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.
Highlights
Since their introduction in [1], neural networks have led to numerous advances across various scientific areas
As is for instance the case in many signal processing or mathematical finance [23, 24] situations the uniform limit theorem from classical topology [25] guarantees that the worst-case approximation error of f by feedforward neural networks (FFNNs) cannot be controlled; the average error incurred by approximating f by FFNNs can be [11, 26, 27]
It illustrates the challenge of learning a piecewise continuous functions with two parts by an FFNN with ReLU activation function 2 hidden layers and 100 neurons in each layer and a comparable PCNN model
Summary
Since their introduction in [1], neural networks have led to numerous advances across various scientific areas. [31] considers an extreme learning machine approach by randomizing all but the network’s final linear layer, which reduces the training task to a classical linear regression problem This approach’s provided approximation results are strictly weaker than the known guarantees for classical feedforward networks with a continuous activation function obtained, as are derived for example in [32]. It illustrates the challenge of learning a piecewise continuous functions with two parts (in grey and orange) by an FFNN with ReLU activation function 2 hidden layers and 100 neurons in each layer (in purple) and a comparable PCNN model (green). The model’s performance is benchmarked against comparable deep neural models trained using conventional training algorithms
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