We prove that, for the fundamental regression task of learning a single neuron, training a one-hidden layer ReLU network of any width by gradient flow from a small initialisation converges to zero loss and is implicitly biased to minimise the rank of network parameters. By assuming that the training points are correlated with the teacher neuron, we complement previous work that considered orthogonal datasets. Our results are based on a detailed non-asymptotic analysis of the dynamics of each hidden neuron throughout the training. We also show and characterise a surprising distinction in this setting between interpolator networks of minimal rank and those of minimal Euclidean norm. Finally we perform a range of numerical experiments, which corroborate our theoretical findings.

We prove that, for the fundamental regression task of learning a single neuron, training a one-hidden layer ReLU network of any width by gradient flow from a small initialisation converges to zero loss and is implicitly biased to minimise the rank of network parameters. By assuming that the training points are correlated with the teacher neuron, we complement previous work that considered orthogonal datasets. Our results are based on a detailed non-asymptotic analysis of the dynamics of each hidden neuron throughout the training. We also show and characterise a surprising distinction in this setting between interpolator networks of minimal rank and those of minimal Euclidean norm. Finally we perform a range of numerical experiments, which corroborate our theoretical findings.

Sparse codes in neuroscience have been suggested to offer certain computational advantages over other neural representations of sensory data. To explore this viewpoint, a sparse code is used to represent natural images in an optimal control task solved with neuro-dynamic programming, and its computational properties are investigated. The central finding is that when feature inputs to a linear network are correlated, an over-complete sparse code increases the memory capacity of the network in an efficient manner beyond that possible for any complete code with the same-sized input, and also increases the speed of learning the network weights. A complete sparse code is found to maximise the memory capacity of a linear network by decorrelating its feature inputs to transform the design matrix of the least-squares problem to one of full rank. It also conditions the Hessian matrix of the least-squares problem, thereby increasing the rate of convergence to the optimal network weights. Other types of decorrelating codes would also achieve this. However, an over-complete sparse code is found to be approximately decorrelated, extracting a larger number of approximately decorrelated features from the same-sized input, allowing it to efficiently increase memory capacity beyond that possible for any complete code: a 2.25 times over-complete sparse code is shown to at least double memory capacity compared with a complete sparse code using the same input. This is used in sequential learning to store a potentially large number of optimal control tasks in the network, while catastrophic forgetting is avoided using a partitioned representation, yielding a cost-to-go function approximator that generalizes over the states in each partition. Sparse code advantages over dense codes and local codes are also discussed.

The reasons behind good DNN generalisation remain an open question. In this paper we explore the problem by looking at the Signal-to-Noise Ratio of nodes in the network. Starting from information theory principles, it is possible to derive an expression for the SNR of a DNN node output. Using this expression we construct figures-of-merit that quantify how well the weights of a node optimise SNR (or, equivalently, information rate). Applying these figures-of-merit, we give examples indicating that weight sets that promote good SNR performance also exhibit good generalisation. In addition, we are able to identify the qualities of weight sets that exhibit good SNR behaviour and hence promote good generalisation. This leads to a discussion of how these results relate to network training and regularisation. Finally, we identify some ways that these observations can be used in training design.