Strong inductive biases provably prevent harmless interpolation
This work addresses a fundamental problem in understanding generalization in modern machine learning, particularly for researchers studying overparameterization and inductive biases, though it is incremental in refining existing theories.
The paper investigates how the strength of an estimator's inductive bias affects harmless interpolation in overparameterized models, showing that strong biases prevent it while weak biases may require fitting noise for good generalization. It provides tight theoretical bounds for high-dimensional kernel regression with convolutional kernels and empirical evidence for deep neural networks.
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.