Parallel and Flexible Dynamic Programming via the Randomized Mini-Batch Operator

TL;DR

This paper introduces a randomized mini-batch operator for dynamic programming that balances convergence speed and parallelization, enabling more flexible and efficient solutions for Markov decision processes.

Contribution

A novel randomized mini-batch operator for DP that combines the convergence benefits of Gauss-Seidel with the parallelism of Bellman, validated through theoretical analysis and extensive experiments.

Findings

The new operator converges faster than Bellman-based methods.

It offers better parallelization than Gauss-Seidel-based methods.

Performance adapts well to different MDP structures and hardware setups.

Abstract

The Bellman operator constitutes the foundation of dynamic programming (DP). An alternative is presented by the Gauss-Seidel operator, whose evaluation, differently from that of the Bellman operator where the states are all processed at once, updates one state at a time, while incorporating into the computation the interim results. The provably better convergence rate of DP methods based on the Gauss-Seidel operator comes at the price of an inherent sequentiality, which prevents the exploitation of modern multi-core systems. In this work we propose a new operator for dynamic programming, namely, the randomized mini-batch operator, which aims at realizing the trade-off between the better convergence rate of the methods based on the Gauss-Seidel operator and the parallelization capability offered by the Bellman operator. After the introduction of the new operator, a theoretical analysis for validating its fundamental properties is conducted. Such properties allow one to successfully deploy the new operator in the main dynamic programming schemes, such as value iteration and modified policy iteration. We compare the convergence of the DP algorithm based on the new operator with its earlier counterparts, shedding light on the algorithmic advantages of the new formulation and the impact of the batch-size parameter on the convergence. Finally, an extensive numerical evaluation of the newly introduced operator is conducted. In accordance with the theoretical derivations, the numerical results show the competitive performance of the proposed operator and its superior flexibility, which allows one to adapt the efficiency of its iterations to different structures of MDPs and hardware setups.