arXiv subjects

Pavel Dvurechensky

Publications and source records attributed to Pavel Dvurechensky.

At least 55 records · Page 3Linked to original sources

arXiv2022

Oracle Complexity Separation in Convex Optimization

Many convex optimization problems have structured objective function written as a sum of functions with different types of oracles (full gradient, coordinate derivative, stochastic gradient) and different evaluation complexity of these oracles. In the strongly convex case these functions also have different condition numbers, which eventually define the iteration complexity of first-order methods and the number of oracle calls required to achieve given accuracy. Motivated by the desire to call more expensive oracle less number of times, in this paper we consider minimization of a sum of two functions and propose a generic algorithmic framework to separate oracle complexities for each component in the sum. As a specific example, for the $μ$-strongly convex problem $\min_{x\in \mathbb{R}^n} h(x) + g(x)$ with $L_h$-smooth function $h$ and $L_g$-smooth function $g$, a special case of our algorithm requires, up to a logarithmic factor, $O(\sqrt{L_h/μ})$ first-order oracle calls for $h$ and $O(\sqrt{L_g/μ})$ first-order oracle calls for $g$. Our general framework covers also the setting of strongly convex objectives, the setting when $g$ is given by coordinate derivative oracle, and the setting when $g$ has a finite-sum structure and is available through stochastic gradient oracle. In the latter two cases we obtain respectively accelerated random coordinate descent and accelerated variance reduction methods with oracle complexity separation.

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arXiv2022

Generalized Mirror Prox for Monotone Variational Inequalities: Universality and Inexact Oracle

We introduce an inexact oracle model for variational inequalities (VI) with monotone operator, propose a numerical method which solves such VI's and analyze its convergence rate. As a particular case, we consider VI's with Hölder-continuous operator and show that our algorithm is universal. This means that without knowing the Hölder parameter $ν$ and Hölder constant $L_ν$ it has the best possible complexity for this class of VI's, namely our algorithm has complexity $O\left( \inf_{ν\in[0,1]}\left(\frac{L_ν}{\varepsilon} \right)^{\frac{2}{1+ν}}R^2 \right)$, where $R$ is the size of the feasible set and $\varepsilon$ is the desired accuracy of the solution. We also consider the case of VI's with strongly monotone operator and generalize our method for VI's with inexact oracle and our universal method for this class of problems. Finally, we show, how our method can be applied to convex-concave saddle point problems with Hölder-continuous partial subgradients.

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arXiv2022

An Approach for Non-Convex Uniformly Concave Structured Saddle Point Problem

Recently, saddle point problems have received much attention due to their powerful modeling capability for a lot of problems from diverse domains. Applications of these problems occur in many applied areas, such as robust optimization, distributed optimization, game theory, and many applications in machine learning such as empirical risk minimization and generative adversarial networks training. Therefore, many researchers have actively worked on developing numerical methods for solving saddle point problems in many different settings. This paper is devoted to developing a numerical method for solving saddle point problems in the non-convex uniformly-concave setting. We study a general class of saddle point problems with composite structure and Hölder-continuous higher-order derivatives. To solve the problem under consideration, we propose an approach in which we reduce the problem to a combination of two auxiliary optimization problems separately for each group of variables, outer minimization problem w.r.t. primal variables, and inner maximization problem w.r.t the dual variables. For solving the outer minimization problem, we use the \textit{Adaptive Gradient Method}, which is applicable for non-convex problems and also works with an inexact oracle that is generated by approximately solving the inner problem. For solving the inner maximization problem, we use the \textit{Restarted Unified Acceleration Framework}, which is a framework that unifies the high-order acceleration methods for minimizing a convex function that has Hölder-continuous higher-order derivatives. Separate complexity bounds are provided for the number of calls to the first-order oracles for the outer minimization problem and higher-order oracles for the inner maximization problem. Moreover, the complexity of the whole proposed approach is then estimated.

