发表机构
HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究SVM对偶问题的循环坐标下降法，提出结合坐标交互与边界几何的全局收缩界，并证明其优于现有方法，且通过数值实验验证了实用性。
AI 中文摘要
循环坐标下降法被广泛用于训练支持向量机，但仅全局线性收敛性对特定实例的速率提供的指导有限。我们研究带箱式约束的凸二次函数的精确循环坐标最小化，并发展出能同时捕捉坐标顺序和边界几何的全局收缩界。我们的收敛性证明结合了两个要素：连续坐标更新之间的相互作用（由归一化Hessian矩阵的三角部分编码）和目标函数增长（由最优性间隙的矩阵下界捕捉）。该证明对奇异Hessian矩阵、非唯一极小点和边界截断更新均保持有效。当两者使用相同的误差界常数时，我们分析的误差界特化严格优于Wang-Lin因子。在正定情形下，分析表明一次Gauss-Seidel扫描的最坏情况收缩在箱式约束下仍是全局上界，且对内部最优点达到等式。我们还通过构造一个具有固定奇异归一化Hessian矩阵的二维坐标族，展示了仅依赖Hessian矩阵的界的局限性，该族的最坏情况收缩随其边界余量趋零而趋近于1。该理论辅以对单个实例最坏情况收缩进行界定的实用程序，并在多个经典SVM数据集上进行了数值评估。
英文摘要
Cyclic coordinate descent is widely used to train support vector machines, but global linear convergence alone gives limited guidance about the rate on a particular instance. We study exact cyclic coordinate minimization for box-constrained convex quadratics and develop global contraction bounds that capture both coordinate order and boundary geometry. Our convergence certificate combines two ingredients: interactions between successive coordinate updates, encoded by the triangular part of the normalized Hessian, and objective growth, captured by a matrix lower bound on the optimality gap. It remains valid for cases with singular Hessians, nonunique minimizers, and updates clipped at the boundary. An error-bound specialization of our analysis strictly sharpens the Wang-Lin factor when both use the same error-bound constant. In the positive-definite case, the analysis shows that the worst-case contraction of one Gauss-Seidel sweep remains a global upper bound under box constraints, with equality for an interior optimizer. We also demonstrate the limitations of Hessian-only bounds by constructing a two-coordinate family with a fixed singular normalized Hessian, whose worst-case contraction approaches one as its boundary margin vanishes. The theory is complemented by practical procedures for bounding the worst-case contraction on individual instances and numerical evaluation on several classical SVM datasets.
Comments27 pages, 5 figures, 3 tables
