Abstract We introduce a novel approach for high-dimensional regression with theoretical guarantees. The new procedure overcomes the challenge of tuning parameter selection of Lasso and possesses several appealing properties. It uses an easily simulated tuning parameter that automatically adapts to both the unknown random error distribution and the correlation structure of the design matrix. It is robust with substantial efficiency gain for heavy-tailed random errors while maintaining high efficiency for normal random errors. Comparing with other alternative robust regression procedures, it also enjoys the property of being equivariant when the response variable undergoes a scale transformation. Computationally, it can be efficiently solved via linear programming. Theoretically, under weak conditions on the random error distribution, we establish a finite-sample error bound with a near-oracle rate for the new estimator with the simulated tuning parameter. Our results make useful contributions to mending the gap between the practice and theory of Lasso and its variants. We also prove that further improvement in efficiency can be achieved by a second-stage enhancement with some light tuning. Our simulation results demonstrate that the proposed methods often outperform cross-validated Lasso in various settings.

中文翻译：

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一种无需调整的稳健高效的高维回归方法

摘要 我们介绍了一种具有理论保证的高维回归的新方法。新程序克服了 Lasso 调整参数选择的挑战，并具有几个吸引人的特性。它使用易于模拟的调谐参数，自动适应未知的随机误差分布和设计矩阵的相关结构。它具有鲁棒性，对于重尾随机错误具有显着的效率增益，同时对正常随机错误保持高效率。与其他替代稳健回归程序相比，当响应变量进行尺度变换时，它还具有等变的特性。在计算上，它可以通过线性规划有效地解决。理论上，在随机误差分布较弱的条件下，我们为具有模拟调整参数的新估计器建立了具有近预言率的有限样本误差界限。我们的结果为弥补 Lasso 及其变体的实践和理论之间的差距做出了有益的贡献。我们还证明，通过带有一些光调谐的第二阶段增强可以进一步提高效率。我们的模拟结果表明，所提出的方法在各种设置中通常优于交叉验证的套索。