矩阵对策的坐标方法
Coordinate Methods for Matrix Games
摘要：
我们开发的原始-对偶坐标方法解决双线性鞍点问题的形式$\min_{x \in \mathcal{X}} \max_{y\in\mathcal{Y}} y^\top A x$，其中包含线性规划，分类，回归作为特殊情况。我们的方法将现有的完全随机次线性方法和方差缩减方法推向其每次迭代复杂度和样本复杂度的极限。我们通过设计有效的数据结构，利用泰勒近似的指数和二项式堆，获得几乎恒定的每次迭代的复杂性。我们通过使用动态采样分布的低方差梯度估计器来提高样本复杂度，该动态采样分布取决于迭代次数和矩阵项的大小。我们的运行时界限改善现有的原始-对偶方法的一个因素取决于稀疏措施的$m$的$n$矩阵$A$。例如，当行和列具有恒定的$\ell_1/\ell_2$范数比时，我们在完全随机设置中提供$m+n$因子的改进，在方差减少设置中提供$\sqrt{m+n}$因子的改进。我们将我们的方法应用于计算几何问题，即最小封闭球，最大内接球，和线性回归，并获得改进的复杂性界。对于线性回归与元素的非负矩阵，我们的保证提高了精确梯度方法的一个因素的$\sqrt{\mathrm{nnz}(A)/(m+n)}$。
英文摘要：
We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form $\min_{x \in \mathcal{X}} \max_{y\in\mathcal{Y}} y^\top A x$ which contain linear programming, classification, and regression as special cases. Our methods push existing fully stochastic sublinear methods and variance-reduced methods towards their limits in terms of per-iteration complexity and sample complexity. We obtain nearly-constant per-iteration complexity by designing efficient data structures leveraging Taylor approximations to the exponential and a binomial heap. We improve sample complexity via low-variance gradient estimators using dynamic sampling distributions that depend on both the iterates and the magnitude of the matrix entries. Our runtime bounds improve upon those of existing primal-dual methods by a factor depending on sparsity measures of the $m$ by $n$ matrix $A$. For example, when rows and columns have constant $\ell_1/\ell_2$ norm ratios, we offer improvements by a factor of $m+n$ in the fully stochastic setting and $\sqrt{m+n}$ in the variance-reduced setting. We apply our methods to computational geometry problems, i.e. minimum enclosing ball, maximum inscribed ball, and linear regression, and obtain improved complexity bounds. For linear regression with an elementwise nonnegative matrix, our guarantees improve on exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/(m+n)}$.
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