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arXiv · 1610.04674

Finite-sum Composition Optimization via Variance Reduced Gradient Descent

Xiangru Lian·Mengdi Wang·Ji Liu
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Abstract

The stochastic composition optimization proposed recently by Wang et al. [2014] minimizes the objective with the compositional expectation form: $\min_x~(\mathbb{E}_iF_i \circ \mathbb{E}_j G_j)(x).$ It summarizes many important applications in machine learning, statistics, and finance. In this paper, we consider the finite-sum scenario for composition optimization: \[\min_x f (x) := \frac{1}{n} \sum_{i = 1}^n F_i \left(\frac{1}{m} \sum_{j = 1}^m G_j (x) \right). \] We propose two algorithms to solve this problem by combining the stochastic compositional gradient descent (SCGD) and the stochastic variance reduced gradient (SVRG) technique. A constant linear convergence rate is proved for strongly convex optimization, which substantially improves the sublinear rate $O(K^{-0.8})$ of the best known algorithm.

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Xiangru Lian, Mengdi Wang, Ji Liu. 2016-10-15. Finite-sum Composition Optimization via Variance Reduced Gradient Descent. https://arxiv.org/abs/1610.04674

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