Related papers: GLL-type Nonmonotone Descent Methods Revisited und…
This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds…
This paper addresses the study of derivative-free smooth optimization problems, where the gradient information on the objective function is unavailable. Two novel general derivative-free methods are proposed and developed for minimizing…
In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method…
In this paper we develop a higher-order method for solving composite (non)convex minimization problems with smooth (non)convex functional constraints. At each iteration our method approximates the smooth part of the objective function and…
In this paper, we study the convergence properties of the Stochastic Gradient Descent (SGD) method for finding a stationary point of a given objective function $J(\cdot)$. The objective function is not required to be convex. Rather, our…
In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the \L{}ojasiewicz inequality proposed in the same year, and it does not…
This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization,…
Nonconvex optimization problems arise in many areas of computational science and engineering and are (approximately) solved by a variety of algorithms. Existing algorithms usually only have local convergence or subsequence convergence of…
In this paper, we study the decentralized optimization problem of minimizing a finite sum of continuously differentiable and possibly nonconvex functions over a fixed-connected undirected network. We propose a unified decentralized…
We study the convergence rate of gradient-based local search methods for solving low-rank matrix recovery problems with general objectives in both symmetric and asymmetric cases, under the assumption of the restricted isometry property.…
We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit…
This paper focuses on the minimization of a sum of a twice continuously differentiable function $f$ and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of $f$…
Zeroth-order optimization is the process of minimizing an objective $f(x)$, given oracle access to evaluations at adaptively chosen inputs $x$. In this paper, we present two simple yet powerful GradientLess Descent (GLD) algorithms that do…
Polyak-{\L}ojasiewicz (PL) [Polyak, 1963] condition is a weaker condition than the strong convexity but suffices to ensure a global convergence for the Gradient Descent algorithm. In this paper, we study the lower bound of algorithms using…
Stochastic gradient descent (SGD) has been studied extensively over the past decades due to its simplicity and broad applicability in machine learning. In this work, we analyze the local behavior of gradient descent and stochastic gradient…
Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of the convergence rate of CR are…
We propose potential-based analyses for first-order algorithms applied to constrained and composite minimization problems. We first propose ``idealized'' frameworks for algorithms in the strongly and non-strongly convex cases and argue…
In this work, we establish that Nesterov's accelerated gradient method, applied to $C^2$ functions satisfying the Polyak--{\L}ojasiewicz inequality around local minimizers, achieves the optimal local linear convergence rate…
Alternating projection method has been used in a wide range of engineering applications since it is a gradient-free method (without requiring tuning the step size) and usually has fast speed of convergence. In this paper, we formalize two…
We study a class of nonconvex-nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka-Lojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or…