  We consider optimization algorithms that successively minimize simple
Taylor-like models of the objective function. Methods of Gauss-Newton type for
minimizing the composition of a convex function and a smooth map are common
examples. Our main result is an explicit relationship between the step-size of
any such algorithm and the slope of the function at a nearby point.
Consequently, we (1) show that the step-sizes can be reliably used to terminate
the algorithm, (2) prove that as long as the step-sizes tend to zero, every
limit point of the iterates is stationary, and (3) show that conditions, akin
to classical quadratic growth, imply that the step-sizes linearly bound the
distance of the iterates to the solution set. The latter so-called error bound
property is typically used to establish linear (or faster) convergence
guarantees. Analogous results hold when the step-size is replaced by the square
root of the decrease in the model's value. We complete the paper with
extensions to when the models are minimized only inexactly.
