Landscape complexity for the empirical risk of generalized linear models: Discrimination between structured data.

Area of Science:

• Statistical Physics
• Machine Learning Theory
• Random Matrix Theory

Background:

• Modern machine learning systems often utilize high-dimensional data with inherent correlations, reflecting complex real-world structures.
• Understanding the loss landscape is crucial for analyzing model training dynamics and generalization capabilities.

Purpose of the Study:

• To determine the average number of critical points in high-dimensional empirical loss functions with correlated Gaussian data.
• To characterize the annealed landscape complexity, providing insights into the structure of the loss landscape.

Main Methods:

• Application of the Kac-Rice formula.
• Leveraging results from random matrix theory.
• Analysis in the large-dimensional limit (large-d) under a technical hypothesis.

Main Results:

• Exact characterization of the annealed landscape complexity for correlated Gaussian vectors.
• Detailed analysis of the loss landscape for a single perceptron.
• Generalization to a two-dataset perceptron model for discrimination tasks.
• Extension to loss functions for generalized linear models with correlated data.

Conclusions:

• The study provides a theoretical framework for understanding the complexity of loss landscapes in machine learning with structured data.
• Results offer insights into the interplay between data structure, model architecture, and training dynamics.
• The findings are applicable to understanding adversarial scenarios and improving model generalization.