Exponential Screening and optimal rates of sparse estimation
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指数筛选与稀疏估计的最优速率
- 研究问题
- 在高斯噪声回归中，如何从候选函数的线性组合中同时追求稀疏性与低均方误差，并适应未知的最优稀疏程度？
- 方法与思路
- 作者提出 Exponential Screening 估计程序，分别以非零系数个数和系数绝对值总量衡量稀疏性，研究适应性与极小极大界。
- 材料陈述的贡献
- 摘要称程序适应误差与稀疏性的最优权衡，达到所讨论聚合问题的最优性及极小极大界，并讨论理论和数值比较。
- 阅读关注点
- 应核对风险定义、候选函数集合及设计条件、适应界与下界的范围，以及计算实现和竞争方法比较的设置。
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原文摘要
In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in this setup is to assume that the linear combination is sparse in some sense, for example, that it involves only few covariates. We consider a general, nonnecessarily linear, regression with Gaussian noise and study a related question, that is, to find a linear combination of approximating functions, which is at the same time sparse and has small mean squared error (MSE). We introduce a new estimation procedure, called Exponential Screening, that shows remarkable adaptation properties. It adapts to the linear combination that optimally balances MSE and sparsity, whether the latter is measured in terms of the number of nonzero entries in the combination (ℓ0 norm) or in terms of the global weight of the combination (ℓ1 norm). The power of this adaptation result is illustrated by showing that Exponential Screening solves optimally and simultaneously all the problems of aggregation in Gaussian regression that have been discussed in the literature. Moreover, we show that the performance of the Exponential Screening estimator cannot be improved in a minimax sense, even if the optimal sparsity is known in advance. The theoretical and numerical superiority of Exponential Screening compared to state-of-the-art sparse procedures is also discussed.