SIAM Journal on Optimization, Volume 30, Issue 3, Page 1878-1904, January 2020.
We analyze the coordinate descent method with a new coordinate selection strategy, called volume sampling. This strategy prescribes selecting subsets of variables of certain size proportionally to the determinants of principal submatrices of the matrix, which bounds the curvature of the objective function. In the particular case when the size of the subsets equals one, volume sampling coincides with the well-known strategy of sampling coordinates proportionally to their Lipschitz constants. For the coordinate descent with volume sampling, we establish the convergence rates for both convex and strongly convex problems. Our theoretical results show that, by increasing the size of the subsets, it is possible to accelerate the method up to the factor which depends on the spectral gap between the corresponding largest eigenvalues of the curvature matrix. Several numerical experiments confirm our theoretical conclusions.

中文翻译：

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具有体积采样的随机坐标下降法

SIAM优化杂志，第30卷，第3期，第1878-1904页，2020年1月。
我们使用一种称为体积采样的新坐标选择策略来分析坐标下降方法。该策略规定选择一定大小的变量子集，该变量子集与矩阵的主要子矩阵的行列式成比例，从而限制目标函数的曲率。在特定情况下，当子集的大小等于1时，体积采样与众所周知的对与其Lipschitz常数成比例的坐标进行采样的策略一致。对于体积采样的坐标下降，我们建立了凸和强凸问题的收敛速度。我们的理论结果表明，通过增加子集的大小，可以将该方法加速到取决于曲率矩阵的相应最大特征值之间的光谱间隙的因数。几个数值实验证实了我们的理论结论。