Rejoinder: Local Rademacher complexities and oracle inequalities in risk minimization
作者回复：局部 Rademacher 复杂度与风险最小化中的 oracle 不等式AI 解读 ↓
← 返回论文列表与筛选
阅读说明
本页整理公开书目信息，不生成学术质量评分。AI 解读根据题名及所注明的摘要或原文片段生成，未完成全文核验，可能有误；具体假设、证明与结论请核对原文。摘要保留来源文本，未经模型改写。
DOI 首次登记不等于接收或发表日期。部分更正、评论等条目没有独立摘要；缺少可靠材料时保留待补状态。
引用量采用 OpenAlex 单条关联记录，不合计多个记录；关联记录可能包含正式发表版本，不等于此版本单独引用。缺失不视为零，引用指标不代表论文质量。
关注作者按书目中的完整姓名匹配，同名和姓名变体需人工辨别。相关论文先按共享细分方向和具体方法排列，再参考通用方法与发表日期。
分类依据与纠错
AI 根据所提供的论文材料归类：回复讨论一般经验过程、学习复杂度与风险界，没有明确涉及所列经济计量应用方向，因此保留空分类。
反馈分类问题
正在核对关联版本…
代码、数据与补充材料
正在核对对应版本的资源…
关注这篇论文的作者
作者回复：局部 Rademacher 复杂度与风险最小化中的 oracle 不等式
- 研究问题
- 这篇作者回复如何解释局部复杂度的定义、其与超额风险界的联系，以及使用近最优函数集合或其切片的不同思路？
- 方法与思路
- 作者围绕近最优集合的几何和经验过程性质解释局部复杂度，讨论固定点思路、Rademacher 过程及集合切片的替代构造。
- 材料陈述的贡献
- 所给回复澄清分布依赖与数据依赖复杂度的联系，并讨论均值—方差关系及星形条件下的相关思路，不能当作主论文的新定理。
- 阅读关注点
- 应区分作者原方法、讨论者的方法及附加条件，核对局部集合与切片的定义，并查阅原文中缺失的变换符号。
- 阅读边界
- 这是作者回复而非主论文；依据版本未注明的 OpenAlex 索引文本，变换记号存在缺失，需查阅原PDF，不能重构公式或外推主论文结论。
原文摘要
I would like to thank the discussants for a number of deep and interesting comments and for their inspiring work on the subject over the years.I will not be able to address all the issues raised in the discussion; I will concentrate just on several of them. Local complexities and excess risk bounds.The first question is about possible ways to define distribution-and data-dependent complexities (such as local Rademacher complexities).The approach taken in my paper is based on geometric and probabilistic properties of the δ-minimal setThe first quantity of interest is the L 2 -diameter of this set, D(F ; δ), and the second one is the function φ n (F ; δ) that is equal to the expected supremum of the empirical process indexed by the differences fg, f, g ∈ F (δ).These two functions are then combined in the expression Ūn (δ; t) that has its roots in Talagrand's concentration inequalities for empirical processes.The -transform of Ūn (•; t) (which is just a way to write solutions of fixed point-type equations) is then used to define the localized complexities that provide upper bounds on the excess risk.Under further assumptions, such as mean-variance relationships discussed in detail by Shen and Wang (Bartlett and Mendelson also discuss this and call the function classes satisfying these relationships "Bernstein classes"), these complexities can be redefined in terms of the local L 2 -continuity modulus of empirical processes.Since the Rademacher process can be used as a data-dependent bootstrap-type "estimate" of the empirical process, this approach also leads to data-dependent local Rademacher complexities.The use of the whole δ-minimal set is not the only possibility.One can also look at its "slices" F (δ 1 , δ 2 ] := F (δ 2 ) \ F (δ 1 ) and define the excess risk bounds in terms of the accuracy of empirical approximation on the slices.One can even make the slices really thin and look at {f ∈ F : Pf -inf g∈F P g = δ}.This was the approach taken by Peter Bartlett and Shahar Mendelson.Under an additional (and relatively innocent) assumption that the class F is star-shaped, they established excess risk bounds (and also ratio-type bounds) in terms of complexities of such "thin slices."