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arXiv2021

Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

In this paper, we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems, and variational inequalities. This framework allows obtaining many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows constructing new methods, which we illustrate by constructing a universal conditional gradient method and a universal method for variational inequalities with a composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem's smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for variational inequalities (VIs) with such operators. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities. This paper is an extended and updated version of [arXiv:1902.00990]. In particular, we add an extension of relative strong convexity for optimization and variational inequalities.

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arXiv2021

Stochastic Saddle-Point Optimization for Wasserstein Barycenters

We consider the population Wasserstein barycenter problem for random probability measures supported on a finite set of points and generated by an online stream of data. This leads to a complicated stochastic optimization problem where the objective is given as an expectation of a function given as a solution to a random optimization problem. We employ the structure of the problem and obtain a convex-concave stochastic saddle-point reformulation of this problem. In the setting when the distribution of random probability measures is discrete, we propose a stochastic optimization algorithm and estimate its complexity. The second result, based on kernel methods, extends the previous one to the arbitrary distribution of random probability measures. Moreover, this new algorithm has a total complexity better than the Stochastic Approximation approach combined with the Sinkhorn algorithm in many cases. We also illustrate our developments by a series of numerical experiments.

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arXiv2021

Recent Theoretical Advances in Non-Convex Optimization

Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical results on global performance guarantees of optimization algorithms for non-convex optimization. We start with classical arguments showing that general non-convex problems could not be solved efficiently in a reasonable time. Then we give a list of problems that can be solved efficiently to find the global minimizer by exploiting the structure of the problem as much as it is possible. Another way to deal with non-convexity is to relax the goal from finding the global minimum to finding a stationary point or a local minimum. For this setting, we first present known results for the convergence rates of deterministic first-order methods, which are then followed by a general theoretical analysis of optimal stochastic and randomized gradient schemes, and an overview of the stochastic first-order methods. After that, we discuss quite general classes of non-convex problems, such as minimization of $α$-weakly-quasi-convex functions and functions that satisfy Polyak--Lojasiewicz condition, which still allow obtaining theoretical convergence guarantees of first-order methods. Then we consider higher-order and zeroth-order/derivative-free methods and their convergence rates for non-convex optimization problems.

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arXiv2021

On a Combination of Alternating Minimization and Nesterov's Momentum

Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms of the objective for convex problems and $1/k$ in terms of the squared gradient norm for non-convex problems, where $k$ is the iteration counter. Our method does not require any knowledge of neither convexity of the problem nor function parameters such as Lipschitz constant of the gradient, i.e. it is adaptive to convexity and smoothness and is uniformly optimal for smooth convex and non-convex problems. Further, we develop its primal-dual modification for strongly convex problems with linear constraints and prove the same $1/k^2$ for the primal objective residual and constraints feasibility.

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arXiv2021

Generalized Self-Concordant Analysis of Frank-Wolfe algorithms

Projection-free optimization via different variants of the Frank-Wolfe (FW) method has become one of the cornerstones in large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant (GSC) functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex. Indeed, in a number of applications, e.g. inverse covariance estimation or distance-weighted discrimination problems in support vector machines, the loss is given by a GSC function having unbounded curvature, implying absence of theoretical guarantees for the existing FW methods. This paper closes this apparent gap in the literature by developing provably convergent FW algorithms with standard O(1/k) convergence rate guarantees. If the problem formulation allows the efficient construction of a local linear minimization oracle, we develop a FW method with linear convergence rate.

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arXiv2021

Stochastic optimization for dynamic pricing

We consider the problem of supply and demand balancing that is stated as a minimization problem for the total expected revenue function describing the behavior of both consumers and suppliers. In the considered market model we assume that consumers follow the discrete choice demand model, while suppliers are equipped with some quantity adjustment costs. The resulting optimization problem is smooth and convex making it amenable for application of efficient optimization algorithms with the aim of automatically setting prices for online marketplaces. We propose to use stochastic gradient methods to solve the above problem. We interpret the stochastic oracle as a response to the behavior of a random market participant, consumer or supplier. This allows us to interpret the considered algorithms and describe a suitable behavior of consumers and suppliers that leads to fast convergence to the equilibrium in a close to the real marketplace environment.

