scholarly journals Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization

2020 ◽  
Author(s):  
Yi Zhou ◽  
Zhe Wang ◽  
Kaiyi Ji ◽  
Yingbin Liang ◽  
Vahid Tarokh
Keyword(s):  
Nonsmooth Optimization ◽  
Nonconvex Optimization ◽  
Convergence Rates ◽  
Linear Convergence ◽  
Gradient Algorithm ◽  
Local Geometry ◽  
Linear Convergence Rate ◽  
Nonsmooth Problems ◽  
Proximal Gradient Algorithm ◽  
Nonconvex And Nonsmooth Optimization

Various types of parameter restart schemes have been proposed for proximal gradient algorithm with momentum to facilitate their convergence in convex optimization. However, under parameter restart, the convergence of proximal gradient algorithm with momentum remains obscure in nonconvex optimization. In this paper, we propose a novel proximal gradient algorithm with momentum and parameter restart for solving nonconvex and nonsmooth problems. Our algorithm is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the convergence and effectiveness of our proposed algorithm.

scholarly journals Asynchronous Delay-Aware Accelerated Proximal Coordinate Descent for Nonconvex Nonsmooth Problems

2019 ◽  
Vol 33 ◽  
pp. 1528-1535
Author(s):  
Ehsan Kazemi ◽  
Liqiang Wang
Keyword(s):  
Large Scale ◽  
Convergence Rates ◽  
Optimization Problems ◽  
Coordinate Descent ◽  
Performance Guarantee ◽  
Nonsmooth Problems ◽  
Descent Property ◽  
Large Scale Problems ◽  
Nonconvex And Nonsmooth Optimization ◽  
Sufficient Descent

Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving optimization problems, but the knowledge of PCD methods in the nonconvex setting is very limited. On the other hand, the asynchronous proximal coordinate descent (APCD) recently have received much attention in order to solve large-scale problems. However, the accelerated variants of APCD algorithms are rarely studied. In this paper, we extend APCD method to the accelerated algorithm (AAPCD) for nonsmooth and nonconvex problems that satisfies the sufficient descent property, by comparing between the function values at proximal update and a linear extrapolated point using a delay-aware momentum value. To the best of our knowledge, we are the first to provide stochastic and deterministic accelerated extension of APCD algorithms for general nonconvex and nonsmooth problems ensuring that for both bounded delays and unbounded delays every limit point is a critical point. By leveraging Kurdyka-Łojasiewicz property, we will show linear and sublinear convergence rates for the deterministic AAPCD with bounded delays. Numerical results demonstrate the practical efficiency of our algorithm in speed.

scholarly journals Convergence rate analysis of an iterative algorithm for solving the multiple-sets split equality problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shijie Sun ◽  
Meiling Feng ◽  
Luoyi Shi
Keyword(s):  
Convergence Rate ◽  
Iterative Algorithm ◽  
Sufficient Conditions ◽  
Linear Convergence ◽  
Regularity Property ◽  
Step Size ◽  
Bounded Linear ◽  
Split Equality Problem ◽  
Linear Convergence Rate ◽  
Equality Problem

Abstract This paper considers an iterative algorithm of solving the multiple-sets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the well-known concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.

Numerical experiments on stochastic block proximal-gradient type method for convex constrained optimization involving coordinatewise separable problems

2019 ◽  
Vol 35 (3) ◽  
pp. 371-378
Author(s):  
PORNTIP PROMSINCHAI ◽  
NARIN PETROT ◽  
◽  
◽  
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Objective Functions ◽  
Constrained Optimization Problems ◽  
Type Method ◽  
Coordinate Descent Algorithm ◽  
Proximal Gradient Algorithm ◽  
Separable Problems ◽  
Gradient Type

In this paper, we consider convex constrained optimization problems with composite objective functions over the set of a minimizer of another function. The main aim is to test numerically a new algorithm, namely a stochastic block coordinate proximal-gradient algorithm with penalization, by comparing both the number of iterations and CPU times between this introduced algorithm and the other well-known types of block coordinate descent algorithm for finding solutions of the randomly generated optimization problems with regularization term.

Sign in / Sign up

Export Citation Format

Share Document