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We examine the last-iterate convergence rate of Bregman proximal methods -from mirror descent to mirror-prox and its optimistic variants - as a functionof the local geometry induced by the prox-mapping defining the method. Forgenerality, we focus on local solutions of constrained, non-monotonevariational inequalities, and we show that the convergence rate of a givenmethod depends sharply on its associated Legendre exponent, a notion thatmeasures the growth rate of the underlying Bregman function (Euclidean,entropic, or other) near a solution. In particular, we show that boundarysolutions exhibit a stark separation of regimes between methods with a zero andnon-zero Legendre exponent: the former converge at a linear rate, while thelatter converge, in general, sublinearly. This dichotomy becomes even morepronounced in linearly constrained problems where methods with entropicregularization achieve a linear convergence rate along sharp directions,compared to convergence in a finite number of steps under Euclideanregularization.

The rate of convergence of Bregman proximal methods: Local geometry vs. regularity vs. sharpness