The Convex Geometry of Backpropagation: Neural Network Gradient Flows Converge to Extreme Points of the Dual Convex Program

• URL: http://arxiv.org/abs/2110.06488v1
• Date: Wed, 13 Oct 2021 04:17:08 GMT
• Title: The Convex Geometry of Backpropagation: Neural Network Gradient Flows Converge to Extreme Points of the Dual Convex Program
• Authors: Yifei Wang, Mert Pilanci
• Abstract summary: We study non- subgradient flows for two-layer ReLULU networks from a convex implicit geometry and duality perspective. We show that we can identify the problem of non- subgradient descent via primal-dual correspondence.
• Score: 26.143558180103334
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study non-convex subgradient flows for training two-layer ReLU neural networks from a convex geometry and duality perspective. We characterize the implicit bias of unregularized non-convex gradient flow as convex regularization of an equivalent convex model. We then show that the limit points of non-convex subgradient flows can be identified via primal-dual correspondence in this convex optimization problem. Moreover, we derive a sufficient condition on the dual variables which ensures that the stationary points of the non-convex objective are the KKT points of the convex objective, thus proving convergence of non-convex gradient flows to the global optimum. For a class of regular training data distributions such as orthogonal separable data, we show that this sufficient condition holds. Therefore, non-convex gradient flows in fact converge to optimal solutions of a convex optimization problem. We present numerical results verifying the predictions of our theory for non-convex subgradient descent.

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