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Alexander Gasnikov

Publications and source records attributed to Alexander Gasnikov.

At least 271 records · Page 15Linked to original sources
arXiv2018

About accelerated randomized methods

We show how one can obtain nonaccelerated randomized coordinate descent method (Yu. Nesterov, 2010) and nonaccelerated method of randomization of sum-type functional (Le Roux-Schmidt-Bach, 2012) from the optimal method for the stochastic optimization problem (SIGMA, Devolder-Glineur-Nesterov-Dvurechensky-Gasnikov, 2014). The main trick is a special restart technique. We considered this trick to be usefull in others contexts. We consider only strongly convex case. We show that accelerated variants of this methods seems to be nontrivial in this context. That is, it is hard (perhaps impossible) to obtain accelerated variants using the same trick. We also propose new approach for accelerated coordinate descent methods. This approach is based on the coupling technique (Allen-Zhu-Orrechia, 2015) and allows us: to generalize accelerated coordinate descent methods for conditional optimization problems, to obtain the dual solution due to the primal-dual nature, to extend Universal method (Yu. Nesterov, 2013) to accelerated coordinate descent methods etc.

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arXiv2018

About universality of entropy in stochastic chemical kinetic models

This paper studies the relationship between the Lyapunov function of a macrosystem whose dynamics is governed by the laws of stochastic chemical kinetics and the invariant measure of this macrosystem arising at large times. A necessary and sufficient condition for the reduction of the search problem for the equilibrium of the macrosystem (the most probable macrostate of the invariant measure of this macrosystem) to an entropy-linear programming problem is given.

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arXiv2018

A universal modification of the linear coupling method

In the late sixties, N. Shor and B. Polyak independently proposed optimal first-order methods for non-smooth convex optimization problems. In 1982 A. Nemirovski proposed optimal first-order methods for smooth convex optimization problems, which utilized auxiliary line search. In 1985 A. Nemirovski and Yu. Nesterov proposed a parametric family of optimal first-order methods for convex optimization problems with intermediate smoothness. In 2013 Yu. Nesterov proposed a universal gradient method which combined all the good properties of the previous methods, except the possibility of using auxiliary line search. One can typically observe that in practice auxiliary line search improves performance for many tasks. In this paper, we propose the apparently first such method of non-smooth convex optimization allowing for the use of the line search procedure. Moreover, it is based on the universal gradient method, which does not require any a priori information about the actual degree of smoothness of the problem. Numerical experiments demonstrate that the proposed method is, in some cases, considerably faster than Nesterov's universal gradient method.

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arXiv2018

Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. This means that neither stepsize nor stopping rule require to know the Lipschitz constant of the objective or constraint. We also construct Mirror Descent for problems with objective function, which is not Lipschitz continuous, e.g. is a quadratic function. Besides that, we address the problem of recovering the solution of the dual problem.

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arXiv2017

Toy model for traffic flow splitting

We propose a simple dynamic model splitting users of the transport network by type of movement: personal transport and public transport. A distinctive feature proposed in the article is the existence of an evolutionary approach component in the justification of the method of splitting.

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arXiv2017

Universal similar triangulars method for searching equilibriums in traffic flow distribution models

We describe how primal-dual version of Nesterov's Universal Method (with one projection) can be applied to solve the dual problem for the problem of searching equilibrium traffic flow distribution in transport networks (BMW-model, Stable Dynamic model). We observe that although the dual problem isn't smooth Universal Methods works as it is Mirror Prox method and we consider saddle-point type problem.

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arXiv2017

Universal fast gradient method for stochastic composit optimization problems

We propose a new simple variant of Fast Gradient Method that requires only one projection per iteration. We called this method Triangle Method (TM) because it has a corresponding geometric description. We generalize TM for convex and strictly convex composite optimization problems. Then we propose Universal Triangle Method (UTM) for convex and strictly convex composite optimization problems (see Yu. Nesterov, Math. Program. 2015. for more details about what is Universal Fast Gradient Method). Finally, based on mini-batch technique we propose Stochastic Universal Triangle Method (SUTM). SUTM can be applied to stochastic convex and strictly convex composite optimization problems. Denote, that all the methods TM, UTM, SUTM are continuous on strictly convexity parameter and all of them reach known lower bounds. With additional assumption about the structure of the problems these methods work better than the lower bounds.

