  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.
