Scalable Design of Robust Neural Network Controllers

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Research description: Research into neural network models is a large field and requires a variety of mathematical techniques to address any relevant research questions. There has been a recent resurgence of interest due to the increase in the prevalence of big-data and computational power available. Examples of such areas include image recognition, weather prediction and natural language processing. One important consideration is the increasing use of neural networks in safety-critical applications, such as autonomous vehicle technology. This accentuates the biggest shortcoming of neural networks, which is their sensitivity to adversarial inputs: small changes in the input set can lead to large changes in the output. Despite considerable effort from the research community to improve our understanding and to allow certification of neural networks, to date guarantees on these systems are not sufficient for their widespread use in safety-critical applications. This doctoral project will build upon the existing research to explore various problems related to the robustness of neural networks. One popular method that has seen a large amount of success is to use bounds on the activation functions within these networks to provide such guarantees. However, due to the large number of possible ways to bound the activation functions, there is a trade-off between conservativeness and complexity. It is possible to improve the scalability of optimization problem by using theory from chordal graphs, where large constraints matrices are split into equivalent smaller constraints matrices. These ideas can also be combined with Sum of Squares programming - a technique that uses semi-definite programming. This technique can be used to obtain tighter bounds on the neural network output, whilst maintaining a computational scalable method of obtaining a solution. These ideas can also be extended to neural network controllers, to provide better control performance and robustness of a feedback system. Aims and objectives: The end goal of this work is to create a framework to design robust neural network controllers in a scalable way. The robustness can be quantified using stability theory and determined through solving a Sum of Squares program. Since neural network structures can become very large, determining the stability can be computational expensive. However, there are ways to reformulate the problem to reduce this computational burden by using ideas from chordal sparsity. These techniques are a key area that is being explored in this project. The main questions that will be addressed in this doctoral project focus on combining Sum of Squares programming and chordal sparsity to the neural network verification problem. Once this is established the next objective is to see how these ideas extend to neural network controllers and then how they can be used to improve the performance of feedback systems. Novelty of the research methodology: Sum of Squares techniques have not yet been applied to problems surrounding neural networks. This has led to gaps in the research area, which will be explored in this DPhil. Combining Sum of Squares and sparsity exploiting methods is an open research area. Neural network controllers are also emerging in the research community and there are many questions that need exploring.