Effective Rates for Iterations Involving Bregman Strongly Nonexpansive Operators

Abstract

We develop the theory of Bregman strongly nonexpansive maps for uniformly Fréchet differentiable Bregman functions from a quantitative perspective. In that vein, we provide moduli witnessing quantitative versions of the central assumptions commonly used in this field on the underlying Bregman function and the Bregman strongly nonexpansive maps. In terms of these moduli, we then compute explicit and effective rates for the asymptotic regularity of Picard iterations of Bregman strongly nonexpansive maps and of the method of cyclic Bregman projections. Further, we also provide similar rates for the asymptotic regularity and metastability of a strongly convergent Halpern-type iteration of a family of such mappings and we use these new results to derive rates for various special instantiations like a Halpern-type proximal point algorithm for monotone operators in Banach spaces as well as Halpern-Mann- and Tikhonov-Mann-type methods.