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Fedor S. Stonyakin

Publications and source records attributed to Fedor S. Stonyakin.

16 recordsLinked to original sources
arXiv2026

Descending Everest with First-Order Oracles: Oracle Complexity of Convex Optimization under $(L_0,L_1)$-Smoothness

For convex $(L_0,L_1)$-smooth functions, the curvature may grow together with the gradient norm. Therefore, even when $\|x^0-x^*\|\le R$, the gradient and the optimality gap at the starting point may be exponentially large in $L_1R$, and the bound $\sqrt{L_0R^2/\varepsilon}$, known for the accelerated method on functions with Lipschitz gradient, no longer describes the complexity of the problem. We determine the oracle complexity of this class for arbitrary deterministic first-order methods: for $L_0R^2/\varepsilon\ge1$, the optimal number of oracle calls $N$ satisfies $1+N=\widetildeΘ(\sqrt{L_0R^2/\varepsilon}+L_1R)$. The upper bound reads $O(\sqrt{L_0R^2/\varepsilon}+(1+L_1R)\log(2+L_1R))$: the method first passes through the large-gradient regime in $O(1+L_1R)$ iterations with an inexact line search, and then the standard accelerated method with a constant stepsize takes over. The lower bound is obtained by means of chains and the orthogonal invariance principle on functions conjugate to strongly convex functions on a polytope. If the initial optimality gap is at most $4\varepsilon$, the optimal bound becomes $1+N=\widetildeΘ(\sqrt{L_0R^2/\varepsilon}+(L_1R)^{2/3})$.

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arXiv2026

Mirror Descent-Type Algorithms for the Variational Inequality Problem with Functional Constraints

Variational inequalities play a key role in machine learning research, such as generative adversarial networks, reinforcement learning, adversarial training, and generative models. This paper is devoted to the constrained variational inequality problems with functional constraints (inequality-type constraints). We propose some mirror descent-type algorithms that switch between productive and non-productive steps depending on the values of the functional constraints at iterations, with many different step size rules and stopping criteria. We analyze the proposed algorithms and prove their optimal convergence rate to achieve a solution with desired accuracy, for problems with bounded and monotone operators and Lipschitz convex functional constraints. In addition, we propose a modification of the proposed algorithms by considering each functional constraint in the calculation when we have a productive step, as well as the first constraint that violates the feasibility. This modification can save the running time of algorithms when we have many functional constraints. In addition, we provide an analysis of the proposed algorithms for $δ$-monotone operators, allowing us to apply the proposed algorithms, as a special case, to constrained minimization problems when we do not have access to the exact information about the subgradient of the objective function. Numerical experiments that illustrate the work and performance of the proposed algorithms are also given.

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arXiv2026

Lipschitz-Free Mirror Descent Methods for Relatively Strongly Convex Functions with/without Absolute and Relative Inexactness

In this paper, we analyze the mirror descent algorithm for non-smooth optimization problems in which the objective function is relatively strongly convex, without relying on the standard Lipschitz continuity assumption commonly used in the literature. We provide convergence analyses for both exact and inexact subgradient information. Furthermore, through numerical experiments, we compare the derived bounds on the quality of the approximate solutions with existing estimates in the literature and demonstrate the effectiveness of the proposed results.

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arXiv2025

Mirror Descent Methods with Weighting Scheme for Outputs for Constrained Variational Inequality Problems

This paper is devoted to the variational inequality problems. We consider two classes of problems, the first is classical constrained variational inequality and the second is the same problem with functional (inequality type) constraints. To solve these problems, we propose mirror descent-type methods with a weighting scheme for the generated points in each iteration of the algorithms. This scheme assigns smaller weights to the initial points and larger weights to the most recent points, thus it improves the convergence rate of the proposed methods. For the variational inequality problem with functional constraints, the proposed method switches between adaptive and non-adaptive steps in the dependence on the values of the functional constraints at iterations. We analyze the proposed methods for the time-varying step sizes and prove the optimal convergence rate for variational inequality problems with bounded and monotone operators. The results of numerical experiments of the proposed methods for classical constrained variational inequality problems show a significant improvement over the modified projection method.

