Minimax Additive Regression under Unknown Dependent Designs

Abstract

We study additive regression when the covariate distribution is unknown and may exhibit dependence, with dimension allowed to increase with sample size. Our function class controls the regularity of marginal densities separately from that of density-weighted additive components.

Under joint-density bounds uniform in dimension and suitable dimension-growth conditions, we establish matching minimax prediction bounds. The classical additive rate remains attainable with known marginals, and with unknown marginals that are sufficiently smooth relative to the weighted components. Otherwise, marginal-density regularity limits the optimal rate over this coupled class. The individual additive components can also be recovered with aggregate squared error of the same order as the prediction error.

Version history

First submitted on 30 September 2026. The latest revision, version 2, was submitted on 3 October 2026.