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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This content will become publicly available on June 25, 2025
A Global Geometric Analysis of Maximal Coding Rate Reduction
The maximal coding rate reduction (MCR2) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima, as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR2 problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR2 a natural choice of objective for learning diverse and discriminative representations via first-order optimization methods. To validate our theoretical findings, we conduct extensive experiments on both synthetic and real data sets.
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- NSF-PAR ID:
- 10526572
- Publisher / Repository:
- International Conference on Machine Learning
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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