论文标题
在平滑凸功能上的循环坐标算法的最坏情况下分析
On the Worst-Case Analysis of Cyclic Coordinate-Wise Algorithms on Smooth Convex Functions
论文作者
论文摘要
我们为在不受限制的平滑凸优化设置中对自动计算机辅助构图坐标算法进行自动化的最坏情况分析提出了一个统一的框架。我们计算了循环坐标下降的精确最坏情况，以及在平滑凸函数类别上的交替最小化算法，并在标准类别的函数上，在具有坐标的LIPSCHITZ梯度的标准函数上，在质量较差的速率上提供了sublinear的上限和下限。我们尤其获得了一种新的循环坐标下降的上限，该坐标下降以优于最佳的数量级优于最佳可用的上限。我们还通过使用通常比通常用于分析块坐标算法的更简单和更自然的假设来提供新的数值界限来证明我们的方法的灵活性。最后，我们提供了以下事实的数值证据：在（确定性的）周期性算法中使用，将随机坐标下降加速到$ O（1/k^2）$的标准方案实际上是效率低下的。
We propose a unifying framework for the automated computer-assisted worst-case analysis of cyclic block coordinate algorithms in the unconstrained smooth convex optimization setup. We compute exact worst-case bounds for the cyclic coordinate descent and the alternating minimization algorithms over the class of smooth convex functions, and provide sublinear upper and lower bounds on the worst-case rate for the standard class of functions with coordinate-wise Lipschitz gradients. We obtain in particular a new upper bound for cyclic coordinate descent that outperforms the best available ones by an order of magnitude. We also demonstrate the flexibility of our approach by providing new numerical bounds using simpler and more natural assumptions than those normally made for the analysis of block coordinate algorithms. Finally, we provide numerical evidence for the fact that a standard scheme that provably accelerates random coordinate descent to a $O(1/k^2)$ complexity is actually inefficient when used in a (deterministic) cyclic algorithm.
