Rank-one Convexification for Sparse Regression

Abstract

Sparse regression models are increasingly prevalent due to their ease of interpretability and superior out-of-sample performance. However, the exact model of sparse regression with an 0 constraint restricting the support of the estimators is a challenging non-convex optimization problem. In this paper, we derive new strong convex relaxations for sparse regression. These relaxations are based on the ideal (convex-hull) formulations for rank-one quadratic terms with indicator variables. The new relaxations can be formulated as semidefinite optimization problems in an extended space and are stronger and more general than the state-of-the-art formulations, including the perspective reformulation and formulations with the reverse Huber penalty and the minimax concave penalty functions. Furthermore, the proposed rankone strengthening can be interpreted as a non-separable, non-convex sparsityinducing regularizer, which dynamically adjusts its penalty according to the shape of the error function. In our computational experiments with benchmark datasets, the proposed conic formulations are solved within seconds and result in near-optimal solutions (with 0.4% optimality gap) for non-convex 0 -problems. Moreover, the resulting estimators also outperform alternative convex approaches from a statistical viewpoint, achieving high prediction accuracy and good interpretability.

On the convex hull of convex quadratic optimization problems with indicators

“…Set X 2×2 corresponds (after scaling) to a generic strictly convex quadratic function of two variables; conic quadratic disjunctive programming representations of cl conv(X 2×2 ) have been used in the literature [4,20], explicit representations of cl conv (X 2×2 ∩ {(x, z, t) : x ≥ 0}) have been given [8,24], and descriptions of the rank-one case d 1 d 2 = 1 were given in [6], but an explicit representation of cl conv(X 2×2 ) with free variables has not been established in the literature.…”

“…We now illustrate Theorem 2 by providing an alternative proof of the main result of [6] using our unifying framework.…”

“…Observation 2 . On the other hand, the strong "optimal" or "dynamic" relaxations [6,17,33], where the decomposition is not fixed but instead is chosen dynamically, are excessive to describe cl conv(X). Indeed, they are of the form (17) for every possible (rank-one, 2 × 2, remainder) matrix, and are not finitely generated.…”

“…In particular, ( 26) is guaranteed to define a high dimensional face of P if |T | is small. Indeed, inequalities (26) were found to be particularly effective computationally if T = {T ⊆ [n] : |T | ≤ κ} for some small κ [6], although a theoretical justification of this observation has been missing until now.…”

“…Similar relaxations based on separable quadratic terms were considered in [17,33]. A generalization of the above approach is to let Γ i = h i h ⊤ i be a rank-one matrix [6,7,23,31,32]; in this case, letting…”