67,789 research outputs found
Exact block-wise optimization in group lasso and sparse group lasso for linear regression
The group lasso is a penalized regression method, used in regression problems
where the covariates are partitioned into groups to promote sparsity at the
group level. Existing methods for finding the group lasso estimator either use
gradient projection methods to update the entire coefficient vector
simultaneously at each step, or update one group of coefficients at a time
using an inexact line search to approximate the optimal value for the group of
coefficients when all other groups' coefficients are fixed. We present a new
method of computation for the group lasso in the linear regression case, the
Single Line Search (SLS) algorithm, which operates by computing the exact
optimal value for each group (when all other coefficients are fixed) with one
univariate line search. We perform simulations demonstrating that the SLS
algorithm is often more efficient than existing computational methods. We also
extend the SLS algorithm to the sparse group lasso problem via the Signed
Single Line Search (SSLS) algorithm, and give theoretical results to support
both algorithms.Comment: We have been made aware of the earlier work by Puig et al. (2009)
which derives the same result for the (non-sparse) group lasso setting. We
leave this manuscript available as a technical report, to serve as a
reference for the previously untreated sparse group lasso case, and for
timing comparisons of various methods in the group lasso setting. The
manuscript is updated to include this referenc
A Unified Successive Pseudo-Convex Approximation Framework
In this paper, we propose a successive pseudo-convex approximation algorithm
to efficiently compute stationary points for a large class of possibly
nonconvex optimization problems. The stationary points are obtained by solving
a sequence of successively refined approximate problems, each of which is much
easier to solve than the original problem. To achieve convergence, the
approximate problem only needs to exhibit a weak form of convexity, namely,
pseudo-convexity. We show that the proposed framework not only includes as
special cases a number of existing methods, for example, the gradient method
and the Jacobi algorithm, but also leads to new algorithms which enjoy easier
implementation and faster convergence speed. We also propose a novel line
search method for nondifferentiable optimization problems, which is carried out
over a properly constructed differentiable function with the benefit of a
simplified implementation as compared to state-of-the-art line search
techniques that directly operate on the original nondifferentiable objective
function. The advantages of the proposed algorithm are shown, both
theoretically and numerically, by several example applications, namely, MIMO
broadcast channel capacity computation, energy efficiency maximization in
massive MIMO systems and LASSO in sparse signal recovery.Comment: submitted to IEEE Transactions on Signal Processing; original title:
A Novel Iterative Convex Approximation Metho
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
This paper develops a general framework for solving a variety of convex cone
problems that frequently arise in signal processing, machine learning,
statistics, and other fields. The approach works as follows: first, determine a
conic formulation of the problem; second, determine its dual; third, apply
smoothing; and fourth, solve using an optimal first-order method. A merit of
this approach is its flexibility: for example, all compressed sensing problems
can be solved via this approach. These include models with objective
functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or
a combination thereof. In addition, the paper also introduces a number of
technical contributions such as a novel continuation scheme, a novel approach
for controlling the step size, and some new results showing that the smooth and
unsmoothed problems are sometimes formally equivalent. Combined with our
framework, these lead to novel, stable and computationally efficient
algorithms. For instance, our general implementation is competitive with
state-of-the-art methods for solving intensively studied problems such as the
LASSO. Further, numerical experiments show that one can solve the Dantzig
selector problem, for which no efficient large-scale solvers exist, in a few
hundred iterations. Finally, the paper is accompanied with a software release.
This software is not a single, monolithic solver; rather, it is a suite of
programs and routines designed to serve as building blocks for constructing
complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This
version has updated reference
A framework for deflated and augmented Krylov subspace methods
We consider deflation and augmentation techniques for accelerating the
convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. Despite some formal similarity, the two techniques are
conceptually different from preconditioning. Deflation (in the sense the term
is used here) "removes" certain parts from the operator making it singular,
while augmentation adds a subspace to the Krylov subspace (often the one that
is generated by the singular operator); in contrast, preconditioning changes
the spectrum of the operator without making it singular. Deflation and
augmentation have been used in a variety of methods and settings. Typically,
deflation is combined with augmentation to compensate for the singularity of
the operator, but both techniques can be applied separately.
We introduce a framework of Krylov subspace methods that satisfy a Galerkin
condition. It includes the families of orthogonal residual (OR) and minimal
residual (MR) methods. We show that in this framework augmentation can be
achieved either explicitly or, equivalently, implicitly by projecting the
residuals appropriately and correcting the approximate solutions in a final
step. We study conditions for a breakdown of the deflated methods, and we show
several possibilities to avoid such breakdowns for the deflated MINRES method.
Numerical experiments illustrate properties of different variants of deflated
MINRES analyzed in this paper.Comment: 24 pages, 3 figure
- …