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Register a new userStochastic Variance Reduction for Variational Inequality Methods
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Yura Malitsky
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We propose stochastic variance reduced algorithms for solving convex-concave saddle point problems, monotone variational inequalities, and monotone inclusions. Our framework applies to extragradient, forward-backward-forward, and forward-reflected-backward methods both in Euclidean and Bregman setups. All proposed methods converge in exactly the same setting as their deterministic counterparts and they either match or improve the best-known complexities for solving structured min-max problems. Our results reinforce the correspondence between variance reduction in variational inequalities and minimization. We also illustrate the improvements of our approach with numerical evaluations on matrix games.
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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.
We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of $tilde{cO}((n+L/mu)ln(1/epsilon))$ for $L$-smooth and $mu$-strongly convex individual functions - one must also know which individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA, SVRG, SAG), it is not possible to get an `accelerated complexity bound of $tilde{cO}((n+sqrt{n L/mu})ln(1/epsilon))$, unless the strong convexity parameter is given explicitly. Lastly, we show that when this class of algorithms is used for minimizing $L$-smooth and convex finite sums, the optimal complexity bound is $tilde{cO}(n+L/epsilon)$, assuming that (on average) the same update rule is used in every iteration, and $tilde{cO}(n+sqrt{nL/epsilon})$, otherwise.
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.
Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and require two evaluations of the map, the former of which could be relatively expensive if $X$ is a complicated set. We consider two related avenues where the per-iteration complexity is significantly reduced: (i) A stochastic projected reflected gradient method requiring a single evaluation of the map and a single projection; and (ii) A stochastic subgradient extragradient method that requires two evaluations of the map, a single projection onto $X$, and a significantly cheaper projection (onto a halfspace) computable in closed form. Under a variance-reduced framework reliant on a sample-average of the map based on an increasing batch-size, we prove almost sure (a.s.) convergence of the iterates to a random point in the solution set for both schemes. Additionally, both schemes display a non-asymptotic rate of $mathcal{O}(1/K)$ where $K$ denotes the number of iterations; notably, both rates match those obtained in deterministic regimes. To address feasibility sets given by the intersection of a large number of convex constraints, we adapt both of the aforementioned schemes to a random projection framework. We then show that the random projection analogs of both schemes also display a.s. convergence under a weak-sharpness requirement; furthermore, without imposing the weak-sharpness requirement, both schemes are characterized by a provable rate of $mathcal{O}(1/sqrt{K})$ in terms of the gap function of the projection of the averaged sequence onto $X$ as well as the infeasibility of this sequence. Preliminary numerics support theoretical findings and the schemes outperform standard extragradient schemes in terms of the per-iteration complexity.
In this work, we propose a distributed algorithm for stochastic non-convex optimization. We consider a worker-server architecture where a set of $K$ worker nodes (WNs) in collaboration with a server node (SN) jointly aim to minimize a global, potentially non-convex objective function. The objective function is assumed to be the sum of local objective functions available at each WN, with each node having access to only the stochastic samples of its local objective function. In contrast to the existing approaches, we employ a momentum based single loop distributed algorithm which eliminates the need of computing large batch size gradients to achieve variance reduction. We propose two algorithms one with adaptive and the other with non-adaptive learning rates. We show that the proposed algorithms achieve the optimal computational complexity while attaining linear speedup with the number of WNs. Specifically, the algorithms reach an $epsilon$-stationary point $x_a$ with $mathbb{E}| abla f(x_a) | leq tilde{O}(K^{-1/3}T^{-1/2} + K^{-1/3}T^{-1/3})$ in $T$ iterations, thereby requiring $tilde{O}(K^{-1} epsilon^{-3})$ gradient computations at each WN. Moreover, our approach does not assume identical data distributions across WNs making the approach general enough for federated learning applications.
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