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Near Optimal Approximations and Finite Memory Policies for POMPDs with Continuous Spaces

Ali Devran Kara, Erhan Bayraktar, Serdar Yuksel

TL;DR

The paper tackles near-optimal control of POMDPs in which state and observation spaces are continuous. It develops two rigorous approximation strategies: discretizing the hidden state to obtain finite-state belief-MDPs with provable near-optimality under regularity and filter-stability assumptions, and discretizing observations together with finite-memory information to yield finite-model approximations with corresponding performance guarantees. It also provides Q-learning algorithms for the approximate models, including finite-memory and discretized-observation variants, and derives bounds that separate discretization effects from memory-effects using Dobrushin and Hilbert-metric based analyses. The results offer scalable, provably near-optimal strategies for continuous-space POMDPs and offer practical guidance on trading off discretization granularity against memory length in learning-based settings.

Abstract

We study an approximation method for partially observed Markov decision processes (POMDPs) with continuous spaces. Belief MDP reduction, which has been the standard approach to study POMDPs requires rigorous approximation methods for practical applications, due to the state space being lifted to the space of probability measures. Generalizing recent work, in this paper we present rigorous approximation methods via discretizing the observation space and constructing a fully observed finite MDP model using a finite length history of the discrete observations and control actions. We show that the resulting policy is near-optimal under some regularity assumptions on the channel, and under certain controlled filter stability requirements for the hidden state process. Furthermore, by quantizing the measurements, we are able to utilize refined filter stability conditions. We also provide a Q learning algorithm that uses a finite memory of discretized information variables, and prove its convergence to the optimality equation of the finite fully observed MDP constructed using the approximation method.

Near Optimal Approximations and Finite Memory Policies for POMPDs with Continuous Spaces

TL;DR

The paper tackles near-optimal control of POMDPs in which state and observation spaces are continuous. It develops two rigorous approximation strategies: discretizing the hidden state to obtain finite-state belief-MDPs with provable near-optimality under regularity and filter-stability assumptions, and discretizing observations together with finite-memory information to yield finite-model approximations with corresponding performance guarantees. It also provides Q-learning algorithms for the approximate models, including finite-memory and discretized-observation variants, and derives bounds that separate discretization effects from memory-effects using Dobrushin and Hilbert-metric based analyses. The results offer scalable, provably near-optimal strategies for continuous-space POMDPs and offer practical guidance on trading off discretization granularity against memory length in learning-based settings.

Abstract

We study an approximation method for partially observed Markov decision processes (POMDPs) with continuous spaces. Belief MDP reduction, which has been the standard approach to study POMDPs requires rigorous approximation methods for practical applications, due to the state space being lifted to the space of probability measures. Generalizing recent work, in this paper we present rigorous approximation methods via discretizing the observation space and constructing a fully observed finite MDP model using a finite length history of the discrete observations and control actions. We show that the resulting policy is near-optimal under some regularity assumptions on the channel, and under certain controlled filter stability requirements for the hidden state process. Furthermore, by quantizing the measurements, we are able to utilize refined filter stability conditions. We also provide a Q learning algorithm that uses a finite memory of discretized information variables, and prove its convergence to the optimality equation of the finite fully observed MDP constructed using the approximation method.
Paper Structure (26 sections, 16 theorems, 108 equations, 1 figure)

This paper contains 26 sections, 16 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convergence Notions for Probability Measures and Regularity Properties of Transition Kernels
  3. Partially Observed Markov Decision Processes and Standard Solution Methods
  4. Reduction to Fully Observed Models Using Belief States
  5. Discretization of Action Sets
  6. Approximation via Quantization of the (Hidden) State Space
  7. Near Optimal Policy Construction Based on Finite State Space
  8. Asymptotic Analysis
  9. A Discussion on the Hidden Space Discretization
  10. Finite Memory Based Compression
  11. An Alternative Finite Window Belief-MDP Reduction
  12. Approximation of the Finite Window Belief-MDP
  13. Quantization of Observations and Loss in Performance
  14. Effect of Observation Discretization
  15. Near Optimality of Finite Memory Feedback with Approximated Discrete Observations
  16. ...and 11 more sections

Key Result

Lemma 1

Under Assumption channel_kernel_reg, we have that where

Figures (1)

  • Figure 1: Construction of the Finite-Window Approximate MDP from the Finite-Window Belief-MDPkara2021convergence. The quantization of the finite window MDP model leads to the collapse of the first coordinate to a fixed measure.

Theorems & Definitions (30)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark
  • ...and 20 more