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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,…
We study the asymptotic behavior of second-order algorithms mixing Newton's method and inertial gradient descent in non-convex landscapes. We show that, despite the Newtonian behavior of these methods, they almost always escape strict…
We consider a generic convex-concave saddle point problem with separable structure, a form that covers a wide-ranged machine learning applications. Under this problem structure, we follow the framework of primal-dual updates for saddle…
One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformu-lation of the problem of interest and solving the resulting problem…
Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided for example by means of stochastic gradient descent methods. In this work, we provide…
The diffusion strategy for distributed learning from streaming data employs local stochastic gradient updates along with exchange of iterates over neighborhoods. In Part I [2] of this work we established that agents cluster around a network…
Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where…
We develop stochastic first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems. When the saddle function is strongly convex in the primal variable, we develop the first stochastic restart scheme for this…
We consider saddle point problems which objective functions are the average of $n$ strongly convex-concave individual components. Recently, researchers exploit variance reduction methods to solve such problems and achieve linear-convergence…
Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, and are typically solved…
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle…
This paper focuses on the distributed optimization of stochastic saddle point problems. The first part of the paper is devoted to lower bounds for the centralized and decentralized distributed methods for smooth (strongly) convex-(strongly)…
The saddle-point optimization problems have a lot of practical applications. This paper focuses on such non-smooth problems in decentralized case. This work contains generalization of recently proposed sliding for centralized problem.…
Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable…
In the paper, we generalize the approach Gasnikov et. al, 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle, to the convex-concave saddle-point problem. The proposed approach works,…
We analyze the behavior of randomized coordinate gradient descent for nonconvex optimization, proving that under standard assumptions, the iterates almost surely escape strict saddle points. By formulating the method as a nonlinear random…
Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms which solve convex optimization problems. When implemented online for streaming data or as dynamic…
In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming…
How does the choice of optimization algorithm shape a model's ability to learn features? To address this question for steepest descent methods --including sign descent, which is closely related to Adam --we introduce steepest mirror flows…
Stochastic gradient descent (SGD) with stochastic momentum is popular in nonconvex stochastic optimization and particularly for the training of deep neural networks. In standard SGD, parameters are updated by improving along the path of the…