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arXiv2021

Newton Method over Networks is Fast up to the Statistical Precision

We propose a distributed cubic regularization of the Newton method for solving (constrained) empirical risk minimization problems over a network of agents, modeled as undirected graph. The algorithm employs an inexact, preconditioned Newton step at each agent's side: the gradient of the centralized loss is iteratively estimated via a gradient-tracking consensus mechanism and the Hessian is subsampled over the local data sets. No Hessian matrices are thus exchanged over the network. We derive global complexity bounds for convex and strongly convex losses. Our analysis reveals an interesting interplay between sample and iteration/communication complexity: statistically accurate solutions are achievable in roughly the same number of iterations of the centralized cubic Newton method, with a communication cost per iteration of the order of $\widetilde{\mathcal{O}}\big(1/\sqrt{1-ρ}\big)$, where $ρ$ characterizes the connectivity of the network. This demonstrates a significant communication saving with respect to that of existing, statistically oblivious, distributed Newton-based methods over networks.

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arXiv2021

On Accelerated Methods for Saddle-Point Problems with Composite Structure

We consider strongly-convex-strongly-concave saddle-point problems with general non-bilinear objective and different condition numbers with respect to the primal and the dual variables. First, we consider such problems with smooth composite terms, one of which having finite-sum structure. For this setting we propose a variance reduction algorithm with complexity estimates superior to the existing bounds in the literature. Second, we consider finite-sum saddle-point problems with composite terms and propose several algorithms depending on the properties of the composite terms. When the composite terms are smooth we obtain better complexity bounds than the ones in the literature, including the bounds of a recently proposed nearly-optimal algorithms which do not consider the composite structure of the problem. If the composite terms are prox-friendly, we propose a variance reduction algorithm that, on the one hand, is accelerated compared to existing variance reduction algorithms and, on the other hand, provides in the composite setting similar complexity bounds to the nearly-optimal algorithm which is designed for non-composite setting. Besides that, our algorithms allow to separate the complexity bounds, i.e. estimate, for each part of the objective separately, the number of oracle calls that is sufficient to achieve a given accuracy. This is important since different parts can have different arithmetic complexity of the oracle, and it is desired to call expensive oracles less often than cheap oracles. The key thing to all these results is our general framework for saddle-point problems, which may be of independent interest. This framework, in turn is based on our proposed Accelerated Meta-Algorithm for composite optimization with probabilistic inexact oracles and probabilistic inexactness in the proximal mapping, which may be of independent interest as well.

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arXiv2021

Tensor methods for strongly convex strongly concave saddle point problems and strongly monotone variational inequalities

In this paper we propose three $p$-th order tensor methods for $μ$-strongly-convex-strongly-concave saddle point problems (SPP). The first method is based on the assumption of $p$-th order smoothness of the objective and it achieves a convergence rate of $O \left( \left( \frac{L_p R^{p - 1}}μ \right)^\frac{2}{p + 1} \log \frac{μR^2}{\varepsilon_G} \right)$, where $R$ is an estimate of the initial distance to the solution, and $\varepsilon_G$ is the error in terms of duality gap. Under additional assumptions of first and second order smoothness of the objective we connect the first method with a locally superlinear converging algorithm and develop a second method with the complexity of $O \left( \left( \frac{L_p R^{p - 1}}μ \right)^\frac{2}{p + 1}\log \frac{L_2 R \max \left\{ 1, \frac{L_1}μ \right\}}μ + \log \frac{\log \frac{L_1^3}{2 μ^2 \varepsilon_G}}{\log \frac{L_1 L_2}{μ^2}} \right)$. The third method is a modified version of the second method, and it solves gradient norm minimization SPP with $\tilde O \left( \left( \frac{L_p R^p}{\varepsilon_\nabla} \right)^\frac{2}{p + 1} \right)$ oracle calls, where $\varepsilon_\nabla$ is an error in terms of norm of the gradient of the objective. Since we treat SPP as a particular case of variational inequalities, we also propose three methods for strongly monotone variational inequalities with the same complexity as the described above.

