Reconciling modern machine-learning practice and the classical bias–variance trade-off

TLDR

Machine‑learning breakthroughs are reshaping science, yet the classic bias–variance trade‑off appears at odds with modern practice where highly expressive models interpolate data and still generalize well. This paper aims to reconcile classical theory with modern practice by unifying them in a single performance curve. The authors introduce a double‑descent curve that extends the textbook U‑shaped bias–variance curve, showing that increasing model capacity beyond interpolation can improve performance. Empirical evidence across diverse models and datasets confirms the ubiquity of double descent, revealing its underlying mechanism and highlighting limits of classical analyses for both theory and practice.

Abstract

Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias-variance trade-off, appears to be at odds with the observed behavior of methods used in modern machine-learning practice. The bias-variance trade-off implies that a model should balance underfitting and overfitting: Rich enough to express underlying structure in data and simple enough to avoid fitting spurious patterns. However, in modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered overfitted, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This "double-descent" curve subsumes the textbook U-shaped bias-variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine-learning models delineates the limits of classical analyses and has implications for both the theory and the practice of machine learning.