19 research outputs found
Generalized Dantzig Selector: Application to the k-support norm
We propose a Generalized Dantzig Selector (GDS) for linear models, in which
any norm encoding the parameter structure can be leveraged for estimation. We
investigate both computational and statistical aspects of the GDS. Based on
conjugate proximal operator, a flexible inexact ADMM framework is designed for
solving GDS, and non-asymptotic high-probability bounds are established on the
estimation error, which rely on Gaussian width of unit norm ball and suitable
set encompassing estimation error. Further, we consider a non-trivial example
of the GDS using -support norm. We derive an efficient method to compute the
proximal operator for -support norm since existing methods are inapplicable
in this setting. For statistical analysis, we provide upper bounds for the
Gaussian widths needed in the GDS analysis, yielding the first statistical
recovery guarantee for estimation with the -support norm. The experimental
results confirm our theoretical analysis.Comment: Updates to boun
Solving Multiple-Block Separable Convex Minimization Problems Using Two-Block Alternating Direction Method of Multipliers
In this paper, we consider solving multiple-block separable convex
minimization problems using alternating direction method of multipliers (ADMM).
Motivated by the fact that the existing convergence theory for ADMM is mostly
limited to the two-block case, we analyze in this paper, both theoretically and
numerically, a new strategy that first transforms a multi-block problem into an
equivalent two-block problem (either in the primal domain or in the dual
domain) and then solves it using the standard two-block ADMM. In particular, we
derive convergence results for this two-block ADMM approach to solve
multi-block separable convex minimization problems, including an improved
O(1/\epsilon) iteration complexity result. Moreover, we compare the numerical
efficiency of this approach with the standard multi-block ADMM on several
separable convex minimization problems which include basis pursuit, robust
principal component analysis and latent variable Gaussian graphical model
selection. The numerical results show that the multiple-block ADMM, although
lacks theoretical convergence guarantees, typically outperforms two-block
ADMMs
Iteration Complexity Analysis of Block Coordinate Descent Methods
In this paper, we provide a unified iteration complexity analysis for a
family of general block coordinate descent (BCD) methods, covering popular
methods such as the block coordinate gradient descent (BCGD) and the block
coordinate proximal gradient (BCPG), under various different coordinate update
rules. We unify these algorithms under the so-called Block Successive
Upper-bound Minimization (BSUM) framework, and show that for a broad class of
multi-block nonsmooth convex problems, all algorithms covered by the BSUM
framework achieve a global sublinear iteration complexity of , where r
is the iteration index. Moreover, for the case of block coordinate minimization
(BCM) where each block is minimized exactly, we establish the sublinear
convergence rate of without per block strong convexity assumption.
Further, we show that when there are only two blocks of variables, a special
BSUM algorithm with Gauss-Seidel rule can be accelerated to achieve an improved
rate of