高维回归的光滑伴随Lasso估计
Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression
摘要：
在高维环境中，稀疏结构对于效率至关重要，无论是在存储器，计算还是性能方面。通常考虑$\ell_1$惩罚以在这种情况下实施稀疏性。稀疏强制方法，Lasso是一个典型的例子，是解决高维问题的热门候选人。为了提高效率，他们依赖于调整参数交易数据拟合与稀疏。对于Lasso理论来说，这个调谐参数应该与噪声电平成比例，但后者在实践中通常是未知的。一种可能的补救措施是联合优化回归参数以及噪声水平。这在文献中被认为有几个名称：例如Scaled-Lasso，Square-root Lasso，Concurrently Lasso estimation，并且可能对置信集或不确定性量化感兴趣。在这项工作中，说明了平滑伴随套索公式的数值困难后，我们提出了一个修改，我们创造了平滑伴随套索，旨在增加数值稳定性。我们提出了一个高效和准确的求解器，导致计算成本不超过一个套索膨胀。我们利用快速Lasso解算器成功背后的标准成分：坐标下降算法，结合安全筛选规则，通过消除早期不相关的特征来实现速度效率。
英文摘要：
In high dimensional settings, sparse structures are crucial for efficiency, both in term of memory, computation and performance. It is customary to consider $\ell_1$ penalty to enforce sparsity in such scenarios. Sparsity enforcing methods, the Lasso being a canonical example, are popular candidates to address high dimension. For efficiency, they rely on tuning a parameter trading data fitting versus sparsity. For the Lasso theory to hold this tuning parameter should be proportional to the noise level, yet the latter is often unknown in practice. A possible remedy is to jointly optimize over the regression parameter as well as over the noise level. This has been considered under several names in the literature: Scaled-Lasso, Square-root Lasso, Concomitant Lasso estimation for instance, and could be of interest for confidence sets or uncertainty quantification. In this work, after illustrating numerical difficulties for the Smoothed Concomitant Lasso formulation, we propose a modification we coined Smoothed Concomitant Lasso, aimed at increasing numerical stability. We propose an efficient and accurate solver leading to a computational cost no more expansive than the one for the Lasso. We leverage on standard ingredients behind the success of fast Lasso solvers: a coordinate descent algorithm, combined with safe screening rules to achieve speed efficiency, by eliminating early irrelevant features.
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。
