Learning Sub-Patterns in Piecewise Continuous Functions

TL;DR

This paper introduces a novel discontinuous deep neural network architecture designed to handle piecewise continuous functions, with theoretical approximation guarantees and a specialized semi-supervised training method, validated on real and synthetic datasets.

Contribution

The paper proposes a new discontinuous neural network model with approximation guarantees and a tailored semi-supervised training procedure for piecewise continuous functions.

Findings

The model achieves universal approximation in the space of piecewise continuous functions.

The semi-supervised training procedure effectively trains the discontinuous neural network.

Experimental results demonstrate the model's performance on financial and synthetic datasets.

Abstract

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies that the neural network's activation function must exhibit a degree of continuity which limits the neural network model's uniform approximation capacity to continuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the network's only and strategically placed, discontinuous unit. We provide approximation guarantees for our architecture in the space of bounded continuous functions and universal approximation guarantees in the space of piecewise continuous functions which we introduced herein. We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, tailored to its structure, and we provide theoretical support for its effectiveness. The performance of our model and trained with the propose procedure is evaluated experimentally on both real-world financial datasets and synthetic datasets.