Convergence of the RMSProp deep learning method with penalty for nonconvex optimization

Area of Science:

• Machine Learning
• Deep Learning Optimization

Background:

• Standard optimization algorithms like RMSProp face challenges in non-convex settings.
• Ensuring convergence and stability is crucial for effective deep learning model training.

Purpose of the Study:

• Introduce a norm-based version of RMSProp with a penalty term (RMSPropW).
• Analyze the convergence properties of RMSPropW in both deterministic and stochastic non-convex settings.

Main Methods:

• Developed RMSPropW by incorporating a penalty into the RMSProp algorithm's cost function.
• Provided analytical proofs for the boundedness and convergence of RMSPropW.
• Conducted numerical simulations on benchmark datasets (MNIST, CIFAR-10, IMDB) for validation.

Main Results:

• Established strict upper bounds on gradient norms and weight parameters using the penalty term.
• Proved gradient sequence convergence to zero in the deterministic case (fixed and diminishing step sizes).
• Demonstrated stochastic weight sequence convergence to a stationary point with probability 1.

Conclusions:

• RMSPropW offers improved stability and convergence guarantees in non-convex deep learning.
• The penalty term effectively bounds gradient and weight norms, enhancing optimization.
• Numerical results support the theoretical findings, showing RMSPropW's efficacy compared to standard RMSProp.