Point-Based Value Iteration for POMDPs with Neural Perception Mechanisms

TL;DR

This paper introduces a novel approach for solving continuous-state POMDPs with neural perception, using a piecewise linear convex representation and two value iteration algorithms, enabling formal policy synthesis in neural-symbolic systems.

Contribution

It proposes a new P-PWLC representation for NS-POMDPs and extends Bellman backups, along with two value iteration algorithms, including an approximate method, for continuous-state neural perception models.

Findings

Demonstrates the approach on case studies with trained neural perception networks.

Proves convexity and continuity of the value functions in the proposed model.

Shows the practical applicability of the algorithms in synthesizing near-optimal policies.

Abstract

The increasing trend to integrate neural networks and conventional software components in safety-critical settings calls for methodologies for their formal modelling, verification and correct-by-construction policy synthesis. We introduce neuro-symbolic partially observable Markov decision processes (NS-POMDPs), a variant of continuous-state POMDPs with discrete observations and actions, in which the agent perceives a continuous-state environment using a neural {\revise perception mechanism} and makes decisions symbolically. The perception mechanism classifies inputs such as images and sensor values into symbolic percepts, which are used in decision making. We study the problem of optimising discounted cumulative rewards for NS-POMDPs. Working directly with the continuous state space, we exploit the underlying structure of the model and the neural perception mechanism to propose a novel piecewise linear and convex representation (P-PWLC) in terms of polyhedra covering the state space and value vectors, and extend Bellman backups to this representation. We prove the convexity and continuity of value functions and present two value iteration algorithms that ensure finite representability. The first is a classical (exact) value iteration algorithm extending the $\alpha$-functions of Porta {\em et al} (2006) to the P-PWLC representation for continuous-state spaces. The second is a point-based (approximate) method called NS-HSVI, which uses the P-PWLC representation and belief-value induced functions to approximate value functions from below and above for two types of beliefs, particle-based and region-based. Using a prototype implementation, we show the practical applicability of our approach on two case studies that employ (trained) ReLU neural networks as perception functions, by synthesising (approximately) optimal strategies.