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）的末点迭代收敛速率，该速率由定义方法的近端映射所诱导的局部几何特性决定。为保持一般性，我们聚焦于带约束的非单调变分不等式的局部解，并证明给定方法的收敛速率严格取决于其关联的Legendre指数——该概念用于度量底层Bregman函数（欧几里得型、熵型或其他类型）在解附近的增长率。特别地，我们证明边界解在Legendre指数为零和非零的方法间呈现明显的机制分离：前者以线性速率收敛，而后者通常以次线性速率收敛。这种二分现象在线性约束问题中更为显著：相较于欧几里得正则化下的有限步收敛，采用熵正则化的方法在锐度方向上可实现线性收敛速率。