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.

翻译：我们研究了具有半定约束的数据驱动最小二乘问题，并在这些约束被松弛时，推导出其最优解谱的有限样本保证。具体而言，我们提供了一个高置信度界限，允许用一个更简单的规划替代完整的半定最小二乘问题，同时确保所得解的特征值在$\varepsilon$范围内接近半定约束所强制要求的特征值。所开发的证书随着数据量的增加而持续缩小，易于计算，与分布无关，且仅需要独立同分布样本。此外，当半定最小二乘用于学习未知二次函数时，我们建立了在无半定约束条件下最小化代理成本得到的梯度下降迭代与真实最小化器之间误差的界限。