ReLU Regression: Complexity, Exact and Approximation Algorithms
Santanu Dey, Guanyi Wang, Yao Xie · arXiv (Cornell University) · 2018
Solving ReLU regression problems is similar to training a neural network with one node with ReLU activation function, which aims to fit a model where the response is related to the linear combination of input feature variables. We study the ReLU regression problem from the algorithmic complexity perspective. We show that the ReLu regression is NP-hard in general, and when the number of features $p$ is fixed, there exists an algorithm that achieves the global optimal solution in $O(n^p)$ running time. We also present an integer programming (IP) framework which can produce dual bounds and feasible upper bounds. Moreover, we present a polynomial-time iterative $n$-Approximation Algorithm, which performs well in practice as demonstrated by numerical studies and can be numerically more stable than IP solutions.