Towards Sharp Minimax Risk Bounds for Operator Learning

TL;DR

This paper establishes fundamental limits on the accuracy of learning unknown operators between Hilbert spaces, revealing a sample complexity curse and providing sharp bounds under various spectral decay conditions.

Contribution

It develops a minimax theory for operator learning, deriving lower and upper bounds that characterize the difficulty of estimating operators with different regularities and spectral properties.

Findings

Minimax risk cannot decay algebraically with sample size for generic Lipschitz operators.

Exponential spectral decay allows for sharper risk bounds.

Higher regularity assumptions do not improve minimax rates, indicating a universal curse of sample complexity.

Abstract

We develop a minimax theory for operator learning, where the goal is to estimate an unknown operator between separable Hilbert spaces from finitely many noisy input-output samples. For uniformly bounded Lipschitz operators, we prove information-theoretic lower bounds together with matching or near-matching upper bounds, covering both fixed and random designs under Hilbert-valued Gaussian noise and Gaussian white noise errors. The rates are controlled by the spectrum of the covariance operator of the measure that defines the error metric. Our setup is very general and allows for measures with unbounded support. A key implication is a curse of sample complexity, which shows that the minimax risk for generic Lipschitz operators cannot decay at any algebraic rate in the sample size. We obtain sharp characterizations when the covariance spectrum decays exponentially and provide general upper and lower bounds in slower-decay regimes. Finally, we show that assuming higher regularity, i.e., H\"older smoothness, does not improve minimax rates over the Lipschitz case, up to potential constants. Thus, we show that learning operators of any finite regularity necessarily suffers a curse of sample complexity.