261 research outputs found
A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima
We introduce Bella, a locally superlinearly convergent Bregman forward
backward splitting method for minimizing the sum of two nonconvex functions,
one of which satisfying a relative smoothness condition and the other one
possibly nonsmooth. A key tool of our methodology is the Bregman
forward-backward envelope (BFBE), an exact and continuous penalty function with
favorable first- and second-order properties, and enjoying a nonlinear error
bound when the objective function satisfies a Lojasiewicz-type property. The
proposed algorithm is of linesearch type over the BFBE along candidate update
directions, and converges subsequentially to stationary points, globally under
a KL condition, and owing to the given nonlinear error bound can attain
superlinear convergence rates even when the limit point is a nonisolated
minimum, provided the directions are suitably selected
Calculus of the exponent of Kurdyka-{\L}ojasiewicz inequality and its applications to linear convergence of first-order methods
In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an
important quantity for analyzing the convergence rate of first-order methods.
Specifically, we develop various calculus rules to deduce the KL exponent of
new (possibly nonconvex and nonsmooth) functions formed from functions with
known KL exponents. In addition, we show that the well-studied Luo-Tseng error
bound together with a mild assumption on the separation of stationary values
implies that the KL exponent is . The Luo-Tseng error bound is known
to hold for a large class of concrete structured optimization problems, and
thus we deduce the KL exponent of a large class of functions whose exponents
were previously unknown. Building upon this and the calculus rules, we are then
able to show that for many convex or nonconvex optimization models for
applications such as sparse recovery, their objective function's KL exponent is
. This includes the least squares problem with smoothly clipped
absolute deviation (SCAD) regularization or minimax concave penalty (MCP)
regularization and the logistic regression problem with
regularization. Since many existing local convergence rate analysis for
first-order methods in the nonconvex scenario relies on the KL exponent, our
results enable us to obtain explicit convergence rate for various first-order
methods when they are applied to a large variety of practical optimization
models. Finally, we further illustrate how our results can be applied to
establishing local linear convergence of the proximal gradient algorithm and
the inertial proximal algorithm with constant step-sizes for some specific
models that arise in sparse recovery.Comment: The paper is accepted for publication in Foundations of Computational
Mathematics: https://link.springer.com/article/10.1007/s10208-017-9366-8. In
this update, we fill in more details to the proof of Theorem 4.1 concerning
the nonemptiness of the projection onto the set of stationary point
A Multi-step Inertial Forward--Backward Splitting Method for Non-convex Optimization
In this paper, we propose a multi-step inertial Forward--Backward splitting
algorithm for minimizing the sum of two non-necessarily convex functions, one
of which is proper lower semi-continuous while the other is differentiable with
a Lipschitz continuous gradient. We first prove global convergence of the
scheme with the help of the Kurdyka-{\L}ojasiewicz property. Then, when the
non-smooth part is also partly smooth relative to a smooth submanifold, we
establish finite identification of the latter and provide sharp local linear
convergence analysis. The proposed method is illustrated on a few problems
arising from statistics and machine learning.Comment: This paper is in company with our recent work on
Forward--Backward-type splitting methods http://arxiv.org/abs/1503.0370
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