高维$L_2$Boosting：收敛速度

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摘要翻译：
Boosting是机器学习中最重要的发展之一。本文研究了高维环境下为回归量身定制的$L_2$Boosting的收敛速度。此外，我们引入了所谓的\textquotedblleft post-boosting\textquotedblright。这是一个后选择估计，它将普通最小二乘应用于第一阶段通过$L_2$Boosting选择的变量。另一个变体是\textquotedblleft正交boosting\textquotedblright\，其中在每一步之后进行一个正交投影。我们证明了在稀疏的高维环境中，后$L_2$boosting和正交boosting都获得了与LASSO相同的收敛速度。我们证明了经典$L_2$Boosting的收敛速度依赖于稀疏特征值常数描述的设计矩阵。为了证明后一结果，我们在分析$L_2$Boosting的重访行为的基础上，给出了纯贪婪算法的新的逼近结果。我们还介绍了可行的早停规则，这些规则易于在实际工作中实现和使用。我们的结果也允许在套索和升压之间进行直接的比较，这在文献中是缺失的。最后，我们通过仿真研究和应用来说明我们的理论结果的相关性，并为Boosting的实际方面提供见解。在这些模拟研究中，后$L_2$Boosting明显优于LASSO。
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英文标题：
《High-Dimensional $L_2$Boosting: Rate of Convergence》
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作者：
Ye Luo and Martin Spindler
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最新提交年份：
2016
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分类信息：

一级分类：Statistics        统计学
二级分类：Machine Learning        机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
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一级分类：Computer Science        计算机科学
二级分类：Machine Learning        机器学习
分类描述：Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.
关于机器学习研究的所有方面的论文（有监督的，无监督的，强化学习，强盗问题，等等），包括健壮性，解释性，公平性和方法论。对于机器学习方法的应用，CS.LG也是一个合适的主要类别。
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一级分类：Economics        经济学
二级分类：Econometrics        计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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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
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一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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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。
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英文摘要：
  Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of $L_2$Boosting, which is tailored for regression, in a high-dimensional setting. Moreover, we introduce so-called \textquotedblleft post-Boosting\textquotedblright. This is a post-selection estimator which applies ordinary least squares to the variables selected in the first stage by $L_2$Boosting. Another variant is \textquotedblleft Orthogonal Boosting\textquotedblright\ where after each step an orthogonal projection is conducted. We show that both post-$L_2$Boosting and the orthogonal boosting achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting. We show that the rate of convergence of the classical $L_2$Boosting depends on the design matrix described by a sparse eigenvalue constant. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of $L_2$Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, post-$L_2$Boosting clearly outperforms LASSO.
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PDF链接：
https://arxiv.org/pdf/1602.08927

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关键词：Boosting boost STI ING Tin post 变量 量身 dimensional setting