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arXiv · 1503.03712

On Graduated Optimization for Stochastic Non-Convex Problems

Elad Hazan·Kfir Y. Levy·Shai Shalev-Shwartz
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Abstract

The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms of theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimiza- tion and analyze its performance. We characterize a parameterized family of non- convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an {\epsilon}-approximate solution within O(1/\epsilon^2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of zero-order optimization, and devise a a variant of our algorithm which converges at rate of O(d^2/\epsilon^4).

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Elad Hazan, Kfir Y. Levy, Shai Shalev-Shwartz. 2015-03-12. On Graduated Optimization for Stochastic Non-Convex Problems. https://arxiv.org/abs/1503.03712

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