Minimax Additive Regression under Unknown Dependent Designs

Baptiste Ferrere, Fabrice Gamboa, Jean-Michel Loubes

We study additive regression under an unknown and potentially non product design distribution, allowing the number of covariates to grow with the sample size. We consider a coupled class that separately controls the smoothness of the marginal densities and of each additive component multiplied by the corresponding marginal density. Under joint-density bounds that hold uniformly in the dimension and suitable dimension-growth conditions, we establish matching minimax bounds for prediction. With known marginal densities, the classical additive rate is attainable. When the marginals are unknown, this rate is preserved if the densities are at least as smooth as the weighted components. Otherwise, marginal-density smoothness determines the minimax rate over the coupled class. Finally, we recover all additive components with total squared error of the same order as the prediction error.