迭代加权最小二乘改进$\ell_\infty$和$\ell_1$回归的收敛性
Improved Convergence for $\ell_\infty$ and $\ell_1$ Regression via Iteratively Reweighted Least Squares
摘要：
迭代重加权最小二乘法（IRLS）是一种在实际中用于解决回归问题的流行技术。这种方法的各种版本已经被提出，但他们的理论分析未能捕捉到良好的实际性能。在本文中，我们提出了一个简单而自然的版本的IRLS解决$\ell_\infty$和$\ell_1$回归，它可证明收敛到一个$(1+ε)$近似解的$O(m^{1/3}\log(1/ε)/ε^{2/3} + \log m/ε^2)$迭代，其中$m$是输入矩阵的行数。有趣的是，该运行时间与输入的条件无关，并且运行时间的主导项在$ε^{-1}$中亚线性地依赖，这对于非光滑函数的优化来说是不典型的。这比Chin等人（ITCS '12）和Christiano等人（STOC '11）的更复杂的算法改进了至少$1/ε^2$的因子，并产生了用于黏菌动力学的真正有效的自然算法（Straszak-Vishnoi，SODA '16，ITCS '16，ITCS '17）。
英文摘要：
The iteratively reweighted least squares method (IRLS) is a popular technique used in practice for solving regression problems. Various versions of this method have been proposed, but their theoretical analyses failed to capture the good practical performance. In this paper we propose a simple and natural version of IRLS for solving $\ell_\infty$ and $\ell_1$ regression, which provably converges to a $(1+ε)$-approximate solution in $O(m^{1/3}\log(1/ε)/ε^{2/3} + \log m/ε^2)$ iterations, where $m$ is the number of rows of the input matrix. Interestingly, this running time is independent of the conditioning of the input, and the dominant term of the running time depends sublinearly in $ε^{-1}$, which is atypical for the optimization of non-smooth functions. This improves upon the more complex algorithms of Chin et al. (ITCS '12), and Christiano et al. (STOC '11) by a factor of at least $1/ε^2$, and yields a truly efficient natural algorithm for the slime mold dynamics (Straszak-Vishnoi, SODA '16, ITCS '16, ITCS '17).
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