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Quadratic regression involves modeling the response as a (generalized) linearfunction of not only the features $x^{j_1}$ but also of quadratic terms$x^{j_1}x^{j_2}$. The inclusion of such higher-order "interaction terms" inregression often provides an easy way to increase accuracy inalready-high-dimensional problems. However, this explodes the problem dimensionfrom linear $O(p)$ to quadratic $O(p^2)$, and it is common to look for sparseinteractions (typically via heuristics). In this paper, we provide a newalgorithm - Interaction Hard Thresholding (IntHT) which is the first one toprovably accurately solve this problem in sub-quadratic time and space. It is avariant of Iterative Hard Thresholding; one that uses the special quadraticstructure to devise a new way to (approx.) extract the top elements of a $p^2$size gradient in sub-$p^2$ time and space. Our main result is to theoreticallyprove that, in spite of the many speedup-related approximations, IntHT linearlyconverges to a consistent estimate under standard high-dimensional sparserecovery assumptions. We also demonstrate its value via synthetic experiments.Moreover, we numerically show that IntHT can be extended to higher-orderregression problems, and also theoretically analyze an SVRG variant of IntHT.

Interaction Hard Thresholding: Consistent Sparse Quadratic Regression in Sub-quadratic Time and Space