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arXiv · 1304.0740

O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions

Lijun Zhang·Tianbao Yang·Rong Jin·Xiaofei He
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Abstract

Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as positive semi-definite cones, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batch, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal $O(1/T)$ rate of convergence with only $O(\log T)$ projections. Our empirical study verifies the theoretical result.

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Lijun Zhang, Tianbao Yang, Rong Jin, Xiaofei He. 2013-04-02. O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions. https://arxiv.org/abs/1304.0740

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