Learning Sub-Patterns in Piece-Wise Continuous Functions

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters, which requires their activation function to exhibit a degree of continuity. However, this continuity constraint on the activation function prevents these neural models from uniformly approximating discontinuous functions. In this paper, we focus on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. Learning such a function involves identifying the partition of the input space, where each part describes a single continuous sub-pattern of the target function, and then uniformly approximating each of these sub-patterns individually. 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 non-differentiable unit. We provide universal approximation guarantees for our architecture. These include a guarantee that its partition component can approximate any partition of the input space in the upper-Kuratowski sense and a guarantee that our architecture is dense in a large non-separable space of discontinuous functions. Quantitative approximation rates and guarantees for the learnability of a performance-optimizing partition are provided. The performance of our architecture is evaluated using the California Housing Market Dataset.