Quasiconvex Shallow Neural Network

CHENHAN XIAO, Yang Weng

Abstract: Deep neural networks generally have highly non-convex structures, resulting in multiple local optima of network weights. The non-convex network is likely to fail, i.e., being trapped in bad local optima with large errors, especially when the task involves convexity (e.g., linearly separable classification). While convexity is essential in training neural networks, designing a convex network structure without strong assumptions (e.g., linearity) of activation or loss function is challenging. To extract and utilize convexity, this paper presents the QuasiConvex shallow Neural Network (QCNN) architecture with mild assumptions. We first decompose the network into building blocks where quasiconvexity is thoroughly studied. Then, we design additional layers to preserve quasiconvexity where such building blocks are integrated into general networks. The proposed QCNN, interpreted as a quasiconvex optimization problem, allows for efficient training with theoretical guarantees. Specifically, we construct equivalent convex feasibility problems to solve the quasiconvex optimization problem. Our theoretical results are verified via extensive experiments on common machine learning tasks. The quasiconvex structure in QCNN demonstrates even better learning ability than non-convex deep networks in some tasks.

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