Strong inductive biases provably prevent harmless interpolation

TL;DR

This paper demonstrates that strong inductive biases in models can prevent harmless interpolation, while weak biases may require fitting noise for good generalization, supported by theoretical bounds and empirical evidence.

Contribution

It provides the first tight non-asymptotic bounds for high-dimensional kernel regression showing how inductive bias strength affects interpolation safety.

Findings

Strong inductive biases prevent harmless interpolation.

Weak biases can require fitting noise to achieve good generalization.

Empirical results show similar behavior in deep neural networks.

Abstract

Classical wisdom suggests that estimators should avoid fitting noise to achieve good generalization. In contrast, modern overparameterized models can yield small test error despite interpolating noise -- a phenomenon often called "benign overfitting" or "harmless interpolation". This paper argues that the degree to which interpolation is harmless hinges upon the strength of an estimator's inductive bias, i.e., how heavily the estimator favors solutions with a certain structure: while strong inductive biases prevent harmless interpolation, weak inductive biases can even require fitting noise to generalize well. Our main theoretical result establishes tight non-asymptotic bounds for high-dimensional kernel regression that reflect this phenomenon for convolutional kernels, where the filter size regulates the strength of the inductive bias. We further provide empirical evidence of the same behavior for deep neural networks with varying filter sizes and rotational invariance.