36 research outputs found
Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization
The Schatten-p quasi-norm is usually used to replace the standard
nuclear norm in order to approximate the rank function more accurately.
However, existing Schatten-p quasi-norm minimization algorithms involve
singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each
iteration, and thus may become very slow and impractical for large-scale
problems. In this paper, we first define two tractable Schatten quasi-norms,
i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove
that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively,
which lead to the design of very efficient algorithms that only need to update
two much smaller factor matrices. We also design two efficient proximal
alternating linearized minimization algorithms for solving representative
matrix completion problems. Finally, we provide the global convergence and
performance guarantees for our algorithms, which have better convergence
properties than existing algorithms. Experimental results on synthetic and
real-world data show that our algorithms are more accurate than the
state-of-the-art methods, and are orders of magnitude faster.Comment: 16 pages, 5 figures, Appears in Proceedings of the 30th AAAI
Conference on Artificial Intelligence (AAAI), Phoenix, Arizona, USA, pp.
2016--2022, 201
Accelerated Variance Reduced Stochastic ADMM
Recently, many variance reduced stochastic alternating direction method of
multipliers (ADMM) methods (e.g.\ SAG-ADMM, SDCA-ADMM and SVRG-ADMM) have made
exciting progress such as linear convergence rates for strongly convex
problems. However, the best known convergence rate for general convex problems
is O(1/T) as opposed to O(1/T^2) of accelerated batch algorithms, where is
the number of iterations. Thus, there still remains a gap in convergence rates
between existing stochastic ADMM and batch algorithms. To bridge this gap, we
introduce the momentum acceleration trick for batch optimization into the
stochastic variance reduced gradient based ADMM (SVRG-ADMM), which leads to an
accelerated (ASVRG-ADMM) method. Then we design two different momentum term
update rules for strongly convex and general convex cases. We prove that
ASVRG-ADMM converges linearly for strongly convex problems. Besides having a
low per-iteration complexity as existing stochastic ADMM methods, ASVRG-ADMM
improves the convergence rate on general convex problems from O(1/T) to
O(1/T^2). Our experimental results show the effectiveness of ASVRG-ADMM.Comment: 16 pages, 5 figures, Appears in Proceedings of the 31th AAAI
Conference on Artificial Intelligence (AAAI), San Francisco, California, USA,
pp. 2287--2293, 201