论文标题
Bregman近端方法的收敛速度：本地几何与规律性与清晰度
The rate of convergence of Bregman proximal methods: Local geometry vs. regularity vs. sharpness
论文作者
论文摘要
我们研究了Bregman近端方法的最后近期收敛速率 - 从镜像到镜像及其乐观的变体 - 作为定义该方法的Prox-Mapping引起的局部几何形状的函数。对于普遍性，我们专注于受约束，非符号变化不平等的局部解决方案，并且我们表明，给定方法的收敛速率急剧取决于其相关的Legendre指数，该概念衡量了基本的Bregman函数（Euclidean，groutopic，groutopic或其他）的概念。特别是，我们表明边界解决方案表现出零和非零legendre指数的方法之间的严格分离：前者以线性速率收敛，而后者通常会收敛。与在欧几里得正则化下有限数量的步骤中相比，在线性约束的问题中，这种二分法变得更加明显，在线性约束的问题中，熵正则化的方法沿着尖锐的方向实现线性收敛速率。
We examine the last-iterate convergence rate of Bregman proximal methods - from mirror descent to mirror-prox and its optimistic variants - as a function of the local geometry induced by the prox-mapping defining the method. For generality, we focus on local solutions of constrained, non-monotone variational inequalities, and we show that the convergence rate of a given method depends sharply on its associated Legendre exponent, a notion that measures the growth rate of the underlying Bregman function (Euclidean, entropic, or other) near a solution. In particular, we show that boundary solutions exhibit a stark separation of regimes between methods with a zero and non-zero Legendre exponent: the former converge at a linear rate, while the latter converge, in general, sublinearly. This dichotomy becomes even more pronounced in linearly constrained problems where methods with entropic regularization achieve a linear convergence rate along sharp directions, compared to convergence in a finite number of steps under Euclidean regularization.
