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Sign Gradient Descent (SignGD) is a simple yet robust optimization method, widely used in machine learning for its resilience to gradient noise and compatibility with low-precision computations. While its empirical performance is well…
The main aim of this paper is to provide an analysis of gradient descent (GD) algorithms with gradient errors that do not necessarily vanish, asymptotically. In particular, sufficient conditions are presented for both stability (almost sure…
Loss functions with non-isolated minima have emerged in several machine learning problems, creating a gap between theory and practice. In this paper, we formulate a new type of local convexity condition that is suitable to describe the…
In this work, we propose an efficient adaptive multilevel preconditioned Jacobi-Davidson (PJD) method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned…
LocalSGD and SCAFFOLD are widely used methods in distributed stochastic optimization, with numerous applications in machine learning, large-scale data processing, and federated learning. However, rigorously establishing their theoretical…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…
We introduce an inexact variant of Stochastic Mirror Descent (SMD), called Inexact Stochastic Mirror Descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling…
In this paper, we revisit stochastic gradient descent (SGD) with AdaGrad-type preconditioning. Our contributions are twofold. First, we develop a unified convergence analysis of SGD with adaptive preconditioning under anisotropic or matrix…
Stochastic gradient descent (SGD) method is popular for solving non-convex optimization problems in machine learning. This work investigates SGD from a viewpoint of graduated optimization, which is a widely applied approach for non-convex…
Block-coordinate descent (BCD) is the method of choice to solve numerous large scale optimization problems, however their theoretical study for non-convex optimization, has received less attention. In this paper, we present a new…
In this thesis, we propose new theoretical frameworks for the analysis of stochastic and distributed methods with error compensation and local updates. Using these frameworks, we develop more than 20 new optimization methods, including the…
The convergence behaviour of first-order methods can be severely slowed down when applied to high-dimensional non-convex functions due to the presence of saddle points. If, additionally, the saddles are surrounded by large plateaus, it is…
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received…
We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent…
The paper explores the differential inclusion of a special form. It is supposed that the support function of the set in the right-hand side of an inclusion may contain the maximum of the finite number of continuously differentiable (in…
In this paper, we propose a Riemannian steepest descent method for solving a blind deconvolution problem. We prove that the proposed algorithm with an appropriate initialization will recover the exact solution with high probability when the…
This work presents the convergence rate analysis of stochastic variants of the broad class of direct-search methods of directional type. It introduces an algorithm designed to optimize differentiable objective functions $f$ whose values can…
Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the…
We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a…
For solving pseudo-convex global optimization problems, we present a novel fully adaptive steepest descent method (or ASDM) without any hard-to-estimate parameters. For the step-size regulation in an $\varepsilon$-normalized direction, we…