A Modern Take on the Bias-Variance Tradeoff in Neural Networks

TL;DR

This paper investigates the bias-variance tradeoff in modern neural networks, revealing that both bias and variance can decrease with increased network width, challenging traditional understanding.

Contribution

It provides empirical measurements and theoretical analysis showing the absence of the classic bias-variance tradeoff in over-parameterized neural networks.

Findings

Bias and variance decrease as network width increases

Introduces a new variance decomposition method

Theoretical analysis supports empirical results

Abstract

The bias-variance tradeoff tells us that as model complexity increases, bias falls and variances increases, leading to a U-shaped test error curve. However, recent empirical results with over-parameterized neural networks are marked by a striking absence of the classic U-shaped test error curve: test error keeps decreasing in wider networks. This suggests that there might not be a bias-variance tradeoff in neural networks with respect to network width, unlike was originally claimed by, e.g., Geman et al. (1992). Motivated by the shaky evidence used to support this claim in neural networks, we measure bias and variance in the modern setting. We find that both bias and variance can decrease as the number of parameters grows. To better understand this, we introduce a new decomposition of the variance to disentangle the effects of optimization and data sampling. We also provide theoretical analysis in a simplified setting that is consistent with our empirical findings.