从调查训练样本中学习：Horvitz-Thompson风险最小化的率界
Learning from Survey Training Samples: Rate Bounds for Horvitz-Thompson Risk Minimizers
摘要：
在二元分类的背景下的经验风险的最小化的泛化能力进行了研究，根据各种各样的复杂性假设的分类器的集合进行优化。相反，致力于这个问题的绝大多数工作规定，用于计算经验风险泛函的训练数据集由i.i.d.意见。除了训练数据在大i.i.d.样本或建模为r. v.的弱依赖序列的实现。在美国，当用于训练分类器的数据是通过更一般的抽样/调查方案提取的并且呈现复杂的依赖结构时，统计保证还没有被记载。本文的主要目的是表明，经验风险最小化理论可以扩展到统计学习是基于调查样本和相关的包含概率的知识的情况下。确切地说，我们证明了，最小化加权版本的经验风险，简称为霍维茨-汤普森风险（HT风险），在一类控制的复杂性导致率的顺序$O_{\mathbb{P}}((κ_N (\log N)/n)^{1/2})$与$κ_N=(n/N)/\min_{i\leq N}π_i$，当数据被抽样的拒绝计划，基数$N\geq n$的统计群体内的（确定性）大小$n$，具有不等概率权重$π_i>0$的基本{无替换的\it采样}的推广。扩展到其他采样方案，然后建立一个耦合参数。除了理论结果，数值实验显示，以显示HT风险最小化的相关性，忽略用于生成训练数据集的采样方案可能会完全危及学习过程。
英文摘要：
The generalization ability of minimizers of the empirical risk in the context of binary classification has been investigated under a wide variety of complexity assumptions for the collection of classifiers over which optimization is performed. In contrast, the vast majority of the works dedicated to this issue stipulate that the training dataset used to compute the empirical risk functional is composed of i.i.d. observations. Beyond the cases where training data are drawn uniformly without replacement among a large i.i.d. sample or modelled as a realization of a weakly dependent sequence of r.v.'s, statistical guarantees when the data used to train a classifier are drawn by means of a more general sampling/survey scheme and exhibit a complex dependence structure have not been documented yet. It is the main purpose of this paper to show that the theory of empirical risk minimization can be extended to situations where statistical learning is based on survey samples and knowledge of the related inclusion probabilities. Precisely, we prove that minimizing a weighted version of the empirical risk, refered to as the Horvitz-Thompson risk (HT risk), over a class of controlled complexity lead to a rate for the excess risk of the order $O_{\mathbb{P}}((κ_N (\log N)/n)^{1/2})$ with $κ_N=(n/N)/\min_{i\leq N}π_i$, when data are sampled by means of a rejective scheme of (deterministic) size $n$ within a statistical population of cardinality $N\geq n$, a generalization of basic {\it sampling without replacement} with unequal probability weights $π_i>0$. Extension to other sampling schemes are then established by a coupling argument. Beyond theoretical results, numerical experiments are displayed in order to show the relevance of HT risk minimization and that ignoring the sampling scheme used to generate the training dataset may completely jeopardize the learning procedure.
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