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arXiv2023

Gradient-Type Method for Optimization Problems with Polyak-Lojasiewicz Condition: Relative Inexactness in Gradient and Adaptive Parameters Setting

We consider minimization problems with the well-known Polya-Lojasievich condition and Lipshitz-continuous gradient. Such problem occurs in different places in machine learning and related fields. Furthermore, we assume that a gradient is available with some relative inexactness. We propose some adaptive gradient-type algorithm, where the adaptivity took place with respect to the smoothness parameter and the level of the gradient inexactness. The theoretical estimate of the the quality of the output point is obtained and backed up by experimental results.

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arXiv2022

Stopping Rules for Gradient Methods for Non-Convex Problems with Additive Noise in Gradient

We study the gradient method under the assumption that an additively inexact gradient is available for, generally speaking, non-convex problems. The non-convexity of the objective function, as well as the use of an inexactness specified gradient at iterations, can lead to various problems. For example, the trajectory of the gradient method may be far enough away from the starting point. On the other hand, the unbounded removal of the trajectory of the gradient method in the presence of noise can lead to the removal of the trajectory of the method from the desired exact solution. The results of investigating the behavior of the trajectory of the gradient method are obtained under the assumption of the inexactness of the gradient and the condition of gradient dominance. It is well known that such a condition is valid for many important non-convex problems. Moreover, it leads to good complexity guarantees for the gradient method. A rule of early stopping of the gradient method is proposed. Firstly, it guarantees achieving an acceptable quality of the exit point of the method in terms of the function. Secondly, the stopping rule ensures a fairly moderate distance of this point from the chosen initial position. In addition to the gradient method with a constant step, its variant with adaptive step size is also investigated in detail, which makes it possible to apply the developed technique in the case of an unknown Lipschitz constant for the gradient. Some computational experiments have been carried out which demonstrate effectiveness of the proposed stopping rule for the investigated gradient methods.

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arXiv2022

Gradient-Type Methods for Optimization Problems with Polyak-Łojasiewicz Condition: Early Stopping and Adaptivity to Inexactness Parameter

Due to its applications in many different places in machine learning and other connected engineering applications, the problem of minimization of a smooth function that satisfies the Polyak-Łojasiewicz condition receives much attention from researchers. Recently, for this problem, the authors of recent work proposed an adaptive gradient-type method using an inexact gradient. The adaptivity took place only with respect to the Lipschitz constant of the gradient. In this paper, for problems with the Polyak-Łojasiewicz condition, we propose a full adaptive algorithm, which means that the adaptivity takes place with respect to the Lipschitz constant of the gradient and the level of the noise in the gradient. We provide a detailed analysis of the convergence of the proposed algorithm and an estimation of the distance from the starting point to the output point of the algorithm. Numerical experiments and comparisons are presented to illustrate the advantages of the proposed algorithm in some examples.

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arXiv2021

Adaptive Gradient-type Methods for Convex Optimization Problems with Relative Accuracy and Sharp Minimum

In this paper, we consider gradient-type methods for convex positively homogeneous optimization problems with relative accuracy. An analogue of the accelerated universal gradient-type method for positively homogeneous optimization problems with relative accuracy is investigated. The second approach is related to subgradient methods with B. T. Polyak stepsize. Result on the linear convergence rate for some methods of this type with adaptive step adjustment is obtained for some class of non-smooth problems. Some generalization to a special class of non-convex non-smooth problems is also considered.

