27 research outputs found
From error bounds to the complexity of first-order descent methods for convex functions
This paper shows that error bounds can be used as effective tools for
deriving complexity results for first-order descent methods in convex
minimization. In a first stage, this objective led us to revisit the interplay
between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can
show the equivalence between the two concepts for convex functions having a
moderately flat profile near the set of minimizers (as those of functions with
H\"olderian growth). A counterexample shows that the equivalence is no longer
true for extremely flat functions. This fact reveals the relevance of an
approach based on KL inequality. In a second stage, we show how KL inequalities
can in turn be employed to compute new complexity bounds for a wealth of
descent methods for convex problems. Our approach is completely original and
makes use of a one-dimensional worst-case proximal sequence in the spirit of
the famous majorant method of Kantorovich. Our result applies to a very simple
abstract scheme that covers a wide class of descent methods. As a byproduct of
our study, we also provide new results for the globalization of KL inequalities
in the convex framework.
Our main results inaugurate a simple methodology: derive an error bound,
compute the desingularizing function whenever possible, identify essential
constants in the descent method and finally compute the complexity using the
one-dimensional worst case proximal sequence. Our method is illustrated through
projection methods for feasibility problems, and through the famous iterative
shrinkage thresholding algorithm (ISTA), for which we show that the complexity
bound is of the form where the constituents of the bound only depend
on error bound constants obtained for an arbitrary least squares objective with
regularization