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arXiv2021

An Accelerated Second-Order Method for Distributed Stochastic Optimization

We consider distributed stochastic optimization problems that are solved with master/workers computation architecture. Statistical arguments allow to exploit statistical similarity and approximate this problem by a finite-sum problem, for which we propose an inexact accelerated cubic-regularized Newton's method that achieves lower communication complexity bound for this setting and improves upon existing upper bound. We further exploit this algorithm to obtain convergence rate bounds for the original stochastic optimization problem and compare our bounds with the existing bounds in several regimes when the goal is to minimize the number of communication rounds and increase the parallelization by increasing the number of workers.

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arXiv2021

Solving smooth min-min and min-max problems by mixed oracle algorithms

In this paper, we consider two types of problems that have some similarity in their structure, namely, min-min problems and min-max saddle-point problems. Our approach is based on considering the outer minimization problem as a minimization problem with inexact oracle. This inexact oracle is calculated via inexact solution of the inner problem, which is either minimization or a maximization problem. Our main assumptions are that the problem is smooth and the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem only zeroth-order oracle is available. To solve the inner problem we use accelerated gradient-free method with zeroth-order oracle. To solve the outer problem we use either inexact variant of Vaydya's cutting-plane method or a variant of accelerated gradient method. As a result, we propose a framework that leads to non-asymptotic complexity bounds for both min-min and min-max problems. Moreover, we estimate separately the number of first- and zeroth-order oracle calls which are sufficient to reach any desired accuracy.

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arXiv2021

Zeroth-Order Algorithms for Smooth Saddle-Point Problems

Saddle-point problems have recently gained increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications only a zeroth-order oracle is available. In this paper, we propose several algorithms to solve stochastic smooth (strongly) convex-concave saddle-point problems using zeroth-order oracles and estimate their convergence rate and its dependence on the dimension $n$ of the variable. In particular, our analysis shows that in the case when the feasible set is a direct product of two simplices, our convergence rate for the stochastic term is only by a $\log n$ factor worse than for the first-order methods. We also consider a mixed setup and develop 1/2th-order methods that use zeroth-order oracle for the minimization part and first-order oracle for the maximization part. Finally, we demonstrate the practical performance of our zeroth-order and 1/2th-order methods on practical problems.

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arXiv2021

Numerical methods for the resource allocation problem in networks

In this paper, we consider the resource allocation problem in a network with a large number of connections which are used by a huge number of users. The resource allocation problem under discussion is a maximization problem with linear inequality constraints. To solve this problem we construct the dual problem and propose to use the following numerical optimization methods for the dual: a fast gradient method, a stochastic projected subgradient method, an ellipsoid method, and a random gradient extrapolation method. A special focus is made on the primal-dual analysis of these methods. For each method we estimate the convergence rate. We also provide some modifications of these methods in the setup of distributed computations, taking into account their application to networks.

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arXiv2021

Zeroth-order methods for noisy Hölder-gradient functions

In this paper, we prove new complexity bounds for zeroth-order methods in non-convex optimization with inexact observations of the objective function values. We use the Gaussian smoothing approach of Nesterov and Spokoiny [2015] and extend their results, obtained for optimization methods for smooth zeroth-order non-convex problems, to the setting of minimization of functions with Hölder-continuous gradient with noisy zeroth-order oracle, obtaining noise upper-bounds as well. We consider finite-difference gradient approximation based on normally distributed random Gaussian vectors and prove that gradient descent scheme based on this approximation converges to the stationary point of the smoothed function. We also consider convergence to the stationary point of the original (not smoothed) function and obtain bounds on the number of steps of the algorithm for making the norm of its gradient small. Additionally, we provide bounds for the level of noise in the zeroth-order oracle for which it is still possible to guarantee that the above bounds hold. We also consider separately the case of $ν= 1$ and show that in this case the dependence of the obtained bounds on the dimension can be improved.

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arXiv2021

First-Order Methods for Convex Optimization

First-order methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. First-order methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in large-scale optimization problems. In this survey we cover a number of key developments in gradient-based optimization methods. This includes non-Euclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projection-free methods, and proximal versions of primal-dual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms.

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