高维统计学习中的最佳子集选择、持久性和$l_1$约束下的optimization
Best subset selection, persistence in high-dimensional statistical learning and optimization under $l_1$ constraint
摘要：
设$(Y,X_1,...,X_m)$为随机向量。期望基于$(X_1,...,X_m)$来预测$Y$。预测方法的例子是回归，分类使用逻辑回归或分离超平面，等等。我们考虑最佳子集选择的问题，并研究它的上下文中$m=n^α$，$α>1$，其中$n$是观察的数量。我们调查的程序是基于经验风险最小化。它示出，在常见的情况下，我们的目标应该是找到最好的子集之间的大小是顺序$o(n/\log(n))$。它还表明，在某种"渐近意义“，当假设一定的稀疏性条件，有没有损失，让$m$远大于$n$，例如，$m=n^α, α>1$。这与从尺寸小于$n$的“最佳”子集开始并且不管$α$的值如何相比。然后，我们研究的条件下，经验风险最小化受$l_1$约束产生近最佳子集。这些结果推广了Greenshtein和Ritov最近的一些结果。最后，我们提出了一个高维模拟研究的“升压型”的分类过程。
英文摘要：
Let $(Y,X_1,...,X_m)$ be a random vector. It is desired to predict $Y$ based on $(X_1,...,X_m)$. Examples of prediction methods are regression, classification using logistic regression or separating hyperplanes, and so on. We consider the problem of best subset selection, and study it in the context $m=n^α$, $α>1$, where $n$ is the number of observations. We investigate procedures that are based on empirical risk minimization. It is shown, that in common cases, we should aim to find the best subset among those of size which is of order $o(n/\log(n))$. It is also shown, that in some ``asymptotic sense,'' when assuming a certain sparsity condition, there is no loss in letting $m$ be much larger than $n$, for example, $m=n^α, α>1$. This is in comparison to starting with the ``best'' subset of size smaller than $n$ and regardless of the value of $α$. We then study conditions under which empirical risk minimization subject to $l_1$ constraint yields nearly the best subset. These results extend some recent results obtained by Greenshtein and Ritov. Finally we present a high-dimensional simulation study of a ``boosting type'' classification procedure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
