Convex neural network synthesis for robustness in the 1-norm

With neural networks being used to control safety-critical systems, they increasingly have to be both accurate (in the sense of matching inputs to outputs) and robust. However, these two properties are often at odds with each other and a trade-off has to be navigated. To address this issue, this paper proposes a method to generate an approximation of a neural network which is certifiably more robust. Crucially, the method is fully convex and posed as a semi-definite programme. An application to robustifying model predictive control is used to demonstrate the results. The aim of this work is to introduce a method to navigate the neural network robustness/accuracy trade-off.

Mapping back and forth between model predictive control and neural networks

Model predictive control (MPC) for linear systems with quadratic costs and linear constraints is shown to admit an exact representation as an implicit neural network. A method to "unravel" the implicit neural network of MPC into an explicit one is also introduced. As well as building links between model-based and data-driven control, these results emphasize the capability of implicit neural networks for representing solutions of optimisation problems, as such problems are themselves implicitly defined functions.

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.

Robust error bounds for quantised and pruned neural networks

With the rise of smartphones and the internet-of-things, data is increasingly getting generated at the edge on local, personal devices. For privacy, latency and energy saving reasons, this shift is causing machine learning algorithms to move towards a decentralised approach, with the data and algorithms stored and even trained locally on devices. The device hardware becomes the main bottleneck for model performance in this set-up, creating a need for slimmed down, more efficient neural networks. Neural network pruning and quantisation are two methods that have been developed to achieve this, with both approaches demonstrating impressive results in reducing the computational cost without sacrificing too much on model performance. However, our understanding behind these methods remains underdeveloped. To address this issue, a semi-definite program to robustly bound the error caused by pruning and quantising a neural network is introduced in this paper. The method can be applied to generic neural networks, accounts for the many nonlinearities of the problem and holds robustly for all inputs in specified sets. It is hoped that the computed bounds will give certainty to software/control/machine learning engineers implementing these algorithms efficiently on limited hardware.