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The gradient descent approach is the key ingredient in variational quantum algorithms and machine learning tasks, which is an optimization algorithm for finding a local minimum of an objective function. The quantum versions of gradient…
Stochastic coordinate descent algorithms are efficient methods in which each iterate is obtained by fixing most coordinates at their values from the current iteration, and approximately minimizing the objective with respect to the remaining…
This paper studies a novel algorithm for nonconvex composite minimization which can be interpreted in terms of dual space nonlinear preconditioning for the classical proximal gradient method. The proposed scheme can be applied to additive…
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger…
Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…
We propose an unconstrained optimization method based on the well-known primal-dual hybrid gradient (PDHG) algorithm. We first formulate the optimality condition of the unconstrained optimization problem as a saddle point problem. We then…
This work proposes A$^2$GD, a novel adaptive accelerated gradient descent method for convex and composite optimization. Smoothness and convexity constants are updated via Lyapunov analysis. Inspired by stability analysis in ODE solvers, the…
We study differentiable strongly quasiconvex functions for providing new properties for algorithmic and monotonicity purposes. Furthemore, we provide insights into the decreasing behaviour of strongly quasiconvex functions, applying this…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…
The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the…
In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial…
Motivated by applications to distributed optimization over networks and large-scale data processing in machine learning, we analyze the deterministic incremental aggregated gradient method for minimizing a finite sum of smooth functions…
We propose new continuous-time formulations for first-order stochastic optimization algorithms such as mini-batch gradient descent and variance-reduced methods. We exploit these continuous-time models, together with simple Lyapunov analysis…
In this paper, we propose a second-order continuous primal-dual dynamical system with time-dependent positive damping terms for a separable convex optimization problem with linear equality constraints. By the Lyapunov function approach, we…
In this work, a nonlinear momentum method is introduced to enhance the convergence performance of momentum-based gradient optimization algorithms. Classical momentum methods, such as the Heavy Ball method, can be viewed as a dynamical…
Adaptive gradient optimizers (AdaGrad), which dynamically adjust the learning rate based on iterative gradients, have emerged as powerful tools in deep learning. These adaptive methods have significantly succeeded in various deep learning…
This paper presents a discrete-time passivity-based analysis of the gradient descent method for a class of functions with sector-bounded gradients. Using a loop transformation, it is shown that the gradient descent method can be interpreted…
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex…
While many solutions for privacy-preserving convex empirical risk minimization (ERM) have been developed, privacy-preserving nonconvex ERM remains a challenge. We study nonconvex ERM, which takes the form of minimizing a finite-sum of…
We study alternating first-order algorithms with no inner loops for solving nonconvex-strongly-concave min-max problems. We show the convergence of the alternating gradient descent--ascent algorithm method by proposing a substantially…