Non-convex Optimization

Jun 11, 2019



PA-GD: On the Convergence of Perturbed Alternating Gradient Descent to Second-Order Stationary Points for Structured Nonconvex Optimization Alternating gradient descent (A-GD) is a simple but popular algorithm in machine learning, which updates two blocks of variables in an alternating manner using gradient descent steps. %, in which a gradient step is taken on one block, while keeping the remaining block fixed. In this paper, we consider a smooth unconstrained nonconvex optimization problem, and propose a {\bf p}erturbed {\bf A}-{\bf GD} (PA-GD) which is able to converge (with high probability) to the second-order stationary points (SOSPs) with a global sublinear rate. {Existing analysis on A-GD type algorithm either only guarantees convergence to first-order solutions, or converges to second-order solutions asymptotically (without rates).} To the best of our knowledge, this is the first alternating type algorithm that takes O(polylog(d)/ϵ2) iterations to achieve an (ϵ,√ϵ)-SOSP with high probability, where polylog(d)denotes the polynomial of the logarithm with respect to problem dimension d. Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al., 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a √ϵ-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification. Faster Stochastic Alternating Direction Method of Multipliers for Nonconvex Optimization In this paper, we propose a faster stochastic alternating direction method of multipliers (ADMM) for nonconvex optimization by using a new stochastic path-integrated differential estimator (SPIDER), called as SPIDER-ADMM. As a major contribution, we give a new theoretical analysis framework for nonconvex stochastic ADMM methods with providing the optimal incremental first-order oracle (IFO) complexity. Specifically, we prove that our SPIDER-ADMM achieves a record-breaking IFO complexity of O(n+n1/2ϵ−1) for finding an ϵ -approximate solution, which improves the deterministic ADMM by a factor O(n1/2), where n denotes the sample size. Based on our new analysis framework, we also prove that the existing ADMM-based non-convex optimization algorithms, SVRG-ADMM and SAGA-ADMM, have the optimal IFO complexity as O(n+n2/3ϵ−1). Thus, SPIDER-ADMM improves the existing stochastic ADMM methods by a factor of O(n1/6). Moreover, we extend SPIDER-ADMM to the online setting, and propose a faster online ADMM, \emph{i.e.}, online SPIDER-ADMM. Our theoretical analysis shows that the online SPIDER-ADMM has the IFO complexity of O(ϵ−32) for finding an ϵ-approximate solution, which improves the existing best results by a factor of O(ϵ12). Finally, the experimental results on benchmark datasets validate that the proposed algorithms have faster convergence rate than the existing ADMM algorithms for nonconvex optimization. Lower Bounds for Smooth Nonconvex Finite-Sum Optimization Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsolved. In this paper, we study the lower bounds for smooth nonconvex finite-sum optimization, where the objective function is the average of n nonconvex component functions. We prove tight lower bounds for the complexity of finding ϵ -suboptimal point and ϵ -approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function). Given our lower bounds, we can show that existing algorithms including {KatyushaX} \citep{allen2018katyushax}, {Natasha} \citep{allen2017natasha} and {StagewiseKatyusha} \citep{yang2018does} have achieved optimal {Incremental First-order Oracle} (IFO) complexity (i.e., number of IFO calls) up to logarithm factors for nonconvex finite-sum optimization. We also point out potential ways to further improve these complexity results, in terms of making stronger assumptions or by a different convergence analysis. Nonconvex Variance Reduced Optimization with Arbitrary Sampling We provide the first importance sampling variants of variance reduced algorithms for empirical risk minimization with non-convex loss functions. In particular, we analyze non-convex versions of \texttt{SVRG}, \texttt{SAGA} and \texttt{SARAH}. Our methods have the capacity to speed up the training process by an order of magnitude compared to the state of the art on real datasets. Moreover, we also improve upon current mini-batch analysis of these methods by proposing importance sampling for minibatches in this setting. Surprisingly, our approach can in some regimes lead to superlinear speedup with respect to the minibatch size, which is not usually present in stochastic optimization. All the above results follow from a general analysis of the methods which works with {\em arbitrary sampling}, i.e., fully general randomized strategy for the selection of subsets of examples to be sampled in each iteration. Finally, we also perform a novel importance sampling analysis of \texttt{SARAH} in the convex setting. Error Feedback Fixes SignSGD and other Gradient Compression Schemes Sign-based algorithms (e.g. signSGD) have been proposed as a biased gradient compression technique to