Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning

Area of Science:

• Deep Learning
• Computational Mathematics
• Optimization Theory

Background:

• Deep learning models often employ loss regularization to improve performance and generalization.
• Local entropy and heat regularization are two such techniques with distinct mathematical underpinnings.
• Understanding their optimization dynamics is crucial for efficient training.

Purpose of the Study:

• To provide new theoretical and computational insights into local entropy and heat regularization.
• To introduce variational characterizations for optimizing these regularized losses.
• To analyze the implications for neural network training and computational cost.

Main Methods:

• Development of variational characterizations for local entropy and heat regularization.
• Introduction of a two-step optimization scheme involving probability density shifts and Gaussian approximations.
• Analysis of the Kullback-Leibler divergence in relation to optimization strategies.
• Exploration of moment matching for local entropy optimization.

Main Results:

• Demonstration of training error monotonicity along optimization iterates under approximation error assumptions.
• Identification of differences in optimization schemes based on Kullback-Leibler divergence arguments.
• Local entropy optimization via moment matching allows for sampling-based, gradient-free training.
• Potential for parallelizable training of neural networks is highlighted.

Conclusions:

• The proposed variational methods offer a new perspective on optimizing regularized deep learning losses.
• Gradient-free training via moment matching is a promising avenue, but computational costs require careful consideration.
• The study presents a more nuanced view of the practical gains from these regularization techniques compared to existing literature.