 Existence of optimal policies
The section establishes conditions under which a finite embedding preserves average-cost behavior for a recurrent policy and identifies when an optimal policy yields a suitable finite approximating set. Under recurrence and finite cycle cost, the embedded model preserves optimal cost and agrees with the original policy on corresponding approximating states.
- Existence of suitable recurrent states: An estimate of the relevant cost condition can identify a finite set containing a recurrent state of a policy.The construction does not require computing the optimal policy when the estimate is obtained by restricting attention to a special policy class.
- Policy-level comparison: Theorem 3.1 compares a recurrent policy in the original MDP with its associated policy in the embedded MDP.The comparison uses a recurrent state z in the approximating set and assumes the cycle cost is well-defined, with nonnegative immediate costs providing an alternative sufficient condition.
- Embedding proof: The proof matches transition probabilities, immediate costs, and cycle costs while the process remains in the approximating set.The Markov property and conditional independence support the correspondence of excursion probabilities, while the remaining cost discrepancy vanishes geometrically through repeated excursions.
- Optimal-policy consequence: Theorem 3.3 states that if an optimal stationary Markov policy has a recurrent state z with finite cycle cost, the embedded MDP has the same optimal cost.The embedded MDP also has an optimal policy that agrees with the original policy on corresponding states of the approximating sets.
- Assumptions: The existence of a stationary Markov optimal policy is an explicit theorem assumption, although the paper notes that it holds for most applications under standard conditions.The average cost is also assumed to be independent of the initial state under standard conditions.
4 Extensions
The extensions apply the embedding to constrained models and specialized applications while preserving exact cost correspondence under the stated construction. They also describe boundaries: approximate excursion quantities lose exactness, constrained models are more sensitive to calculation errors, and multidimensional applications may require further approximations.
- Approximate embeddings: Approximate excursion costs and mean excursion times produce a non-exact embedding, but embedded optimal costs approach the original costs as those estimates improve.Exact equality of optimal costs is therefore replaced by continuity with respect to the approximation errors.
- Constrained MDPs: Exact embedding extends to constrained MDPs, where the original objective is minimized subject to long-run average-cost constraints.The extension introduces additional immediate cost functions and corresponding constrained average-cost functionals.
- Constrained MDPs: Hard constraints make standard constrained approximations more difficult because feasibility can be lost even when the original problem is feasible.The paper states that the exact approximation avoids this infeasibility difficulty.
- Applications: For the stated application model, the embedding preserves all costs of the original model and permits approximation by a finite chain.A specialized action representation is defined before applying the same embedding arguments.
- Limitations: In constrained models, errors in cycle-cost calculations may cause infeasibility, whereas in the optimization problem they may lead only to sub-optimality.Thus, without exact cycle-cost computation, the constrained extension shares the infeasibility problem of traditional approximation methods.
5 Examples
The examples show how finite embedding applies to reservoir-control and queueing models, often reducing computation to a finite set while preserving exact costs under stated conditions. They also identify settings where dimensionality or approximate excursion calculations limit practicality or exactness.
- Water reservoirs: Reservoir control becomes a discrete-state, finite-action MDP after discretizing inflows, water levels, and control variables.The state is (L_t, D_t), the action is (Y_t, Z_t), and bounded controls yield finitely many actions.
- Water reservoirs: Embedding can restrict reservoir optimization to a finite set F when states outside F permit only maximal controls.This uses Y_t = ȳ and Z_t = z̄ outside F, although the omitted state set may be significant in two dimensions.
- Water reservoirs: When inflows are i.i.d., random-walk results can calculate entrance distributions and mean times outside F, simplifying the embedding.If c(l, z̄) is constant outside F, the cycle cost is a constant multiple of the average return time.
- Single queues: For a controlled single queue, service is controlled below threshold I and fixed at maximal rate μ for queue sizes x ≥ I.With sub-linear costs for x ≥ I, state 0 is recurrent under any policy and cycle costs are finite, allowing computation of the embedded model.
- Single queues: Skip-free one-dimensional queues permit easy decoupling of behavior below and above I, while batch arrivals or service require implicit cost-flow expressions.The resulting embedded-model expression uses costs accumulated until the first hit of {0, 1, ..., I−1}.
- Multidimensional queues: Multidimensional queueing models use queue sizes as states and service-queue selection as controls, with positive-coefficient conditions making hitting-cost computations feasible.The model has one infinite queue per customer type and a single server choosing which queue to serve.