  In this paper we introduce the concept of modulus of regularity as a tool to
analyze the speed of convergence, including the finite termination, for classes
of Fej\'er monotone sequences which appear in fixed point theory, monotone
operator theory, and convex optimization. This concept allows for a unified
approach to several notions such as weak sharp minima, error bounds, metric
subregularity, H\"older regularity, etc., as well as to obtain rates of
convergence for Picard iterates, the Mann algorithm, the proximal point
algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative
version of the well-known fact that for a convex lower semi-continuous function
the set of minimizers coincides with the set of zeros of its subdifferential
and the set of fixed points of its resolvent.
