A Use of Even Activation Functions in Neural Networks

TL;DR

This paper demonstrates that using even activation functions in neural networks, specifically designed to reflect data symmetries, can significantly improve performance in tasks with symmetric target functions.

Contribution

The study introduces a novel approach of integrating data structure knowledge into neural networks via custom even activation functions, validated through theoretical and experimental analysis.

Findings

Using even activation functions improves neural network performance.

Replacing a standard activation with the 'Seagull' function enhances results.

Even activation functions are underutilized but highly promising.

Abstract

Despite broad interest in applying deep learning techniques to scientific discovery, learning interpretable formulas that accurately describe scientific data is very challenging because of the vast landscape of possible functions and the "black box" nature of deep neural networks. The key to success is to effectively integrate existing knowledge or hypotheses about the underlying structure of the data into the architecture of deep learning models to guide machine learning. Currently, such integration is commonly done through customization of the loss functions. Here we propose an alternative approach to integrate existing knowledge or hypotheses of data structure by constructing custom activation functions that reflect this structure. Specifically, we study a common case when the multivariate target function $f$ to be learned from the data is partially exchangeable, \emph{i.e.} $f(u,v,w)=f(v,u,w)$ for $u,v\in \mathbb{R}^d$. For instance, these conditions are satisfied for the classification of images that is invariant under left-right flipping. Through theoretical proof and experimental verification, we show that using an even activation function in one of the fully connected layers improves neural network performance. In our experimental 9-dimensional regression problems, replacing one of the non-symmetric activation functions with the designated "Seagull" activation function $\log(1+x^2)$ results in substantial improvement in network performance. Surprisingly, even activation functions are seldom used in neural networks. Our results suggest that customized activation functions have great potential in neural networks.