alleviate the communication bottleneck in training large neural networks across multiple workers. We show simple convex counter-examples where signSGD does not converge to the optimum. Further, even when it does converge, signSGD may generalize poorly when compared with SGD. These issues arise because of the biased nature of the sign compression operator. Finally we show that using error-feedback, i.e. incorporating the error made by the compression operator into the next step, overcomes these issues. We prove that our algorithm, EF-SGD, with arbitrary compression operator, achieves the \emph{same rate of convergence} as SGD without any additional assumptions, indicating that we get gradient compression \emph{for free}. Our experiments thoroughly substantiate the theory showing the superiority of our algorithm. A Composite Randomized Incremental Gradient Method We consider the problem of minimizing the composition of a smooth function (which can be nonconvex) and a smooth vector mapping, where both of them can be express as the average of a large number of components. We propose a composite randomized incremental gradient method by extending the SAGA framework. The gradient sample complexity of our method matches that of several recently developed methods based on SVRG in the general case. However, for structured problems where linear convergence rates can be obtained, our method can be much better for ill-conditioned problems. In addition, when the finite-sum structure only appear for the inner mapping, the sample complexity of our method is the same as that of SAGA for minimizing finite sum of smooth nonconvex functions, despite the additional outer composition and the stochastic composite gradients being biased in our case. Optimal Continuous DR-Submodular Maximization and Applications to Provable Mean Field Inference Mean field inference for discrete graphical models is generally a highly nonconvex problem, which also holds for the class of probabilistic log-submodular models. Existing optimization methods, e.g., coordinate ascent algorithms, can only generate local optima. In this work we propose provable mean filed methods for probabilistic log-submodular models and its posterior agreement (PA) with strong approximation guarantees. The main algorithmic technique is a new Double Greedy scheme, termed DR-DoubleGreedy, for continuous DR-submodular maximization with box-constraints. It is a one-pass algorithm with linear time complexity, reaching the optimal 1/2 approximation ratio, which may be of independent interest. We validate the superior performance of our algorithms against baselines on both synthetic and real-world datasets. Multiplicative Weights Updates as a distributed constrained optimization algorithm: Convergence to second-order stationary points almost always Non-concave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers ([Ge et al. 2015, Lee et al 2017] and references therein) indicate that first order methods work well and avoid saddles points. Results as in [Lee \etal 2017], however, are limited to the \textit{unconstrained} case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on \textit{constrained} non-concave maximization. We analyze a variant of a well-established algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem max x∈DP(x), where P is non-concave, twice continuously differentiable and D is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order KKT conditions. We combine techniques from dynamical systems as well as taking advantage of a recent connection between Baum Eagon inequality and MWU [Palaiopanos et al 2017]. Katalyst: Boosting Convex Katayusha for Non-Convex Problems with a Large Condition Number An important class of non-convex objectives that has wide applications in machine learning consists of a sum of n smooth functions and a non-smooth convex function. Tremendous studies have been devoted to conquering these problems by leveraging one of the two types of variance reduction techniques, i.e., SVRG-type that computes a full gradient occasionally and SAGA-type that maintains n stochastic gradients at every iteration. In practice, SVRG-type is preferred to SAGA-type due to its potentially less memory costs. An interesting question that has been largely ignored is how to improve the complexity of variance reduction methods for problems with a large condition number that measures the degree to which the objective is close to a convex function. In this paper, we present a simple but non-trivial boosting of a state-of-the-art SVRG-type method for convex problems (namely Katyusha) to enjoy an improved complexity for solving non-convex problems with a large condition number (that is close to a convex function). To the best of our knowledge, its complexity has the best dependence on n and the degree of non-convexity, and also matches that of a recent SAGA-type accelerated stochastic algorithm for a constrained non-convex smooth optimization problem.



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