摘要翻译:
设$\mathcal{F}$是定义在概率空间$(S,\mathcal{a},P)$上的一类可测函数$F:S\mapsto[0,1]$。给定一个I.I.D.的样本(X_1,...,X_n)。取值于S中且具有公共分布P的随机变量,设P_n表示基于(X_1,...,X_n)的经验测度。我们研究了一个经验风险最小化问题$p_nf\to\min$,$F\in\mathcal{F}$。给出这个问题的一个解$\hat{f}_n$,其目标是得到它的超额风险\[\mathcal{E}_p(\hat{f}_n):=P\hat{f}_n-\inf_{f\in\mathcal{f}}Pf,\]用类$\mathcal{f}$的相关几何参数表示的非常一般的上界。利用集中不等式和其他经验过程工具,我们得到了超额风险的分布相关上界和数据相关上界,这些上界在许多例子中都是渐近正确的。这些边界包括由类中的函数索引的经验和Rademacher过程的局部超范数。我们使用这些边界来开发抽象风险最小化问题中的模型选择技术,这些技术可以应用于更专业的回归和分类框架。
---
英文标题:
《2004 IMS Medallion Lecture: Local Rademacher complexities and oracle
inequalities in risk minimization》
---
作者:
Vladimir Koltchinskii
---
最新提交年份:
2007
---
分类信息:
一级分类:Mathematics 数学
二级分类:Statistics Theory 统计理论
分类描述:Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计:例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
--
一级分类:Statistics 统计学
二级分类:Statistics Theory 统计理论
分类描述:stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近,贝叶斯推论,决策理论,估计,基础,推论,检验。
--
---
英文摘要:
Let $\mathcal{F}$ be a class of measurable functions $f:S\mapsto [0,1]$ defined on a probability space $(S,\mathcal{A},P)$. Given a sample (X_1,...,X_n) of i.i.d. random variables taking values in S with common distribution P, let P_n denote the empirical measure based on (X_1,...,X_n). We study an empirical risk minimization problem $P_nf\to \min$, $f\in \mathcal{F}$. Given a solution $\hat{f}_n$ of this problem, the goal is to obtain very general upper bounds on its excess risk \[\mathcal{E}_P(\hat{f}_n):=P\hat{f}_n-\inf_{f\in \mathcal{F}}Pf,\] expressed in terms of relevant geometric parameters of the class $\mathcal{F}$. Using concentration inequalities and other empirical processes tools, we obtain both distribution-dependent and data-dependent upper bounds on the excess risk that are of asymptotically correct order in many examples. The bounds involve localized sup-norms of empirical and Rademacher processes indexed by functions from the class. We use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification.
---
PDF链接:
https://arxiv.org/pdf/708.0083