Related papers: A Derivation of Nesterov's Accelerated Gradient Al…
We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov's accelerated gradient descent. The new algorithm has a simple geometric…
We present a coupled system of ODEs which, when discretized with a constant time step/learning rate, recovers Nesterov's accelerated gradient descent algorithm. The same ODEs, when discretized with a decreasing learning rate, leads to novel…
Iterative gradient-based optimization algorithms are widely used to solve difficult or large-scale optimization problems. There are many algorithms to choose from, such as gradient descent and its accelerated variants such as Polyak's Heavy…
We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves…
A number of optimization algorithms have been inspired by the physics of Newtonian motion. Here, we ask the question: do algorithms themselves obey some ``natural laws of motion,'' and can they be derived by an application of these laws? We…
The optimized gradient method (OGM) provides a factor-$\sqrt{2}$ speedup upon Nesterov's celebrated accelerated gradient method in the convex (but non-strongly convex) setup. However, this improved acceleration mechanism has not been well…
We study distributed optimization problems when $N$ nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant $L$), and bounded gradient. We…
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…
We study the behavior of first-order methods applied to a lower-unbounded convex function $f$, i.e., $\inf f = -\infty$. Such a setting has received little attention since the trajectories of gradient descent and Nesterov's accelerated…
This paper delves into the investigation of a distributed aggregative optimization problem within a network. In this scenario, each agent possesses its own local cost function, which relies not only on the local state variable but also on…
In this paper, we try to uncover the second-order essence of several first-order optimization methods. For Nesterov Accelerated Gradient, we rigorously prove that the algorithm makes use of the difference between past and current gradients,…
We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation…
We consider a distributed stochastic optimization problem that is solved by a decentralized network of agents with only local communication between neighboring agents. The goal of the whole system is to minimize a global objective function…
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if…
Gradient-based minimax optimal algorithms have greatly promoted the development of continuous optimization and machine learning. One seminal work due to Yurii Nesterov [Nes83a] established $\tilde{\mathcal{O}}(\sqrt{L/\mu})$ gradient…
We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex or, respectively, smooth and convex functions. We establish sufficient conditions, via new discrete Lyapunov…
We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent…
We present a unifying Nearly Asymptotically Invariant Manifold (NAIM) framework for understanding Nesterovs Accelerated Gradient (NAG) method. By lifting the first-order gradient flow into a second-order phase space we construct a NAIM a…
In this paper, we propose a unified view of gradient-based algorithms for stochastic convex composite optimization by extending the concept of estimate sequence introduced by Nesterov. This point of view covers the stochastic gradient…
In this thesis we develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive…