Learning Sub-Patterns in Piecewise Continuous Functions
This work addresses a fundamental problem in machine learning for approximating functions with discontinuities, offering a novel model with theoretical and experimental validation, though it is incremental in extending neural network capabilities.
The paper tackles the limitation of neural networks in approximating piecewise continuous functions due to activation function continuity, proposing a discontinuous deep neural network model with a two-step training procedure. It provides approximation guarantees for bounded continuous and piecewise continuous functions, and demonstrates performance on financial and synthetic datasets.
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.