Related papers: A perturbed preconditioned gradient descent method…

Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints. First- and second-order necessary optimality conditions state that the Riemannian…

Stochastic Gradient Descent (SGD) plays a central role in modern machine learning. While there is extensive work on providing error upper bound for SGD, not much is known about SGD error lower bound. In this paper, we study the convergence…

Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing…

We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD)…

The stochastic gradient descent (SGD) method is a widely used approach for solving stochastic optimization problems, but its convergence is typically slow. Existing variance reduction techniques, such as SAGA, improve convergence by…

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G.…

The convergence behavior of gradient methods for minimizing convex differentiable functions is one of the core questions in convex optimization. This paper shows that their well-known complexities can be achieved under conditions weaker…

Achieving optimal rates for stochastic composite convex optimization without prior knowledge of problem parameters remains a central challenge. In the deterministic setting, the auto-conditioned fast gradient method has recently been…

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and…

We prove the first convergence guarantees for a subgradient method minimizing a generic Lipschitz function over generic Lipschitz inequality constraints. No smoothness or convexity (or weak convexity) assumptions are made. Instead, we…

The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A…

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as…

Composite optimization offers a powerful modeling tool for a variety of applications and is often numerically solved by means of proximal gradient methods. In this paper, we consider fully nonconvex composite problems under only local…

In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called Conditioned SGD, based on a preconditioning of the gradient direction. Using a discrete-time approach with martingale tools, we establish…

Matrix completion has attracted much interest in the past decade in machine learning and computer vision. For low-rank promotion in matrix completion, the nuclear norm penalty is convenient due to its convexity but has a bias problem.…

In this paper we consider large-scale composite optimization problems having the objective function formed as a sum of two terms (possibly nonconvex), one has (block) coordinate-wise Lipschitz continuous gradient and the other is…

In this work, we investigate the margin-maximization bias exhibited by gradient-based algorithms in classifying linearly separable data. We present an in-depth analysis of the specific properties of the velocity field associated with…

We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global…

Stochastic Gradient Descent (SGD) is widely used in machine learning research. Previous convergence analyses of SGD under the vanishing step-size setting typically require Robbins-Monro conditions. However, in practice, a wider variety of…

We propose $\textsf{ScaledGD($\lambda$)}$, a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor…