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Online POMDP Planning with Anytime Deterministic Optimality Guarantees

Moran Barenboim, Vadim Indelman

TL;DR

The paper tackles online POMDP planning by deriving deterministic bounds that relate a tractable simplified POMDP to the original, enabling certificates of policy quality at any planning node. It introduces two algorithmic instantiations, DB-POMCP and RB-POMCP, which attach these deterministic bounds to decision-making and, in the case of RB-POMCP, exploration as well. The approach yields finite-time convergence guarantees and enables pruning of suboptimal actions via bound intervals, offering potential safety benefits in uncertain environments. Empirical evaluations across classic POMDP benchmarks demonstrate improved decision-making and scalable planning under finite horizons, while also highlighting limitations in very large observation spaces and opportunities for tighter, more informed bounds. Overall, the work bridges theoretical guarantees with practical online planning, paving the way for certifiably safe and efficient autonomous decision-making under uncertainty.

Abstract

Decision-making under uncertainty is a critical aspect of many practical autonomous systems due to incomplete information. Partially Observable Markov Decision Processes (POMDPs) offer a mathematically principled framework for formulating decision-making problems under such conditions. However, finding an optimal solution for a POMDP is generally intractable. In recent years, there has been a significant progress of scaling approximate solvers from small to moderately sized problems, using online tree search solvers. Often, such approximate solvers are limited to probabilistic or asymptotic guarantees towards the optimal solution. In this paper, we derive a deterministic relationship for discrete POMDPs between an approximated and the optimal solution. We show that at any time, we can derive bounds that relate between the existing solution and the optimal one. We show that our derivations provide an avenue for a new set of algorithms and can be attached to existing algorithms that have a certain structure to provide them with deterministic guarantees with marginal computational overhead. In return, not only do we certify the solution quality, but we demonstrate that making a decision based on the deterministic guarantee may result in superior performance compared to the original algorithm without the deterministic certification.

Online POMDP Planning with Anytime Deterministic Optimality Guarantees

TL;DR

The paper tackles online POMDP planning by deriving deterministic bounds that relate a tractable simplified POMDP to the original, enabling certificates of policy quality at any planning node. It introduces two algorithmic instantiations, DB-POMCP and RB-POMCP, which attach these deterministic bounds to decision-making and, in the case of RB-POMCP, exploration as well. The approach yields finite-time convergence guarantees and enables pruning of suboptimal actions via bound intervals, offering potential safety benefits in uncertain environments. Empirical evaluations across classic POMDP benchmarks demonstrate improved decision-making and scalable planning under finite horizons, while also highlighting limitations in very large observation spaces and opportunities for tighter, more informed bounds. Overall, the work bridges theoretical guarantees with practical online planning, paving the way for certifiably safe and efficient autonomous decision-making under uncertainty.

Abstract

Decision-making under uncertainty is a critical aspect of many practical autonomous systems due to incomplete information. Partially Observable Markov Decision Processes (POMDPs) offer a mathematically principled framework for formulating decision-making problems under such conditions. However, finding an optimal solution for a POMDP is generally intractable. In recent years, there has been a significant progress of scaling approximate solvers from small to moderately sized problems, using online tree search solvers. Often, such approximate solvers are limited to probabilistic or asymptotic guarantees towards the optimal solution. In this paper, we derive a deterministic relationship for discrete POMDPs between an approximated and the optimal solution. We show that at any time, we can derive bounds that relate between the existing solution and the optimal one. We show that our derivations provide an avenue for a new set of algorithms and can be attached to existing algorithms that have a certain structure to provide them with deterministic guarantees with marginal computational overhead. In return, not only do we certify the solution quality, but we demonstrate that making a decision based on the deterministic guarantee may result in superior performance compared to the original algorithm without the deterministic certification.
Paper Structure (43 sections, 8 theorems, 87 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 43 sections, 8 theorems, 87 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Simplified POMDP
  5. Anytime Deterministic Guarantees for Simplified POMDPs
  6. Simplified Observation Space
  7. Fixed Policy Guarantees for Simplified Observation Spaces
  8. Optimality Guarantees for Simplified Observation Spaces
  9. Simplified State and Observation Spaces
  10. Fixed policy guarantees
  11. Optimality Guarantees
  12. Early Stopping Criteria
  13. Exploration Strategies
  14. Impact of POMDP Characteristics on Deterministic Bounds
  15. Algorithms
  16. ...and 28 more sections

Key Result

Theorem 1

Let $b_t$ belief state at time $t$, and $T$ be the last time step of the POMDP. Let $V^{\pi}(b_t)$ be the theoretical value function by following a policy $\pi$, and let $\bar{V}^{\pi}(b_t)$ be the simplified value function, as defined in def:simplifiedValueFunc, by following the same policy. Then,

Figures (3)

  • Figure 1: The figure depicts two search trees: a complete tree (left) that considers all states and observations at each planning step, and a simplified tree (right) that incorporates only a subset of states and observations, linked to simplified models. Our methodology establishes a deterministic link between these two trees.
  • Figure 2: Bound intervals for different actions. The optimal value function is guaranteed to be between the maximal lower and upper bounds. As a result, actions $a^2$ and $a^4$ are suboptimal and can be pruned safely.
  • Figure 3: The graphs show the measured planning time for RB-POMCP and DB-POMCP to find the optimal action for Rock Sample under different UCT coefficient values. Guaranteeing the optimal action made possible by using the bounds in corollary \ref{['lemma:udbOptimality']}. All simulation runs were capped at 3,600 seconds.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1.1
  • proof
  • Theorem 2
  • proof
  • Corollary 2.1
  • proof
  • ...and 11 more