Finite-sum Composition Optimization via Variance Reduced Gradient Descent
Xiangru Lian, Mengdi Wang, Ji Liu
Abstract
The stochastic composition optimization proposed recently by Wang et al. [2014] minimizes the objective with the composite expectation form: $\min_x (\mathbbE_iF_i ∘\mathbbE_j G_j)(x).$ It summarizes many important applications in machine learning, statistics, and finance. In this paper, we consider the finite-sum scenario for composition optimization: $\min_x f (x) := \frac1n \sum_i = 1^n F_i \left( \frac1m \sum_j = 1^m G_j (x) \right)$. In this paper, two algorithms are proposed 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.
BibTeX
@InProceedings{pmlr-v54-lian17a,
title = {{Finite-sum Composition Optimization via Variance Reduced Gradient Descent}},
author = {Lian, Xiangru and Wang, Mengdi and Liu, Ji},
booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics},
pages = {1159--1167},
year = {2017},
editor = {Singh, Aarti and Zhu, Jerry},
volume = {54},
series = {Proceedings of Machine Learning Research},
month = {20--22 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v54/lian17a/lian17a.pdf},
url = {https://proceedings.mlr.press/v54/lian17a.html},
abstract = {The stochastic composition optimization proposed recently by Wang et al. [2014] minimizes the objective with the composite expectation form: $\min_x (\mathbbE_iF_i ∘\mathbbE_j G_j)(x).$ It summarizes many important applications in machine learning, statistics, and finance. In this paper, we consider the finite-sum scenario for composition optimization: $\min_x f (x) := \frac1n \sum_i = 1^n F_i \left( \frac1m \sum_j = 1^m G_j (x) \right)$. In this paper, two algorithms are proposed 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. }
}