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arXiv2017

Newton-Kantorovich-Vietoris method

In this short work we consider nonlinear equation in Banach space $x = A(x)$. We combine classical Newton method with Vietoris method to propose new more general method. Under rather general conditions about the noise we investigate local convergence of the method.

math.FA↗
arXiv2017

Adaptive Similar Triangles Method: a Stable Alternative to Sinkhorn's Algorithm for Regularized Optimal Transport

In this paper, we are motivated by two important applications: entropy-regularized optimal transport problem and road or IP traffic demand matrix estimation by entropy model. Both of them include solving a special type of optimization problem with linear equality constraints and objective given as a sum of an entropy regularizer and a linear function. It is known that the state-of-the-art solvers for this problem, which are based on Sinkhorn's method (also known as RSA or balancing method), can fail to work, when the entropy-regularization parameter is small. We consider the above optimization problem as a particular instance of a general strongly convex optimization problem with linear constraints. We propose a new algorithm to solve this general class of problems. Our approach is based on the transition to the dual problem. First, we introduce a new accelerated gradient method with adaptive choice of gradient's Lipschitz constant. Then, we apply this method to the dual problem and show, how to reconstruct an approximate solution to the primal problem with provable convergence rate. We prove the rate $O(1/k^2)$, $k$ being the iteration counter, both for the absolute value of the primal objective residual and constraints infeasibility. Our method has similar to Sinkhorn's method complexity of each iteration, but is faster and more stable numerically, when the regularization parameter is small. We illustrate the advantage of our method by numerical experiments for the two mentioned applications. We show that there exists a threshold, such that, when the regularization parameter is smaller than this threshold, our method outperforms the Sinkhorn's method in terms of computation time.

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arXiv2017

Gradient-free two-points optimal method for non smooth stochastic convex optimization problem with additional small noise

Using double-smoothing technique and stochastic mirror descent with inexact oracle we built an optimal algorithm (up to a multiplicative factor) for two-points gradient-free non-smooth stochastic convex programming. We investigate how much can be the level of noise (the nature of this noise isn't necessary stochastic) for the rate of convergence to be maintained (up to a multiplicative factor).

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arXiv2017

Dual approaches to the strongly convex simple function minimization problem under affine restrictions

We consider strongly convex optimization problems with affine-type restrictions. We build dual problem and solve dual problem by Fast Gradient Method. We use primal-dual structure of this method to construct the solution of the primal problem. The paper contain a lot of different tricks that allows to generalize mentioned above results for almost all methods we would like to choose to solve the dual problem.

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arXiv2017

Searching equillibriums in large transport networks

In this Doctoral Dissertation we propose new variational principles for traffic assignment problems. So to find equillibrium we have to solve large-scale convex optimization problem of special (multilevel) type. We propose different numerical approaches for this problem.

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arXiv2017

Nonlinear stochastic multiarmed bandit problems with inexact oracle

In the paper we consider one point and two point multiarmed bamdit problems. In other words we consider the online stochastic convex optimization problems with oracle that return the value (realization) of the function at one point or at two points. We allow these values to be inexact, but the level of noise should be small enough. We generalize well known results for inexact oracle case. And we also generalize classical results to prox-structures differ from euclidian.

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arXiv2017

Around power law for PageRank components in Buckley-Osthus model of web graph

In the paper we investigate power law for PageRank components for the Buckley-Osthus model for web graph. We compare different numerical methods for PageRank calculation. With the best method we do a lot of numerical experiments. These experiments confirm the hypothesis about power law. At the end we discuss real model of web-ranking based on the classical PageRank approach.

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arXiv2017

Efficient numerical methods for searching equillibriums in large transport networks

We propose new variational principles for traffic assignment problems. So to find equillibrium we have to solve large-scale convex optimization problem of special type. We propose some kind of "algebra" on different models and corresponding numerical algorithms that allows us to combine the optimal method from the basis ones and predescribed operations on methods.

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