发表机构
The University of Tokyo; RIKEN（东京大学; 理化学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对紧凸集上带正定 Hessian 的凸优化，提出阻尼牛顿 Frank--Wolfe 方法，结合残差回溯与切换规则，实现全局线性及局部高阶收敛，实验验证其稳健性与最优性能。
AI 中文摘要
我们研究在紧凸集上针对具有正定 Hessian 矩阵的二次可微目标函数的凸优化问题，假设通过线性最小化预言机访问可行集。现有的二阶 Frank--Wolfe 方法要么在一般凸集上仅提供局部线性收敛速率，要么仅在多面体上实现全局线性和局部二次收敛。我们提出了阻尼牛顿 Frank--Wolfe 方法，该方法使用 Frank--Wolfe 或 away-step Frank--Wolfe 内迭代近似求解约束阻尼牛顿子问题。我们开发了残差回溯技术，以根牛顿步长作为安全参考，并结合一个最终采取全步长的切换规则。在 $p$ 阶导数的 Hölder 光滑性假设下，我们建立了全局线性收敛性和至少 $1+\ u$ 阶的局部收敛（$p=2$），以及局部二次收敛（$p=3$）。在矩阵感知和岭正则化逻辑回归上的实验表明，残差回溯变体始终稳健，并且通常在测试的一阶和二阶基线中取得最佳性能。
英文摘要
We study convex optimization over compact convex sets for twice differentiable objective functions with positive definite Hessians, assuming access to the feasible set through a linear minimization oracle. Existing second-order Frank--Wolfe methods either provide only a local linear rate over general convex sets or achieve global linear and local quadratic convergence only over polytopes. We propose damped Newton Frank--Wolfe methods that approximately solve constrained damped Newton subproblems using Frank--Wolfe or away-step Frank--Wolfe inner iterations. We develop residual backtracking, using a root Newton stepsize as a safe reference, together with a switching rule that eventually takes full steps. Under Hölder smoothness assumptions for $p$th derivatives, we establish global linear convergence and local convergence with Q-order at least $1+ν$ ($p=2$) and local quadratic convergence ($p=3$). Experiments on matrix sensing and ridge-regularized logistic regression show that the residual-backtracking variant is consistently robust and often achieves the best performance among the tested first- and second-order baselines.
Comments37 pages, 11 figures
