We consider the stochastic composition optimization problem proposed in wang2017stochastic , which has applications ranging from estimation to statistical and machine learning. We propose the first ADMM-based algorithm named com-SVR-ADMM, and show that com-SVR-ADMM converges linearly for strongly convex and Lipschitz smooth objectives, and has a convergence rate of @math , which improves upon the @math rate in wang2016accelerating when the objective is convex and Lipschitz smooth. Moreover, com-SVR-ADMM possesses a rate of @math when the objective is convex but without Lipschitz smoothness. We also conduct experiments and show that it outperforms existing algorithms.
The stochastic composition optimization problem was first proposed in @cite_13 , where two solution algorithms, Basic SCGD and accelerated SCGD, were proposed. The algorithms were shown to achieve a sublinear convergence rate for convex and strongly convex cases, and were also shown to converge to a stationary point in the nonconvex case. Later, @cite_0 proposed a proximal gradient algorithm called ASC-PG to improve the convergence rate when both inner and outer functions are smooth. However, the convergence rate is sublinear and their results do not include the regularizer when the objective functions are not strongly convex. In @cite_4 , the authors solved the finite sample case of stochastic composition optimization and obtained two linear-convergent algorithms based on the stochastic variance reduction gradient technique (SVRG) proposed in @cite_15 . However, the algorithms do not handle the regularizer either.
@cite_0 @cite_15 @cite_13 @cite_4
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Consider the stochastic composition optimization problem where the objective is a composition of two expected-value functions. We propose a new stochastic first-order method, namely the accelerated stochastic compositional proximal gradient (ASC-PG) method, which updates based on queries to the sampling oracle using two different timescales. The ASC-PG is the first proximal gradient method for the stochastic composition problem that can deal with nonsmooth regularization penalty. We show that the ASC-PG exhibits faster convergence than the best known algorithms, and that it achieves the optimal sample-error complexity in several important special cases. We further demonstrate the application of ASC-PG to reinforcement learning and conduct numerical experiments. Consider the stochastic composition optimization problem where the objective is a composition of two expected-value functions. We propose a new stochastic first-order method, namely the accelerated stochastic compositional proximal gradient (ASC-PG) method, which updates based on queries to the sampling oracle using two different timescales. The ASC-PG is the first proximal gradient method for the stochastic composition problem that can deal with nonsmooth regularization penalty. We show that the ASC-PG exhibits faster convergence than the best known algorithms, and that it achieves the optimal sample-error complexity in several important special cases. We further demonstrate the application of ASC-PG to reinforcement learning and conduct numerical experiments. The stochastic composition optimization proposed recently by [2014] minimizes the objective with the compositional expectation form: @math It summarizes many important applications in machine learning, statistics, and finance. In this paper, we consider the finite-sum scenario for composition optimization: [ f (x) := 1 n i = 1 ^n F_i ( 1 m j = 1 ^m G_j (x) ). ] 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 @math of the best known algorithm. Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to the inherent variance in the gradient computation. To accelerate the convergence, some variance-reduced stochastic gradient algorithms, e.g., proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently been proposed to solve strongly convex problems. Under the strongly convex condition, these variance-reduced stochastic gradient algorithms achieve a linear convergence rate. However, many machine learning problems are convex but not strongly convex. In this paper, we introduce Prox-SVRG and its projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of non-strongly convex optimization problems widely used in machine learning. As the main technical contribution of this paper, we show that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is the first to be rigorously proved for a class of non-strongly convex problems in both constrained and regularized settings. Moreover, the SSC inequality is independent of algorithms and may be applied to analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which may be of independent interest. To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithms on solving both constrained and regularized problems without strong convexity.