Finite Model Approximations and Asymptotic Optimality of Quantized Policies in Decentralized Stochastic Control

TL;DR

This paper demonstrates that finite model approximations via quantization can closely approximate optimal strategies in decentralized stochastic control problems, establishing asymptotic optimality and applying results to classical problems like Witsenhausen's counterexample.

Contribution

It introduces a method for constructing finite, quantized models that approximate optimal policies in decentralized control, proving asymptotic optimality and providing new solutions for classical problems.

Findings

Quantized policies can approximate optimal strategies arbitrarily closely.

Finite models through uniform quantization are asymptotically optimal.

First rigorous construction of ε-optimal strategies for Witsenhausen's problem.

Abstract

In this paper, we consider finite model approximations of a large class of static and dynamic team problems where these models are constructed through uniform quantization of the observation and action spaces of the agents. The strategies obtained from these finite models are shown to approximate the optimal cost with arbitrary precision under mild technical assumptions. In particular, quantized team policies are asymptotically optimal. This result is then applied to Witsenhausen's celebrated counterexample and the Gaussian relay channel problem. For the Witsenhausen's counterexample, our approximation approach provides, to our knowledge, the first rigorously established result that one can construct an $\varepsilon$-optimal strategy for any $\varepsilon > 0$ through a solution of a simpler problem.