Optimal errors and phase transitions in high-dimensional generalized linear models.

Area of Science:

• Machine Learning
• Statistics
• Signal Processing
• Information Theory

Background:

• Generalized linear models (GLMs) are foundational in various fields, including machine learning and statistics.
• Analysis of GLMs with random data matrices is crucial for applications like compressed sensing and neural networks.
• Previous nonrigorous predictions for optimal errors in specific GLM cases existed, often derived from statistical physics methods.

Purpose of the Study:

• To rigorously analyze generalized linear models (GLMs) with random data matrices in the high-dimensional limit.
• To evaluate mutual information for deriving Bayes-optimal estimation and generalization errors.
• To rigorously establish and algorithmically interpret decades-old conjectures regarding optimal errors in GLMs.

Main Methods:

• Analysis of GLMs in the high-dimensional limit (large samples and dimension with a fixed ratio).
• Evaluation of mutual information (free entropy) to determine optimal errors.
• Rigorous mathematical derivation and algorithmic interpretation of results.

Main Results:

• Rigorous establishment of Bayes-optimal estimation and generalization errors for random GLMs.
• Algorithmic interpretation of results via the generalized approximate message-passing (GAMP) algorithm.
• Characterization of parameter regions where GAMP achieves optimal performance and identification of sharp phase transitions.

Conclusions:

• The random GLM framework provides a rigorous benchmark for evaluating algorithms in high-dimensional settings.
• The generalized approximate message-passing algorithm is shown to achieve optimal performance in specific regimes.
• This work bridges statistical physics predictions with rigorous mathematical analysis and algorithmic applications.