  A major goal in Algorithmic Game Theory is to justify equilibrium concepts
from an algorithmic and complexity perspective. One appealing approach is to
identify natural distributed algorithms that converge quickly to an
equilibrium. This paper established new convergence results for two
generalizations of Proportional Response in Fisher markets with buyers having
CES utility functions. The starting points are respectively a new convex and a
new convex-concave formulation of such markets. The two generalizations
correspond to suitable mirror descent algorithms applied to these formulations.
Several of our new results are a consequence of new notions of strong Bregman
convexity and of strong Bregman convex-concave functions, and associated linear
rates of convergence, which may be of independent interest.
  Among other results, we analyze a damped generalized Proportional Response
and show a linear rate of convergence in a Fisher market with buyers whose
utility functions cover the full spectrum of CES utilities aside the extremes
of linear and Leontief utilities; when these utilities are included, we obtain
an empirical O(1/T) rate of convergence.
