arXiv subjects

Alexander Gasnikov

Publications and source records attributed to Alexander Gasnikov.

At least 181 records · Page 10Linked 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

Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling

In this paper we study the convex-concave saddle-point problem $\min_x \max_y f(x) + y^T \mathbf{A} x - g(y)$, where $f(x)$ and $g(y)$ are smooth and convex functions. We propose an Accelerated Primal-Dual Gradient Method (APDG) for solving this problem, achieving (i) an optimal linear convergence rate in the strongly-convex-strongly-concave regime, matching the lower complexity bound (Zhang et al., 2021), and (ii) an accelerated linear convergence rate in the case when only one of the functions $f(x)$ and $g(y)$ is strongly convex or even none of them are. Finally, we obtain a linearly convergent algorithm for the general smooth and convex-concave saddle point problem $\min_x \max_y F(x,y)$ without the requirement of strong convexity or strong concavity.

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arXiv2022

On the relations of stochastic convex optimization problems with empirical risk minimization problems on $p$-norm balls

In this paper, we consider convex stochastic optimization problems arising in machine learning applications (e.g., risk minimization) and mathematical statistics (e.g., maximum likelihood estimation). There are two main approaches to solve such kinds of problems, namely the Stochastic Approximation approach (online approach) and the Sample Average Approximation approach, also known as the Monte Carlo approach, (offline approach). In the offline approach, the problem is replaced by its empirical counterpart (the empirical risk minimization problem). The natural question is how to define the problem sample size, i.e., how many realizations should be sampled so that the quite accurate solution of the empirical problem be the solution of the original problem with the desired precision. This issue is one of the main issues in modern machine learning and optimization. In the last decade, a lot of significant advances were made in these areas to solve convex stochastic optimization problems on the Euclidean balls (or the whole space). In this work, we are based on these advances and study the case of arbitrary balls in the $\ell_p$-norms. We also explore the question of how the parameter $p$ affects the estimates of the required number of terms as a function of empirical risk.

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arXiv2022

Primal-Dual Stochastic Mirror Descent for MDPs

We consider the problem of learning the optimal policy for infinite-horizon Markov decision processes (MDPs). For this purpose, some variant of Stochastic Mirror Descent is proposed for convex programming problems with Lipschitz-continuous functionals. An important detail is the ability to use inexact values of functional constraints and compute the value of dual variables. We analyze this algorithm in a general case and obtain an estimate of the convergence rate that does not accumulate errors during the operation of the method. Using this algorithm, we get the first parallel algorithm for mixing average-reward MDPs with a generative model without reduction to discounted MDP. One of the main features of the presented method is low communication costs in a distributed centralized setting, even with very large networks.

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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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arXiv2022

A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization and Coordinate Descent

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.

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arXiv2022

Vaidya's method for convex stochastic optimization in small dimension

This paper considers a general problem of convex stochastic optimization in a relatively low-dimensional space (e.g., 100 variables). It is known that for deterministic convex optimization problems of small dimensions, the fastest convergence is achieved by the center of gravity type methods (e.g., Vaidya's cutting plane method). For stochastic optimization problems, the question of whether Vaidya's method can be used comes down to the question of how it accumulates inaccuracy in the subgradient. The recent result of the authors states that the errors do not accumulate on iterations of Vaidya's method, which allows proposing its analog for stochastic optimization problems. The primary technique is to replace the subgradient in Vaidya's method with its probabilistic counterpart (the arithmetic mean of the stochastic subgradients). The present paper implements the described plan, which ultimately leads to an effective (if parallel computations for batching are possible) method for solving convex stochastic optimization problems in relatively low-dimensional spaces.

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arXiv2021

Mirror Descent and Constrained Online Optimization Problems

We consider the following class of online optimization problems with functional constraints. Assume, that a finite set of convex Lipschitz-continuous non-smooth functionals are given on a closed set of $n$-dimensional vector space. The problem is to minimize the arithmetic mean of functionals with a convex Lipschitz-continuous non-smooth constraint. In addition, it is allowed to calculate the (sub)gradient of each functional only once. Using some recently proposed adaptive methods of Mirror Descent the method is suggested to solve the mentioned constrained online optimization problem with optimal estimate of accuracy. For the corresponding non-Euclidean prox-structure the case of a set of $n$-dimensional vectors lying on the standard $n$-dimensional simplex is considered.

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arXiv2021

Mirror Descent for Constrained Optimization Problems with Large Subgradient Values

Based on the ideas of arXiv:1710.06612, we consider the problem of minimization of the Holder-continuous non-smooth functional $f$ with non-positive convex (generally, non-smooth) Lipschitz-continuous functional constraint. We propose some novel strategies of step-sizes and adaptive stopping rules in Mirror Descent algorithms for the considered class of problems. It is shown that the methods are applicable to the objective functionals of various levels of smoothness. Applying the restart technique to the Mirror Descent Algorithm there was proposed an optimal method to solve optimization problems with strongly convex objective functionals. Estimates of the rate of convergence of the considered algorithms are obtained depending on the level of smoothness of the objective functional. These estimates indicate the optimality of considered methods from the point of view of the theory of lower oracle bounds. In addition, the case of a quasi-convex objective functional and constraint was considered.

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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 decentralized distributed convex optimization

In the last few years, the theory of decentralized distributed convex optimization has made significant progress. The lower bounds on communications rounds and oracle calls have appeared, as well as methods that reach both of these bounds. In this paper, we focus on how these results can be explained based on optimal algorithms for the non-distributed setup. In particular, we provide our recent results that have not been published yet and that could be found in details only in arXiv preprints.

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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

Convex optimization

This textbook is based on lectures given by the authors at MIPT (Moscow), HSE (Moscow), FEFU (Vladivostok), V.I. Vernadsky KFU (Simferopol), ASU (Republic of Adygea), and the University of Grenoble-Alpes (Grenoble, France). First of all, the authors focused on the program of a two-semester course of lectures on convex optimization, which is given to students of MIPT. The first chapter of this book contains the materials of the first semester ("Fundamentals of convex analysis and optimization"), the second and third chapters contain the materials of the second semester ("Numerical methods of convex optimization"). The textbook has a number of features. First, in contrast to the classic manuals, this book does not provide proofs of all the theorems mentioned. This allowed, on one side, to describe more themes, but on the other side, made the presentation less self-sufficient. The second important point is that part of the material is advanced and is published in the Russian educational literature, apparently for the first time. Third, the accents that are given do not always coincide with the generally accepted accents in the textbooks that are now popular. First of all, we talk about a sufficiently advanced presentation of conic optimization, including robust optimization, as a vivid demonstration of the capabilities of modern convex analysis.

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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

Near-Optimal Decentralized Algorithms for Saddle Point Problems over Time-Varying Networks

Decentralized optimization methods have been in the focus of optimization community due to their scalability, increasing popularity of parallel algorithms and many applications. In this work, we study saddle point problems of sum type, where the summands are held by separate computational entities connected by a network. The network topology may change from time to time, which models real-world network malfunctions. We obtain lower complexity bounds for algorithms in this setup and develop near-optimal methods which meet the lower bounds.

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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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