通用鲁棒性定律的四项扩展：对数凹协变量、异质混合、向量预测与马尔可夫数据
原标题：Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data
AI 摘要
该论文由 Tobias J. Peter 撰写，对 Bubeck 和 Sellke 提出的「通用鲁棒性定律」进行四项推广，分别放宽了原定律的四个结构假设：高斯型等周集中、协变量分布同质、标量输出和样本独立。作者证明，在指数集中、异质混合、向量值预测器和平稳马尔可夫数据下，该定律仍以相同速率成立，并给出显式常数。这一工作将鲁棒性定律的适用范围从理想化 i.i.d. 标量场景扩展到更贴近实际的分布与数据结构。
正文节选
Tobias J. Peter Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data Abstract Bubeck and Sellke’s universal law of robustness says that fitting \(n\) noisy samples below the noise level with a class of \(p\) real parameters forces a Lipschitz constant of order \(\sqrt{nd/p}\), when the covariate distribution on \(\mathbb{R}^{d}\) is isoperimetric: smooth interpolation costs a factor \(d\) of overparametri