Understanding Double Descent Requires a Fine-Grained Bias-Variance Decomposition

TL;DR

This paper introduces a detailed bias-variance decomposition for deep learning models, revealing complex behaviors of variance components and their divergence at the interpolation boundary, which can be mitigated by ensemble methods.

Contribution

It provides a novel, interpretable bias-variance decomposition accounting for multiple sources of randomness and analyzes its high-dimensional asymptotics in kernel regression.

Findings

Bias decreases monotonically with network width.

Variance terms show non-monotonic behavior and can diverge.

Divergence caused by interaction of sampling and initialization, mitigated by ensemble methods.

Abstract

Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, with simpler models exhibiting high bias and more complex models exhibiting high variance of the predictive function. However, such a simple trade-off does not adequately describe deep learning models that simultaneously attain low bias and variance in the heavily overparameterized regime. A primary obstacle in explaining this behavior is that deep learning algorithms typically involve multiple sources of randomness whose individual contributions are not visible in the total variance. To enable fine-grained analysis, we describe an interpretable, symmetric decomposition of the variance into terms associated with the randomness from sampling, initialization, and the labels. Moreover, we compute the high-dimensional asymptotic behavior of this decomposition for random feature kernel regression, and analyze the strikingly rich phenomenology that arises. We find that the bias decreases monotonically with the network width, but the variance terms exhibit non-monotonic behavior and can diverge at the interpolation boundary, even in the absence of label noise. The divergence is caused by the \emph{interaction} between sampling and initialization and can therefore be eliminated by marginalizing over samples (i.e. bagging) \emph{or} over the initial parameters (i.e. ensemble learning).