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

Quadratic regression involves modeling the response as a (generalized) linear function 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" in regression often provides an easy way to increase accuracy in already-high-dimensional problems. However, this explodes the problem dimension from linear \(O(p)\) to quadratic \(O(p^2)\), and it is common to look for sparse interactions (typically via heuristics). In this paper, we provide a new algorithm - Interaction Hard Thresholding (IntHT) which is the first one to provably accurately solve this problem in sub-quadratic time and space. It is a variant of Iterative Hard Thresholding; one that uses the special quadratic structure 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 theoretically prove that, in spite of the many speedup-related approximations, IntHT linearly converges to a consistent estimate under standard high-dimensional sparse recovery assumptions. We also demonstrate its value via synthetic experiments. Moreover, we numerically show that IntHT can be extended to higher-order regression problems, and also theoretically analyze an SVRG variant of IntHT.