Polynomial-Time Solutions for ReLU Network Training: A Complexity Classification via Max-Cut and Zonotopes

• URL: http://arxiv.org/abs/2311.10972v1
• Date: Sat, 18 Nov 2023 04:41:07 GMT
• Title: Polynomial-Time Solutions for ReLU Network Training: A Complexity Classification via Max-Cut and Zonotopes
• Authors: Yifei Wang and Mert Pilanci
• Abstract summary: We prove that the hardness of approximation of ReLU networks not only mirrors the complexity of the Max-Cut problem but also, in certain special cases, exactly corresponds to it. In particular, when $epsilonleqsqrt84/83-1approx 0.006$, we show that it is NP-hard to find an approximate global dataset of the ReLU network objective with relative error $epsilon$ with respect to the objective value.
• Score: 70.52097560486683
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate the complexity of training a two-layer ReLU neural network with weight decay regularization. Previous research has shown that the optimal solution of this problem can be found by solving a standard cone-constrained convex program. Using this convex formulation, we prove that the hardness of approximation of ReLU networks not only mirrors the complexity of the Max-Cut problem but also, in certain special cases, exactly corresponds to it. In particular, when $\epsilon\leq\sqrt{84/83}-1\approx 0.006$, we show that it is NP-hard to find an approximate global optimizer of the ReLU network objective with relative error $\epsilon$ with respect to the objective value. Moreover, we develop a randomized algorithm which mirrors the Goemans-Williamson rounding of semidefinite Max-Cut relaxations. To provide polynomial-time approximations, we classify training datasets into three categories: (i) For orthogonal separable datasets, a precise solution can be obtained in polynomial-time. (ii) When there is a negative correlation between samples of different classes, we give a polynomial-time approximation with relative error $\sqrt{\pi/2}-1\approx 0.253$. (iii) For general datasets, the degree to which the problem can be approximated in polynomial-time is governed by a geometric factor that controls the diameter of two zonotopes intrinsic to the dataset. To our knowledge, these results present the first polynomial-time approximation guarantees along with first hardness of approximation results for regularized ReLU networks.

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