  Error bounds, which refer to inequalities that bound the distance of vectors
in a test set to a given set by a residual function, have proven to be
extremely useful in analyzing the convergence rates of a host of iterative
methods for solving optimization problems. In this paper, we present a new
framework for establishing error bounds for a class of structured convex
optimization problems, in which the objective function is the sum of a smooth
convex function and a general closed proper convex function. Such a class
encapsulates not only fairly general constrained minimization problems but also
various regularized loss minimization formulations in machine learning, signal
processing, and statistics. Using our framework, we show that a number of
existing error bound results can be recovered in a unified and transparent
manner. To further demonstrate the power of our framework, we apply it to a
class of nuclear-norm regularized loss minimization problems and establish a
new error bound for this class under a strict complementarity-type regularity
condition. We then complement this result by constructing an example to show
that the said error bound could fail to hold without the regularity condition.
Consequently, we obtain a rather complete answer to a question raised by Tseng.
We believe that our approach will find further applications in the study of
error bounds for structured convex optimization problems.
