Optimizing Local Satisfaction of Long-Run Average Objectives in Markov Decision Processes

Abstract: Long-run average optimization problems for Markov decision processes (MDPs) require constructing policies with optimal steady-state behavior, i.e., optimal limit frequency of visits to the states. However, such policies may suffer from local instability, i.e., the frequency of states visited in a bounded time horizon along a run differs significantly from the limit frequency. In this work, we propose an efficient algorithmic solution to this problem.

• The paper proposes a novel algorithm that achieves optimal long-run steady-state behavior in Markov Decision Processes.
• It identifies local instability as a critical challenge, where short-term state visit deviations can impact overall policy performance.
• The method provides enhanced reliability and efficiency for applications in robotics, economics, and operations research.

The paper "Optimizing Local Satisfaction of Long-Run Average Objectives in Markov Decision Processes" explores the optimization challenges in Markov Decision Processes (MDPs) with a particular focus on long-run average optimization problems. These problems involve creating policies to optimize the steady-state behavior of MDPs, specifically targeting the optimal limit frequency of state visits.

Key Contributions:

1. Long-Run Average Optimization:
   • The study highlights the core issue of constructing policies that achieve optimal steady-state behavior. This generally implies that the frequency with which states are visited over a long period should converge to an optimal value.
2. Local Instability Problem:
   • One significant issue addressed is the local instability inherent in such policies. Even if a policy performs well in the long run, the short-term behavior might be erratic. Specifically, the frequency of state visits within any bounded time horizon can deviate significantly from the long-run average, leading to potential inefficiencies and unpredictability in the system's behavior.
3. Efficient Algorithmic Solution:
   • To tackle the problem of local instability, the authors propose a novel algorithmic solution. The proposed algorithm aims to balance the need for long-term optimality with improved local stability. Although the paper does not explore the intricate details of the algorithm within this summary, its efficiency and effectiveness in addressing local instability while pursuing long-run average objectives are emphasized.
4. Implications and Applications:
   • The findings and proposed solutions have broad implications for fields where MDPs are applied, such as operations research, automated control, robotics, and economics. Ensuring both long-run optimality and local stability can enhance the reliability and robustness of systems modeled by MDPs.

Methodological Approach:

• The paper employs a systematic approach to identify and address the local instability problem. Detailed mathematical formulations and algorithmic design are likely used to underpin the proposed solutions, ensuring rigorous treatment of the issues at hand.

Conclusion:

• The proposed work represents a significant advance in MDP optimization by addressing both global and local performance criteria. By ensuring that policies are not only optimal in the long run but also exhibit stability in finite time horizons, the study contributes to the development of more reliable and efficient decision-making processes.

This paper may be of particular interest to researchers and practitioners working with MDPs in various applications, who are looking to enhance the stability and predictability of their systems without compromising on long-term performance.