Robust Deep Network Learning of Nonlinear Regression Tasks by Parametric Leaky Exponential Linear Units (LELUs) and a Diffusion Metric

This document proposes a parametric activation function (ac.f.) aimed at improving multidimensional nonlinear data regression. It is a established knowledge that nonlinear ac.f.'s are required for learning nonlinear datasets. This work shows that smoothness and gradient properties of the ac.f. further impact the performance of large neural networks in terms of overfitting and sensitivity to model parameters. Smooth but vanishing-gradient ac.f.'s such as ELU or SiLU have limited performance and non-smooth ac.f.'s such as RELU and Leaky-RELU further impart discontinuity in the trained model. Improved performance is demonstrated with a smooth "Leaky Exponential Linear Unit", with non-zero gradient that can be trained. A novel diffusion-loss metric is also proposed to gauge the performance of the trained models in terms of overfitting.

Prevention of Overfitting on Mesh-Structured Data Regressions with a Modified Laplace Operator

This document reports on a method for detecting and preventing overfitting on data regressions, herein applied to mesh-like data structures. The mesh structure allows for the straightforward computation of the Laplace-operator second-order derivatives in a finite-difference fashion for noiseless data. Derivatives of the training data are computed on the original training mesh to serve as a true label of the entropy of the training data. Derivatives of the trained data are computed on a staggered mesh to identify oscillations in the interior of the original training mesh cells. The loss of the Laplace-operator derivatives is used for hyperparameter optimisation, achieving a reduction of unwanted oscillation through the minimisation of the entropy of the trained model. In this setup, testing does not require the splitting of points from the training data, and training is thus directly performed on all available training points. The Laplace operator applied to the trained data on a staggered mesh serves as a surrogate testing metric based on diffusion properties.