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High-dimensional linear regression with interaction effects is broadlyapplied in research fields such as bioinformatics and social science. In thispaper, we first investigate the minimax rate of convergence for regressionestimation in high-dimensional sparse linear models with two-way interactions.We derive matching upper and lower bounds under three types of heredityconditions: strong heredity, weak heredity and no heredity. From the results:(i) A stronger heredity condition may or may not drastically improve theminimax rate of convergence. In fact, in some situations, the minimax rates ofconvergence are the same under all three heredity conditions; (ii) The minimaxrate of convergence is determined by the maximum of the total price ofestimating the main effects and that of estimating the interaction effects,which goes beyond purely comparing the order of the number of non-zero maineffects $r_1$ and non-zero interaction effects $r_2$; (iii) Under any of thethree heredity conditions, the estimation of the interaction terms may be thedominant part in determining the rate of convergence for two different reasons:1) there exist more interaction terms than main effect terms or 2) a largeambient dimension makes it more challenging to estimate even a small number ofinteraction terms. Second, we construct an adaptive estimator that achieves theminimax rate of convergence regardless of the true heredity condition and thesparsity indices $r_1, r_2$.

High-dimensional Adaptive Minimax Sparse Estimation with Interactions