The Role of Neural Network Activation Functions
This provides new theoretical insights for researchers in machine learning, explaining the role of activation functions and justifying common design strategies like weight decay and skip connections, though it is incremental as it builds on existing spline theory.
The paper tackled the theoretical understanding of activation functions in neural networks, showing that training problems relate to infinite-dimensional optimizations over Banach spaces whose solutions are fractional and polynomial splines, with the spline order depending on the activation function.
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as "weight decay" and "path-norm" regularization, and provide a new justification for the use of "skip connections" in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function.