Learning Rates for Stochastic Gradient Descent With Nonconvex Objectives.

Area of Science:

• Machine Learning
• Optimization Algorithms
• Statistical Learning Theory

Background:

• Stochastic Gradient Descent (SGD) is widely used for training complex, nonconvex models, balancing error minimization and generalization.
• Existing research often analyzes computational and statistical properties of SGD separately.
• A unified understanding of these properties in nonconvex settings is lacking.

Purpose of the Study:

• To develop novel learning rates for SGD in nonconvex learning settings.
• To provide high-probability bounds for both computational and statistical errors.
• To offer insights into implicit regularization through balancing these errors.

Main Methods:

• Development of new learning rate schedules for SGD.
• Derivation of high-probability bounds for computational and statistical errors.
• Analysis of the complexity of SGD iterates with respect to iteration number.

Main Results:

• Novel learning rates for SGD in nonconvex settings are introduced.
• Controllable growth in the complexity of SGD iterates is demonstrated.
• Insights into achieving implicit regularization by tuning the number of passes are provided.
• Refined understanding of uniform convergence of gradients via Rademacher chaos complexities.

Conclusions:

• The proposed methods offer a way to jointly analyze and control computational and statistical errors in SGD for nonconvex models.
• Tuning the number of passes can effectively balance these errors, leading to implicit regularization.
• This work enhances the theoretical understanding of SGD in complex learning scenarios.