Asymptotic analysis of online training algorithms in deep learning

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Neural networks have achieved immense practical success in various fields of science, engineering and finance due to their ability to learn high-dimensional, nonlinear relationships from large datasets. The parameters of the neural network are calibrated by 'training' the neural network to minimise an appropriate objective function for a dataset using stochastic gradient descent (SGD) methods. Current mathematical understanding of why SGD methods can successfully train a neural network and how such a neural network generalizes to new out-of-sample data is limited to cases where the neural network has a simple architecture. The objectives of the planned research is twofold. First, we will develop mathematical theory for more sophisticated neural network architectures. A notable example is the family of recurrent neural networks (RNNs), which include a hidden state with the "memory" of the data sequence at previous time steps. Novel approaches are required to analyse RNNs. The memory updates depend strongly on the distribution of the input data sequence, which will be correlated across time. Another example is deep reinforcement learning algorithms such as actor-critic neural network algorithms. These algorithms are challenging to mathematically analyse since they are simultaneously learning the dynamics as well as an optimal policy. Furthermore, the distribution of the data changes as the reinforcement learning model changes during training. In our analysis, we plan to study both single-layer and multi-layer (deep) neural networks. Secondly, our analysis will attempt to study important fundamental questions for the implementation of deep learning models in applications, including how information propagates (the vanishing/exploding gradient problem). Our research will contribute to the mathematical theory of deep learning. Convergence and generalization theory for deep learning models is important to guarantee the reliability and accuracy of deep learning when implemented in applications. This project falls within the following EPSRC research areas: non-linear systems, statistics and applied probability, numerical analysis, and mathematical sciences.