Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory
This provides a theoretical foundation for understanding why residual networks work well, which is important for researchers in machine learning and control theory, though it is incremental as it builds on existing links between networks and control systems.
The paper tackles the problem of explaining the universal approximation capabilities of deep residual neural networks by using geometric nonlinear control theory, showing that a residual network with n+1 neurons per layer can approximate any continuous function from ℝⁿ to ℝⁿ arbitrarily well on compact sets under a condition on the activation function.
In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a general sufficient condition for a residual network to have the power of universal approximation by asking the activation function, or one of its derivatives, to satisfy a quadratic differential equation. Many activation functions used in practice satisfy this assumption, exactly or approximately, and we show this property to be sufficient for an adequately deep neural network with $n+1$ neurons per layer to approximate arbitrarily well, on a compact set and with respect to the supremum norm, any continuous function from $\mathbb{R}^n$ to $\mathbb{R}^n$. We further show this result to hold for very simple architectures for which the weights only need to assume two values. The first key technical contribution consists of relating the universal approximation problem to controllability of an ensemble of control systems corresponding to a residual network and to leverage classical Lie algebraic techniques to characterize controllability. The second technical contribution is to identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.