  This paper reveals that a common and central role, played in many error bound
(EB) conditions and a variety of gradient-type methods, is a residual measure
operator. On one hand, by linking this operator with other optimality measures,
we define a group of abstract EB conditions, and then analyze the interplay
between them; on the other hand, by using this operator as an ascent direction,
we propose an abstract gradient-type method, and then derive EB conditions that
are necessary and sufficient for its linear convergence. The former provides a
unified framework that not only allows us to find new connections between many
existing EB conditions, but also paves a way to construct new EB conditions.
The latter allows us to claim the weakest conditions guaranteeing linear
convergence for a number of fundamental algorithms, including the gradient
method, the proximal point algorithm, and the forward-backward splitting
algorithm. In addition, we show linear convergence for the proximal alternating
linearized minimization algorithm under a group of equivalent EB conditions,
which are strictly weaker than the traditional strongly convex condition.
Moreover, by defining a new EB condition, we show Q-linear convergence of
Nesterov's accelerated forward-backward algorithm without strong convexity.
Finally, we verify EB conditions for a class of dual objective functions.
