Robust Least Squares Problems with Binary Uncertain Data

Abstract

We propose a Binary Robust Least Squares (BRLS) model that encompasses key robust least squares formulations, such as those involving uncertain binary labels and adversarial noise constrained within a hypercube. {To develop algorithms with theoretical guarantees for the BRLS problem, we exploit the structure of the inner binary maximization problem with a convex quadratic objective function. Refined guarantees are obtained when the noise correlations are sign-structured, in which case the inner problem admits sharper submodular or supermodular oracles. For the supermodular linear BRLS problem, we establish a link between saddle points of its continuous relaxation and global minimax points of BRLS, and propose a projected-gradient algorithm to find an $\epsilon$-global minimax point in $O(\epsilon^{-2})$ iterations. For the supermodular nonlinear BRLS problem, we develop a Moreau-envelope-based framework that finds an $\epsilon$-stationary point in expectation within $O(\epsilon^{-4})$ iterations. For the linear submodular case and the linear general case, we utilize a double-greedy algorithm and a semidefinite relaxation as the respective subsolvers; the latter attains an approximation ratio below $2/\pi$. Coupled with the projected-gradient framework, these oracles yield approximate minimax guarantees within $O(\epsilon^{-2})$ iterations. Numerical experiments on health status prediction with candidate label-corruption sets, synthetic linear BRLS, and thresholded phase retrieval with missing binary labels illustrate the behavior and robustness gains of the BRLS model under structured noise compared with classical least-squares-based baselines.