Understanding Pathologies of Deep Heteroskedastic Regression

TL;DR

This paper investigates the overfitting behaviors of deep heteroskedastic regression models, revealing a phase transition influenced by regularization and providing a theoretical framework to optimize hyperparameter tuning and improve calibration performance.

Contribution

It introduces a phase transition perspective in heteroskedastic regression, develops a statistical field theory model, and simplifies hyperparameter tuning while enhancing calibration results.

Findings

Identification of a phase transition in model overfitting behavior.

Theoretical framework aligns qualitatively with empirical results.

Reduced hyperparameter search complexity improves calibration performance.

Abstract

Deep, overparameterized regression models are notorious for their tendency to overfit. This problem is exacerbated in heteroskedastic models, which predict both mean and residual noise for each data point. At one extreme, these models fit all training data perfectly, eliminating residual noise entirely; at the other, they overfit the residual noise while predicting a constant, uninformative mean. We observe a lack of middle ground, suggesting a phase transition dependent on model regularization strength. Empirical verification supports this conjecture by fitting numerous models with varying mean and variance regularization. To explain the transition, we develop a theoretical framework based on a statistical field theory, yielding qualitative agreement with experiments. As a practical consequence, our analysis simplifies hyperparameter tuning from a two-dimensional to a one-dimensional search, substantially reducing the computational burden. Experiments on diverse datasets, including UCI datasets and the large-scale ClimSim climate dataset, demonstrate significantly improved performance in various calibration tasks.