Concentration inequalities for semidefinite least squares based on data

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 $\varepsilon$-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.

💡 Research Summary

This paper investigates data‑driven least‑squares (LS) problems that are subject to semidefinite (SD) constraints, a class often referred to as semidefinite least‑squares (SDLS). The presence of SD constraints dramatically increases computational complexity, especially when the problem must be solved repeatedly on large data sets. The authors ask whether one can safely drop the SD constraints, solve a simpler program (often a quadratic program, QP), and still obtain a solution whose spectrum (i.e., eigenvalues) is close to that enforced by the original constraints.

The core contribution is a finite‑sample, distribution‑free concentration inequality that quantifies exactly how close the eigenvalues of the relaxed‑constraint solution are to the target interval (