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.

翻译：我们研究了Bregman近端方法——从镜像下降法到镜像近端法及其乐观变体——的末次迭代收敛速度，并将其作为定义该方法近端映射所诱导的局部几何的函数进行分析。为兼顾一般性，我们聚焦于约束非单调变分不等式的局部解，并表明给定方法的收敛速度高度依赖于其相关勒让德指数——这一概念衡量的是底层Bregman函数（欧几里得型、熵型或其他）在解附近的增长速率。特别地，我们指出边界解在具有零与非零勒让德指数的方法之间呈现出截然不同的区间分离：前者以线性速率收敛，而后者通常以次线性速率收敛。在线性约束问题中，这种二分法更为显著：与欧几里得正则化下有限步收敛相比，采用熵正则化的方法能沿尖锐方向实现线性收敛速率。