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arXiv2021

Adaptation to Inexactness for some Gradient-type Methods

We introduce a notion of inexact model of a convex objective function, which allows for errors both in the function and in its gradient. For this situation, a gradient method with an adaptive adjustment of some parameters of the model is proposed and an estimate for the convergence rate is found. This estimate is optimal on a class of sufficiently smooth problems in the presence of errors. We consider a special class of convex nonsmooth optimization problems. In order to apply the proposed technique to this class, an artificial error should be introduced. We show that the method can be modified for such problems to guarantee a convergence in the function with a nearly optimal rate on the class of convex nonsmooth optimization problems. An adaptive gradient method is proposed for objective functions with some relaxation of the Lipschitz condition for the gradient that satisfy the Polyak--Lojasievicz gradient dominance condition. Here, the objective function and its gradient can be given inexactly. The adaptive choice of the parameters is performed during the operation of the method with respect to both the Lipschitz constant of the gradient and a value corresponding to the error of the gradient and the objective function. The linear convergence of the method is justified up to a value associated with the errors.

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arXiv2020

One Method for Minimization a Convex Lipschitz-Continuous Function of 2 Variables on a Fixed Square

In the article we have obtained some estimates of the rate of convergence for the recently proposed by Yu.E. Nesterov method of minimization of a convex Lipschitz-continuous function of two variables on a square with a fixed side. The method consists in solving auxiliary problems of one-dimensional minimization along the separating segments and does not imply the calculation of the exact value of the gradient of the objective functional. Experiments have shown that the method under consideration can achieve the desired accuracy of solving the problem in less time than the other methods (gradient descent and ellipsoid method) considered, both in the case of a known exact solution and using estimates of the convergence rate of the methods.

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arXiv2018

Some Analogue of Quadratic Interpolation for a Special Class of Non-Smooth Functionals and One Application to Adaptive Mirror Descent for Constrained Optimization Problems

Theoretical estimates of the convergence rate of many well-known gradient-type optimization methods are based on quadratic interpolation, provided that the Lipschitz condition for the gradient is satisfied. In this article we obtain a possibility of constructing an analogue of such interpolation in the class of locally Lipschitz quasi-convex functionals with the special conditions of non-smoothness (Lipshitz-continuous subgradient) introduced in this paper. As an application, estimates are obtained for the rate of convergence of the previously proposed adaptive mirror descent method for the problems of minimizing a quasi-convex locally Lipschitz functional with several convex functional constraints.

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arXiv2018

Adaptive algorithms for mirror descent in convex programming problems with Lipschitz constraints

The paper is devoted to new modifications of recently proposed adaptive methods of Mirror Descent for convex minimization problems in the case of several convex functional constraints. Methods for problems of two classes are considered. The first type of problems with Lipschitz-continuous objective (generally speaking, nonsmooth) functional. The second one is for problems with a Lipschitz-continuous gradient of the objective smooth functional. We consider the class of problems with a non-smooth objective functional equal to the maximum of smooth functionals with a Lipschitz-continuous gradient. Note that functional constraints, generally speaking, are non-smooth and Lipschitz-contionuous. The proposed modifications allow saving the algorithm running time due to consideration of not all functional constraints on non-productive steps. Estimates for the rate of convergence of the methods under consideration are obtained. The methods proposed are optimal from the point of view of lower oracle estimates. The results of numerical experiments illustrating the advantages of the proposed procedure for some examples are given.

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arXiv2018

One Mirror Descent Algorithm for Convex Constrained Optimization Problems with non-standard growth properties

The paper is devoted to a special Mirror Descent algorithm for problems of convex minimization with functional constraints. The objective function may not satisfy the Lipschitz condition, but it must necessarily have the Lipshitz-continuous gradient. We assume, that the functional constraint can be non-smooth, but satisfying the Lipschitz condition. In particular, such functionals appear in the well-known Truss Topology Design problem. Also we have applied the technique of restarts in the mentioned version of Mirror Descent for strongly convex problems. Some estimations for a rate of convergence are investigated for considered Mirror Descent algorithms.

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