ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees
This work addresses a fundamental open problem in certifiable neural network-based control for applications like robotics and autonomous systems, though it is incremental in providing theoretical guarantees.
The authors tackled the problem of determining the required complexity of ReLU neural networks to approximate model predictive control policies with guaranteed closed-loop performance, deriving explicit bounds on network width and depth and proposing a non-uniform error framework to reduce complexity and enhance performance.
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.