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Alexander Gasnikov

Publications and source records attributed to Alexander Gasnikov.

At least 199 records · Page 11Linked to original sources
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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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

Lower Bounds and Optimal Algorithms for Smooth and Strongly Convex Decentralized Optimization Over Time-Varying Networks

We consider the task of minimizing the sum of smooth and strongly convex functions stored in a decentralized manner across the nodes of a communication network whose links are allowed to change in time. We solve two fundamental problems for this task. First, we establish the first lower bounds on the number of decentralized communication rounds and the number of local computations required to find an $ε$-accurate solution. Second, we design two optimal algorithms that attain these lower bounds: (i) a variant of the recently proposed algorithm ADOM (Kovalev et al., 2021) enhanced via a multi-consensus subroutine, which is optimal in the case when access to the dual gradients is assumed, and (ii) a novel algorithm, called ADOM+, which is optimal in the case when access to the primal gradients is assumed. We corroborate the theoretical efficiency of these algorithms by performing an experimental comparison with existing state-of-the-art methods.

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arXiv2021

Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model

In this paper we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank--Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages, and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows (i.e.,\ meeting the capacity constraints and induced by flows on the paths) in the stable dynamics model, which provides us with a computable duality gap.

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arXiv2021

Analogues of Switching Subgradient Schemes for Relatively Lipschitz-Continuous Convex Programming Problems

Recently some specific classes of non-smooth and non-Lipschitz convex optimization problems were selected by Yu.~Nesterov along with H.~Lu. We consider convex programming problems with similar smoothness conditions for the objective function and functional constraints. We introduce a new concept of an inexact model and propose some analogues of switching subgradient schemes for convex programming problems for the relatively Lipschitz-continuous objective function and functional constraints. Some class of online convex optimization problems is considered. The proposed methods are optimal in the class of optimization problems with relatively Lipschitz-continuous objective and functional constraints.

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arXiv2021

Flexible Modification of Gauss-Newton Method and Its Stochastic Extension

This work presents a novel version of recently developed Gauss-Newton method for solving systems of nonlinear equations, based on upper bound of solution residual and quadratic regularization ideas. We obtained for such method global convergence bounds and under natural non-degeneracy assumptions we present local quadratic convergence results. We developed stochastic optimization algorithms for presented Gauss-Newton method and justified sub-linear and linear convergence rates for these algorithms using weak growth condition (WGC) and Polyak-Lojasiewicz (PL) inequality. We show that Gauss-Newton method in stochastic setting can effectively find solution under WGC and PL condition matching convergence rate of the deterministic optimization method. The suggested method unifies most practically used Gauss-Newton method modifications and can easily interpolate between them providing flexible and convenient method easily implementable using standard techniques of convex optimization.

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arXiv2021

Improved Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandit

We consider $β$-smooth (satisfies the generalized Holder condition with parameter $β> 2$) stochastic convex optimization problem with zero-order one-point oracle. The best known result was arXiv:2006.07862: $\mathbb{E} \left[f(\overline{x}_N) - f(x^*)\right] = \tilde{\mathcal{O}} \left(\dfrac{n^{2}}{γN^{\frac{β-1}β}} \right)$ in $γ$-strongly convex case, where $n$ is the dimension. In this paper we improve this bound: $\mathbb{E} \left[f(\overline{x}_N) - f(x^*)\right] = \tilde{\mathcal{O}} \left(\dfrac{n^{2-\frac{1}β}}{γN^{\frac{β-1}β}} \right).$

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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

Non-convex optimization in digital pre-distortion of the signal

In this paper, we give some observation of applying modern optimization methods for functionals describing digital predistortion (DPD) of signals with orthogonal frequency division multiplexing (OFDM) modulation. The considered family of model functionals is determined by the class of cascade Wiener--Hammerstein models, which can be represented as a computational graph consisting of various nonlinear blocks. To assess optimization methods with the best convergence depth and rate as a properties of this models family we multilaterally consider modern techniques used in optimizing neural networks and numerous numerical methods used to optimize non-convex multimodal functions. The research emphasizes the most effective of the considered techniques and describes several useful observations about the model properties and optimization methods behavior.

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arXiv2021

Adaptive Catalyst for Smooth Convex Optimization

In this paper, we present a generic framework that allows accelerating almost arbitrary non-accelerated deterministic and randomized algorithms for smooth convex optimization problems. The main approach of our envelope is the same as in Catalyst (Lin et al., 2015): an accelerated proximal outer gradient method, which is used as an envelope for a non-accelerated inner method for the $\ell_2$ regularized auxiliary problem. Our algorithm has two key differences: 1) easily verifiable stopping criteria for inner algorithm; 2) the regularization parameter can be tunned along the way. As a result, the main contribution of our work is a new framework that applies to adaptive inner algorithms: Steepest Descent, Adaptive Coordinate Descent, Alternating Minimization. Moreover, in the non-adaptive case, our approach allows obtaining Catalyst without a logarithmic factor, which appears in the standard Catalyst (Lin et al., 2015, 2018).

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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

One-Point Gradient-Free Methods for Smooth and Non-Smooth Saddle-Point Problems

In this paper, we analyze gradient-free methods with one-point feedback for stochastic saddle point problems $\min_{x}\max_{y} φ(x, y)$. For non-smooth and smooth cases, we present analysis in a general geometric setup with arbitrary Bregman divergence. For problems with higher-order smoothness, the analysis is carried out only in the Euclidean case. The estimates we have obtained repeat the best currently known estimates of gradient-free methods with one-point feedback for problems of imagining a convex or strongly convex function. The paper uses three main approaches to recovering the gradient through finite differences: standard with a random direction, as well as its modifications with kernels and residual feedback. We also provide experiments to compare these approaches for the matrix game.

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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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