Solving Smooth Min-Min and Min-Max Problems by Mixed Oracle Algorithms

Egor, Gladin,; Sadiev, Abdurakhmon; Gasnikov, Alexander; Dvurechensky, Pavel; Beznosikov, Aleksandr; Alkousa, Mohammad

doi:10.1007/978-3-030-86433-0_2

Cited by 10 publications

(3 citation statements)

References 16 publications

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“…Further deterministic "cutting-plane" improvements are connected with the additional assumptions about small dimension of the involved vectors x or/and y (see [43,44,91]) or with different structural (e.g., SPP on balls in 1-or ∞-norms) and sparsity assumptions, see e.g., [21,111,112] and references therein. Here lower bounds are mostly unknown.…”

Section: Recent Advancesmentioning

confidence: 99%

Smooth monotone stochastic variational inequalities and saddle point problems: A survey

Beznosikov

Polyak²,

Gorbunov

et al. 2023

Eur. Math. Soc. Mag.

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This paper is a survey of methods for solving smooth, (strongly) monotone stochastic variational inequalities. To begin with, we present the deterministic foundation from which the stochastic methods eventually evolved. Then we review methods for the general stochastic formulation, and look at the finite-sum setup. The last parts of the paper are devoted to various recent (not necessarily stochastic) advances in algorithms for variational inequalities.

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Section: Recent Advancesmentioning

confidence: 99%

Smooth monotone stochastic variational inequalities and saddle point problems: A survey

Beznosikov

Polyak²,

Gorbunov

et al. 2023

Eur. Math. Soc. Mag.

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“…A detailed theoretical analysis of Vaidya's method can be found in [19] and [20]. It was proved in [21] that Vaidya's method can use δ-subgradients instead of the exact subgradient. Below we present a variant of this method that uses δ-subgradients (Algorithm 3).…”

Section: 12mentioning

confidence: 99%

Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variable

Alkousa,

Gasnikov,

Gladin

et al. 2023

Sb. Math.

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Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional. Bibliography: 28 titles.

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“…Gradient-free methods for convex-concave saddle-point problems were studied in [6,5,29,57] with the main focus on the complexity in terms of the number of zeroth-order oracle calls. Unlike these papers, we focus here also on the iteration complexity.…”

Section: Saddle-point Problemsmentioning

confidence: 99%

The Power of First-Order Smooth Optimization for Black-Box Non-Smooth Problems

Gasnikov¹,

Novitskii²,

Novitskii³

et al. 2022

Preprint

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Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.1 Note that, for most of the algorithms in this paper, we can make these assumptions only on the intersection of Qγ and the ball x 0 + B d p (R) for some p ∈ [1, 2], where x 0 is the starting point of the algorithm and R = O x 0 − x * p ln d with x * being a solution of (1) closest to x 0 [32].

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Solving Smooth Min-Min and Min-Max Problems by Mixed Oracle Algorithms

Cited by 10 publications

References 16 publications

Smooth monotone stochastic variational inequalities and saddle point problems: A survey

Smooth monotone stochastic variational inequalities and saddle point problems: A survey

Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variable

The Power of First-Order Smooth Optimization for Black-Box Non-Smooth Problems

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