Projective Proximal Gradient Descent for A Class of Nonconvex Nonsmooth
Optimization Problems: Fast Convergence Without Kurdyka-Lojasiewicz (KL)
Property
- URL: http://arxiv.org/abs/2304.10499v1
- Date: Thu, 20 Apr 2023 17:39:24 GMT
- Title: Projective Proximal Gradient Descent for A Class of Nonconvex Nonsmooth
Optimization Problems: Fast Convergence Without Kurdyka-Lojasiewicz (KL)
Property
- Authors: Yingzhen Yang, Ping Li
- Abstract summary: Non and nonsmooth optimization problems are important and challenging for learning.
In this paper, we show a new analysis showing fast convergence of PPGD.
- Score: 22.232788341658924
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Nonconvex and nonsmooth optimization problems are important and challenging
for statistics and machine learning. In this paper, we propose Projected
Proximal Gradient Descent (PPGD) which solves a class of nonconvex and
nonsmooth optimization problems, where the nonconvexity and nonsmoothness come
from a nonsmooth regularization term which is nonconvex but piecewise convex.
In contrast with existing convergence analysis of accelerated PGD methods for
nonconvex and nonsmooth problems based on the Kurdyka-\L{}ojasiewicz (K\L{})
property, we provide a new theoretical analysis showing local fast convergence
of PPGD. It is proved that PPGD achieves a fast convergence rate of
$\cO(1/k^2)$ when the iteration number $k \ge k_0$ for a finite $k_0$ on a
class of nonconvex and nonsmooth problems under mild assumptions, which is
locally Nesterov's optimal convergence rate of first-order methods on smooth
and convex objective function with Lipschitz continuous gradient. Experimental
results demonstrate the effectiveness of PPGD.
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