Related papers: Heavy Ball and Nesterov Accelerations with Hessian…

Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method…

We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of…

We consider the problem of minimizing a strongly convex smooth function where the gradients are subject to additive worst-case deterministic errors that are square-summable. We study the trade-offs between the convergence rate and…

Optimization in Deep Learning is mainly dominated by first-order methods which are built around the central concept of backpropagation. Second-order optimization methods, which take into account the second-order derivatives are far less…

First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive…

We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit…

This paper concerns the adaptive control of a class of discrete-time nonlinear systems with all states accessible. Recently, a high-order tuner algorithm was developed for the minimization of convex loss functions with time-varying…

We introduce a generic scheme to solve nonconvex optimization problems using gradient-based algorithms originally designed for minimizing convex functions. Even though these methods may originally require convexity to operate, the proposed…

We study stochastic nonconvex optimization under heavy-tailed noise. In this setting, the stochastic gradients only have bounded $p$-th central moment ($p$-BCM) for some $p \in (1,2]$. Building on the foundational work of Arjevani et al.…

In this paper, an efficient modified Newton type algorithm is proposed for nonlinear unconstrianed optimization problems. The modified Hessian is a convex combination of the identity matrix (for steepest descent algorithm) and the Hessian…

We consider a distributionally robust second-order stochastic dominance constrained optimization problem. We require the dominance constraints hold with respect to all probability distributions in a Wasserstein ball centered at the…

Due to its simplicity and efficiency, the first-order gradient method has been extensively employed in training neural networks. Although the optimization problem of the neural network is non-convex, recent research has proved that the…

We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the…

We propose a novel algorithm that extends the methods of ball smoothing and Gaussian smoothing for noisy derivative-free optimization by accounting for the heterogeneous curvature of the objective function. The algorithm dynamically adapts…

We propose and analyze a class of second-order dynamical systems for continuous-time optimization that incorporate fractional-order gradient terms. The system is given by \begin{equation} \ddot{x}(t) + \frac{\alpha}{t}\dot{x}(t) +…

While the superior performance of second-order optimization methods such as Newton's method is well known, they are hardly used in practice for deep learning because neither assembling the Hessian matrix nor calculating its inverse is…

We propose a class of discrete state sampling algorithms based on Nesterov's accelerated gradient method, which extends the classical Metropolis-Hastings (MH) algorithm. The evolution of the discrete states probability distribution governed…

We study accelerated optimization methods in the Gaussian phase retrieval problem. In this setting, we prove that gradient methods with Polyak or Nesterov momentum have similar implicit regularization to gradient descent. This implicit…

We study gradient-based optimization methods obtained by direct Runge-Kutta discretization of the ordinary differential equation (ODE) describing the movement of a heavy-ball under constant friction coefficient. When the function is high…

This article presents a second-order fully distributed optimization algorithm, HBNET-GIANT, driven by heavy-ball momentum, for $L$-smooth and $\mu$-strongly convex objective functions. A rigorous convergence analysis is performed, and we…