Reduced-Order Neural Network Synthesis with Robustness Guarantees

In the wake of the explosive growth in smartphones and cyberphysical systems, there has been an accelerating shift in how data is generated away from centralised data towards on-device generated data. In response, machine learning algorithms are being adapted to run locally on board, potentially hardware limited, devices to improve user privacy, reduce latency and be more energy efficient. However, our understanding of how these device orientated algorithms behave and should be trained is still fairly limited. To address this issue, a method to automatically synthesize reduced-order neural networks (having fewer neurons) approximating the input/output mapping of a larger one is introduced. The reduced-order neural network's weights and biases are generated from a convex semi-definite programme that minimises the worst-case approximation error with respect to the larger network. Worst-case bounds for this approximation error are obtained and the approach can be applied to a wide variety of neural networks architectures. What differentiates the proposed approach to existing methods for generating small neural networks, e.g. pruning, is the inclusion of the worst-case approximation error directly within the training cost function, which should add robustness. Numerical examples highlight the potential of the proposed approach. The overriding goal of this paper is to generalise recent results in the robustness analysis of neural networks to a robust synthesis problem for their weights and biases.