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Register a new userImproved Oracle Complexity of Variance Reduced Methods for Nonsmooth Convex Stochastic Composition Optimization
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Tianyi Lin
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We consider the nonsmooth convex composition optimization problem where the objective is a composition of two finite-sum functions and analyze stochastic compositional variance reduced gradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much attention given their edge over stochastic compositional gradient descent (SCGD); but the theoretical analysis exclusively assumes strong convexity of the objective, which excludes several important examples such as Lasso, logistic regression, principle component analysis and deep neural nets. In contrast, we prove non-asymptotic incremental first-order oracle (IFO) complexity of SCVRG or its novel variants for nonsmooth convex composition optimization and show that they are provably faster than SCGD and gradient descent. More specifically, our method achieves the total IFO complexity of $Oleft((m+n)logleft(1/epsilonright)+1/epsilon^3right)$ which improves that of $Oleft(1/epsilon^{3.5}right)$ and $Oleft((m+n)/sqrt{epsilon}right)$ obtained by SCGD and accelerated gradient descent (AGD) respectively. Experimental results confirm that our methods outperform several existing methods, e.g., SCGD and AGD, on sparse mean-variance optimization problem.
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Convex composition optimization is an emerging topic that covers a wide range of applications arising from stochastic optimal control, reinforcement learning and multi-stage stochastic programming. Existing algorithms suffer from unsatisfactory sample complexity and practical issues since they ignore the convexity structure in the algorithmic design. In this paper, we develop a new stochastic compositional variance-reduced gradient algorithm with the sample complexity of $O((m+n)log(1/epsilon)+1/epsilon^3)$ where $m+n$ is the total number of samples. Our algorithm is near-optimal as the dependence on $m+n$ is optimal up to a logarithmic factor. Experimental results on real-world datasets demonstrate the effectiveness and efficiency of the new algorithm.
We consider a generic empirical composition optimization problem, where there are empirical averages present both outside and inside nonlinear loss functions. Such a problem is of interest in various machine learning applications, and cannot be directly solved by standard methods such as stochastic gradient descent. We take a novel approach to solving this problem by reformulating the original minimization objective into an equivalent min-max objective, which brings out all the empirical averages that are originally inside the nonlinear loss functions. We exploit the rich structures of the reformulated problem and develop a stochastic primal-dual algorithm, SVRPDA-I, to solve the problem efficiently. We carry out extensive theoretical analysis of the proposed algorithm, obtaining the convergence rate, the computation complexity and the storage complexity. In particular, the algorithm is shown to converge at a linear rate when the problem is strongly convex. Moreover, we also develop an approximate version of the algorithm, named SVRPDA-II, which further reduces the memory requirement. Finally, we evaluate our proposed algorithms on several real-world benchmarks, and experimental results show that the proposed algorithms significantly outperform existing techniques.
In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To address these limitations, this paper proposes fast randomized stochastic algorithms for non-convex SBO problems. First, we present a stochastic method for non-convex SBO with only one lower problem and establish its sample complexity of $O(1/epsilon^3)$ for finding an $epsilon$-stationary point under Lipschitz continuous conditions of stochastic oracles, matching the lower bound for stochastic smooth non-convex optimization. Second, we present a randomized stochastic method for non-convex SBO with $m>1$ lower level problems (multi-task SBO) by processing a constant number of lower problems at each iteration, and establish its sample complexity no worse than $O(m/epsilon^3)$, which could be a better complexity than that of simply processing all $m$ lower problems at each iteration. Lastly, we establish even faster convergence results for gradient-dominant functions. To the best of our knowledge, this is the first work considering multi-task SBO and developing state-of-the-art sample complexity results.
This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $epsilon$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the existing approaches in terms of both the oracle complexity and practical implementation. The GT-HSGD algorithm implements specialized local hybrid stochastic gradient estimators that are fused over the network to track the global gradient. Remarkably, GT-HSGD achieves a network topology-independent oracle complexity of $O(n^{-1}epsilon^{-3})$ when the required error tolerance $epsilon$ is small enough, leading to a linear speedup with respect to the centralized optimal online variance-reduced approaches that operate on a single node. Numerical experiments are provided to illustrate our main technical results.
Stochastic gradient methods (SGMs) have been extensively used for solving stochastic problems or large-scale machine learning problems. Recent works employ various techniques to improve the convergence rate of SGMs for both convex and nonconvex cases. Most of them require a large number of samples in some or all iterations of the improved SGMs. In this paper, we propose a new SGM, named PStorm, for solving nonconvex nonsmooth stochastic problems. With a momentum-based variance reduction technique, PStorm can achieve the optimal complexity result $O(varepsilon^{-3})$ to produce a stochastic $varepsilon$-stationary solution, if a mean-squared smoothness condition holds and $Theta(varepsilon^{-1})$ samples are available for the initial update. Different from existing optimal methods, PStorm can still achieve a near-optimal complexity result $tilde{O}(varepsilon^{-3})$ by using only one or $O(1)$ samples in every update. With this property, PStorm can be applied to online learning problems that favor real-time decisions based on one or $O(1)$ new observations. In addition, for large-scale machine learning problems, PStorm can generalize better by small-batch training than other optimal methods that require large-batch training and the vanilla SGM, as we demonstrate on training a sparse fully-connected neural network and a sparse convolutional neural network.
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