Functional Linear Regression of Cumulative Distribution Functions

The estimation of cumulative distribution functions (CDFs) is an important learning task with a great variety of downstream applications, such as risk assessments in predictions and decision making. In this paper, we study functional regression of contextual CDFs where each data point is sampled from a linear combination of context dependent CDF basis functions. We propose functional ridge-regression-based estimation methods that estimate CDFs accurately everywhere. In particular, given $n$ samples with $d$ basis functions, we show estimation error upper bounds of $\widetilde{O}(\sqrt{d/n})$ for fixed design, random design, and adversarial context cases. We also derive matching information theoretic lower bounds, establishing minimax optimality for CDF functional regression. Furthermore, we remove the burn-in time in the random design setting using an alternative penalized estimator. Then, we consider agnostic settings where there is a mismatch in the data generation process. We characterize the error of the proposed estimators in terms of the mismatched error, and show that the estimators are well-behaved under model mismatch. Finally, to complete our study, we formalize infinite dimensional models where the parameter space is an infinite dimensional Hilbert space, and establish self-normalized estimation error upper bounds for this setting.