ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees

TL;DR

This paper develops explicit bounds on ReLU neural network complexity needed to accurately approximate MPC policies while guaranteeing closed-loop performance, addressing a key trade-off in control applications.

Contribution

It introduces a projection-based method and a state-aware error framework to determine network size and improve approximation accuracy for ReLU-based MPC.

Findings

Derived explicit bounds on network width and depth for guaranteed performance

Proposed a non-uniform error framework with adaptive scaling

Established a Lipschitz continuity property for the MPC cost function

Abstract

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we propose a projection-based method to enforce hard constraints and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with guaranteed closed-loop performance. To further reduce network complexity and enhance closed-loop performance, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.