  When I is an ideal of a standard graded algebra S with homogeneous maximal
ideal \mm, it is known by the work of several authors that the
Castelnuovo-Mumford regularity of I^m ultimately becomes a linear function dm +
e for m \gg 0.
  We give several constraints on the behavior of what may be termed the
\emph{regularity defect} (the sequence e_m = \reg I^m - dm). When I is
\mm-primary we give a family of bounds on the first differences of the e_m,
including an upper bound on the increasing part of the sequence; for example,
we show that the e_i cannot increase for i \geq \dim(S). When I is a monomial
ideal, we show that the e_i become constant for i \geq n(n-1)(d-1), where n =
\dim(S).
