The Mathematics of Deep Learning

Machine learning (ML), in particular Deep Learning (DL) is one of the fastest growing areas of modern science and technology, which has potentially enormous and transformative impact on all areas of our life. The applications of DL embrace many disciplines such as (bio-)medical sciences, computer vision, the physical sciences, the social sciences, speech recognition, gaming, music and finance. DL based algorithms are now used to play chess and GO at the highest level, diagnose illness, drive cars, recruit staff and even make legal judgements. The possible applications in the future are almost unlimited. Perhaps DL methods will be used in the future to predict the weather and climate, of even human behaviour. However, alongside this explosive growth has been a concern that there is a lack of explainability behind DL and the way that DL based algorithms make their decisions. This leads to a lack of trustworthiness in the use of the algorithms. A reason for this is that the huge successes of deep learning is not well understood, the results are mysterious, and there is a lack of a clear link between the data training DL algorithms (which is often vague and unstructured) and the decisions made by these algorithms. Part of the reason for this is that DL has advanced so fast, that there is a lack of understanding of its foundations. According to the leading computer scientist Ali Rahimi at NIPS 2017: 'We say things like "machine learning is the new electricity". I'd like to offer another analogy. Machine learning has become alchemy!' Indeed, despite the roots of ML lying in mathematics, statistics and computer science there currently is hardly any rigorous mathematical theory for the setup, training and application performance of deep neural networks. We urgently need the opportunity to change machine learning from alchemy into science. This programme grant aims to rise to this challenge, and, by doing so, to unlock the future potential of artificial intelligence. It aims to put deep learning onto a firm mathematical basis, and will combine theory, modelling, data, computation to unlock the next generation of deep learning. The grant will comprise an interlocked set of work packages aimed to address both the theoretical development of DL (so that it becomes explainable) and the algorithmic development (so that it becomes trustworthy). These will then be linked to the development of DL in a number of key application areas including image processing, partial differential equations and environmental problems. For example we will explore the question of whether it is possible to use DL based algorithms to forecast the weather and climate faster and more accurately than the existing physics based algorithms. The investigators on the grant will be doing both theoretical investigations and will work with end-users of DL in many application areas. Mindful that policy makers are trying to address the many issues raised by DL, the investigators will also reach out to them through a series of workshops and conferences. The results of the work will also be presented to the public at science festivals and other open events.