An analysis of penalized interaction models

An important consideration for variable selection in interaction models is to design an appropriate penalty that respects hierarchy of the importance of the variables. A common theme is to include an interaction term only after the corresponding main effects are present. In this paper, we study several recently proposed approaches and present a unified analysis on the convergence rate for a class of estimators, when the design satisfies the restricted eigenvalue condition. In particular, we show that with probability tending to one, the resulting estimates have a rate of convergence $s\sqrt{\log p_1/n}$ in the $\ell_1$ error, where $p_1$ is the ambient dimension, $s$ is the true dimension and $n$ is the sample size. We give a new proof that the restricted eigenvalue condition holds with high probability, when the variables in the main effects and the errors follow sub-Gaussian distributions. Under this setup, the interactions no longer follow Gaussian or sub-Gaussian distributions even if the main effects follow Gaussian, and thus existing works are not applicable. This result is of independent interest.