  Modern statistical applications often involve minimizing an objective
function that may be nonsmooth and/or nonconvex. This paper focuses on a broad
Bregman-surrogate algorithm framework including the local linear approximation,
mirror descent, iterative thresholding, DC programming and many others as
particular instances. The recharacterization via generalized Bregman functions
enables us to construct suitable error measures and establish global
convergence rates for nonconvex and nonsmooth objectives in possibly high
dimensions. For sparse learning problems with a composite objective, under some
regularity conditions, the obtained estimators as the surrogate's fixed points,
though not necessarily local minimizers, enjoy provable statistical guarantees,
and the sequence of iterates can be shown to approach the statistical truth
within the desired accuracy geometrically fast. The paper also studies how to
design adaptive momentum based accelerations without assuming convexity or
smoothness by carefully controlling stepsize and relaxation parameters.
