Provably data-driven projection method for quadratic programming
Anh Tuan Nguyen, Viet Anh Nguyen
TL;DR
The paper addresses the scalability of convex quadratic programs by learning data-driven projection matrices that map high-dimensional problems to a lower-dimensional space while preserving near-optimal objectives. It extends the LP-based data-driven projection framework to QPs by introducing a perturbed, strongly convex formulation, localizing the QP solution via Carathéodory’s theorem, and modeling the computation with a unrolled active-set method that fits the Goldberg–Jerrum framework. The authors prove a tight generalization bound on the pseudo-dimension of the resulting loss class, show a matching lower bound, and extend the theory to settings where the focus is on matching the optimal solution or learning an input-aware projection. Together, these results provide provable guarantees for data-driven projection learning in large-scale QPs and offer practical avenues for solution-informed projection design and input-aware mappings.
Abstract
Projection methods aim to reduce the dimensionality of the optimization instance, thereby improving the scalability of high-dimensional problems. Recently, Sakaue and Oki proposed a data-driven approach for linear programs (LPs), where the projection matrix is learned from observed problem instances drawn from an application-specific distribution of problems. We analyze the generalization guarantee for the data-driven projection matrix learning for convex quadratic programs (QPs). Unlike in LPs, the optimal solutions of convex QPs are not confined to the vertices of the feasible polyhedron, and this complicates the analysis of the optimal value function. To overcome this challenge, we demonstrate that the solutions of convex QPs can be localized within a feasible region corresponding to a special active set, utilizing Caratheodory's theorem. Building on such observation, we propose the unrolled active set method, which models the computation of the optimal value as a Goldberg-Jerrum (GJ) algorithm with bounded complexities, thereby establishing learning guarantees. We then further extend our analysis to other settings, including learning to match the optimal solution and input-aware setting, where we learn a mapping from QP problem instances to projection matrices.