Memorizing without overfitting: Bias, variance, and interpolation in over-parameterized models

TL;DR

This paper uses statistical physics to analyze bias and variance in over-parameterized models, revealing phase transitions and challenging classical bias-variance trade-off assumptions in deep learning.

Contribution

It provides analytic expressions for bias and variance in minimal over-parameterized models, elucidating their behavior and generalization properties beyond classical theory.

Findings

Increasing parameters causes a phase transition with zero training error and diverging test error due to variance.

In over-parameterized neural networks, test error decreases as both bias and variance decrease.

Overfitting can occur without noise, and bias persists even when student and teacher models match.

Abstract

The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using "over-parameterized models" where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in over-parameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of over-parameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold, the test error of the two-layer neural network decreases due to a monotonic decrease in \emph{both} the bias and variance in contrast with the classical bias-variance trade-off. We also show that in contrast with classical intuition, over-parameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in over-parameterized models and relate our results to random matrix theory.