提出早停镜像下降法，实现高维回归最优风险界。
Sharp Risk Bounds for Early-Stopping in Gaussian Linear Regression
- 基于局部高斯宽度，推广最小二乘估计的风险界到早停镜像下降。
- 在ℓ₁约束下得到当前最紧的风险上界，证明方法通用性。
- 适合研究高维统计推断与优化算法的学者参考。
我们研究在任意凸体和设计矩阵下的高维高斯线性回归中的早停镜像下降（ESMD），目标是最小化样本内均方误差。主要结果表明，基于局部高斯宽度的一些最紧风险界可推广至ESMD。我们推导了势函数需满足的充分条件，通过Minkowski泛函表达，从而构造新势函数并分析已有势函数。这些结果给出ESMD极小极大最优性的通用充分条件，系统比较了其与最小二乘估计（LSE）的性能，并在ℓ₁-约束情形下建立了目前已知最紧的风险上界。
原文摘要 · Abstract (English)
We study early-stopped mirror descent (ESMD) for high-dimensional Gaussian linear regression over arbitrary convex bodies and design matrices, where the task is to minimize the in-sample mean squared error. Our main result shows that some of the sharpest risk bounds for the least squares estimator (LSE), based on the local Gaussian width, extend to ESMD. We derive sufficient conditions on the potential, expressed via the Minkowski functional, under which our result holds. These conditions allow us to construct new potentials and analyze existing ones. Our results then yield general sufficient conditions for minimax optimality of ESMD, provide a systematic comparison with the LSE, and establish the tightest known risk bound in the $\ell_1$-constrained setting.
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