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Author: Dvurechensky, Pavel × End date: 2024 × Start date: 2020 × Type: report × Version: publishedVersion ×

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Now showing 1 - 10 of 14
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    Gradient methods for problems with inexact model of the objective
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Dvinskikh, Darina; Dvurechensky, Pavel; Kroshnin, Alexey; Kuznetsova, Olesya; Agafonov, Artem; Gasnikov, Alexander; Tyurin, Alexander; Uribe, Cesar A.; Pasechnyuk, Dmitry; Artamonov, Sergei
    We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.
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    On the optimal combination of tensor optimization methods
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Kamzolov, Dmitry; Gasnikov, Alexander; Dvurechensky, Pavel
    We consider the minimization problem of a sum of a number of functions having Lipshitz p -th order derivatives with different Lipschitz constants. In this case, to accelerate optimization, we propose a general framework allowing to obtain near-optimal oracle complexity for each function in the sum separately, meaning, in particular, that the oracle for a function with lower Lipschitz constant is called a smaller number of times. As a building block, we extend the current theory of tensor methods and show how to generalize near-optimal tensor methods to work with inexact tensor step. Further, we investigate the situation when the functions in the sum have Lipschitz derivatives of a different order. For this situation, we propose a generic way to separate the oracle complexity between the parts of the sum. Our method is not optimal, which leads to an open problem of the optimal combination of oracles of a different order.
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    Inexact tensor methods and their application to stochastic convex optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Agafonov, Artem; Kamzolov, Dmitry; Dvurechensky, Pavel; Gasnikov, Alexander
    We propose a general non-accelerated tensor method under inexact information on higher- order derivatives, analyze its convergence rate, and provide sufficient conditions for this method to have similar complexity as the exact tensor method. As a corollary, we propose the first stochastic tensor method for convex optimization and obtain sufficient mini-batch sizes for each derivative.
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    Tensor methods for strongly convex strongly concave saddle point problems and strongly monotone variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Ostroukhov, Petr; Kamalov, Rinat; Dvurechensky, Pavel; Gasnikov, Alexander
    In this paper we propose three tensor methods for strongly-convex-strongly-concave saddle point problems (SPP). The first method is based on the assumption of higher-order smoothness (the derivative of the order higher than 2 is Lipschitz-continuous) and achieves linear convergence rate. Under additional assumptions of first and second order smoothness of the objective we connect the first method with a locally superlinear converging algorithm in the literature and develop a second method with global convergence and local superlinear convergence. The third method is a modified version of the second method, but with the focus on making the gradient of the objective small. Since we treat SPP as a particular case of variational inequalities, we also propose two methods for strongly monotone variational inequalities with the same complexity as the described above.
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    Generalized self-concordant Hessian-barrier algorithms
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Dvurechensky, Pavel; Staudigl, Mathias; Uribe , Casar A.
    Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized selfconcordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and Lp-minimization are discussed to given the efficiency of the method.
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    Inexact model: A framework for optimization and variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Gasnikov, Alexander; Tyurin, Alexander; Pasechnyuk, Dmitry; Agafonov, Artem; Dvurechensky, Pavel; Dvinskikh, Darina; Piskunova, Victorya
    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 to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, 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 to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.
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    Distributed optimization with quantization for computing Wasserstein barycenters
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Krawchenko, Roman; Uribe, César A.; Gasnikov, Alexander; Dvurechensky, Pavel
    We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validate our algorithmic analysis.
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    Alternating minimization methods for strongly convex optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Tupitsa, Nazarii; Dvurechensky, Pavel; Gasnikov, Alexander
    We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under Polyak-Łojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in many-blocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root of the condition number as opposed to condition number for the non-accelerated method.
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    Zeroth-order algorithms for smooth saddle-point problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Sadiev, Abdurakhmon; Beznosikov, Aleksandr; Dvurechensky, Pavel; Gasnikov, Alexander
    Saddle-point problems have recently gained an 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 which 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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    Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Stonyakin, Fedor; Gasnikov, Alexander; Tyurin, Alexander; Pasechnyuk, Dmitry; Agafonov, Artem; Dvurechensky, Pavel; Dvinskikh, Darina; Artamonov, Sergei; Piskunova, Victorya
    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 to obtain 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 to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.