Concentration inequalities for semidefinite least squares based on data
Abstract
We study data-driven least squares (LS) problems with semidefinite (SD) constraints and derive finite-sample guarantees on the spectrum of their optimal solutions when these constraints are relaxed. In particular, we provide a high confidence bound allowing one to solve a simpler program in place of the full SDLS problem, while ensuring that the eigenvalues of the resulting solution are -close of those enforced by the SD constraints. The developed certificate, which consistently shrinks as the number of data increases, turns out to be easy-to-compute, distribution-free, and only requires independent and identically distributed samples. Moreover, when the SDLS is used to learn an unknown quadratic function, we establish bounds on the error between a gradient descent iterate minimizing the surrogate cost obtained with no SD constraints and the true minimizer.
Keywords
Cite
@article{arxiv.2509.13166,
title = {Concentration inequalities for semidefinite least squares based on data},
author = {Filippo Fabiani and Andrea Simonetto},
journal= {arXiv preprint arXiv:2509.13166},
year = {2026}
}