1,849 research outputs found
Bregman Proximal Gradient Algorithm with Extrapolation for a class of Nonconvex Nonsmooth Minimization Problems
In this paper, we consider an accelerated method for solving nonconvex and
nonsmooth minimization problems. We propose a Bregman Proximal Gradient
algorithm with extrapolation(BPGe). This algorithm extends and accelerates the
Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive
global Lipschitz gradient continuity assumption needed in Proximal Gradient
algorithms (PG). The BPGe algorithm has higher generality than the recently
introduced Proximal Gradient algorithm with extrapolation(PGe), and besides,
due to the extrapolation step, BPGe converges faster than BPG algorithm.
Analyzing the convergence, we prove that any limit point of the sequence
generated by BPGe is a stationary point of the problem by choosing parameters
properly. Besides, assuming Kurdyka-{\'L}ojasiewicz property, we prove the
whole sequences generated by BPGe converges to a stationary point. Finally, to
illustrate the potential of the new method BPGe, we apply it to two important
practical problems that arise in many fundamental applications (and that not
satisfy global Lipschitz gradient continuity assumption): Poisson linear
inverse problems and quadratic inverse problems. In the tests the accelerated
BPGe algorithm shows faster convergence results, giving an interesting new
algorithm.Comment: Preprint submitted for publication, February 14, 201
Catalyst Acceleration for Gradient-Based Non-Convex Optimization
We introduce a generic scheme to solve nonconvex optimization problems using
gradient-based algorithms originally designed for minimizing convex functions.
Even though these methods may originally require convexity to operate, the
proposed approach allows one to use them on weakly convex objectives, which
covers a large class of non-convex functions typically appearing in machine
learning and signal processing. In general, the scheme is guaranteed to produce
a stationary point with a worst-case efficiency typical of first-order methods,
and when the objective turns out to be convex, it automatically accelerates in
the sense of Nesterov and achieves near-optimal convergence rate in function
values. These properties are achieved without assuming any knowledge about the
convexity of the objective, by automatically adapting to the unknown weak
convexity constant. We conclude the paper by showing promising experimental
results obtained by applying our approach to incremental algorithms such as
SVRG and SAGA for sparse matrix factorization and for learning neural networks
Efficient Inexact Proximal Gradient Algorithm for Nonconvex Problems
The proximal gradient algorithm has been popularly used for convex
optimization. Recently, it has also been extended for nonconvex problems, and
the current state-of-the-art is the nonmonotone accelerated proximal gradient
algorithm. However, it typically requires two exact proximal steps in each
iteration, and can be inefficient when the proximal step is expensive. In this
paper, we propose an efficient proximal gradient algorithm that requires only
one inexact (and thus less expensive) proximal step in each iteration.
Convergence to a critical point %of the nonconvex problem is still guaranteed
and has a convergence rate, which is the best rate for nonconvex
problems with first-order methods. Experiments on a number of problems
demonstrate that the proposed algorithm has comparable performance as the
state-of-the-art, but is much faster
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