提升最小二乘估计的策略
Tactics for Improving Least Squares Estimation
AI总结：
本文探讨了高维最小二乘回归中加速计算的策略，包括MM原理、Moreau包络 smoothing 和约束估计的proximal距离原理，通过迭代加权最小二乘等方法提高计算效率。
摘要：
本文讨论了在高维最小二乘回归中加速计算的策略。这些策略包括：(a) 主要化-最小化 (MM) 原理，(b) 通过Moreau包络进行平滑，以及(c) 用于约束估计的proximal距离原理。在迭代加权最小二乘中，MM原理可以创建一个替代函数，通过交换案例权重来调整响应值。将其减少到普通最小二乘允许在迭代中重用Gram矩阵及其Cholesky分解。此策略适用于L2E回归和广义线性模型的估计。对于如分位数回归等问题，非光滑目标函数项可以被其Moreau包络近似替代，并通过球面二次函数进行主要化。最后，具有距离到集合惩罚的惩罚回归也受益于这种视角。我们的数值实验验证了去权重和Moreau包络近似的速度和实用性。Julia软件实现这些实验可在我们的网页上获得。
英文摘要：
This paper deals with tactics for fast computation in least squares regression in high dimensions. These tactics include: (a) the majorization-minimization (MM) principle, (b) smoothing by Moreau envelopes, and (c) the proximal distance principle for constrained estimation. In iteratively reweighted least squares, the MM principle can create a surrogate function that trades case weights for adjusted responses. Reduction to ordinary least squares then permits the reuse of the Gram matrix and its Cholesky decomposition across iterations. This tactic is pertinent to estimation in L2E regression and generalized linear models. For problems such as quantile regression, non-smooth terms of an objective function can be replaced by their Moreau envelope approximations and majorized by spherical quadratics. Finally, penalized regression with distance-to-set penalties also benefits from this perspective. Our numerical experiments validate the speed and utility of deweighting and Moreau envelope approximations. Julia software implementing these experiments is available on our web page.
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