Convex Relaxations of ReLU Neural Networks Approximate Global Optima in Polynomial Time

• URL: http://arxiv.org/abs/2402.03625v3
• Date: Fri, 12 Jul 2024 12:55:53 GMT
• Title: Convex Relaxations of ReLU Neural Networks Approximate Global Optima in Polynomial Time
• Authors: Sungyoon Kim, Mert Pilanci,
• Abstract summary: In this paper, we study the optimality gap between two-layer ReLULU networks regularized with weight decay and their convex relaxations. Our study sheds new light on understanding why local methods work well.
• Score: 45.72323731094864
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we study the optimality gap between two-layer ReLU networks regularized with weight decay and their convex relaxations. We show that when the training data is random, the relative optimality gap between the original problem and its relaxation can be bounded by a factor of O(log n^0.5), where n is the number of training samples. A simple application leads to a tractable polynomial-time algorithm that is guaranteed to solve the original non-convex problem up to a logarithmic factor. Moreover, under mild assumptions, we show that local gradient methods converge to a point with low training loss with high probability. Our result is an exponential improvement compared to existing results and sheds new light on understanding why local gradient methods work well.

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