scholarly journals Convexification procedures and decomposition methods for nonconvex optimization problems

1979 ◽  
Vol 29 (2) ◽  
pp. 169-197 ◽  
Author(s):  
D. P. Bertsekas
Keyword(s):  
Nonconvex Optimization ◽  
Optimization Problems ◽  
Decomposition Methods ◽  
Nonconvex Optimization Problems ◽  
Convexification Procedures

scholarly journals A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

2021 ◽  
Author(s):  
Tianyi Liu ◽  
Zhehui Chen ◽  
Enlu Zhou ◽  
Tuo Zhao
Keyword(s):  
Neural Networks ◽  
Nonconvex Optimization ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Optimization Problems ◽  
Saddle Points ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Nonconvex Optimization Problems ◽  
Empirical Success

Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.

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