54 research outputs found
Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis
Recently several methods were proposed for sparse optimization which make
careful use of second-order information [10, 28, 16, 3] to improve local
convergence rates. These methods construct a composite quadratic approximation
using Hessian information, optimize this approximation using a first-order
method, such as coordinate descent and employ a line search to ensure
sufficient descent. Here we propose a general framework, which includes
slightly modified versions of existing algorithms and also a new algorithm,
which uses limited memory BFGS Hessian approximations, and provide a novel
global convergence rate analysis, which covers methods that solve subproblems
via coordinate descent
Global convergence rate analysis of unconstrained optimization methods based on probabilistic models
We present global convergence rates for a line-search method which is based
on random first-order models and directions whose quality is ensured only with
certain probability. We show that in terms of the order of the accuracy, the
evaluation complexity of such a method is the same as its counterparts that use
deterministic accurate models; the use of probabilistic models only increases
the complexity by a constant, which depends on the probability of the models
being good. We particularize and improve these results in the convex and
strongly convex case.
We also analyze a probabilistic cubic regularization variant that allows
approximate probabilistic second-order models and show improved complexity
bounds compared to probabilistic first-order methods; again, as a function of
the accuracy, the probabilistic cubic regularization bounds are of the same
(optimal) order as for the deterministic case
Sparse Inverse Covariance Selection via Alternating Linearization Methods
Gaussian graphical models are of great interest in statistical learning.
Because the conditional independencies between different nodes correspond to
zero entries in the inverse covariance matrix of the Gaussian distribution, one
can learn the structure of the graph by estimating a sparse inverse covariance
matrix from sample data, by solving a convex maximum likelihood problem with an
-regularization term. In this paper, we propose a first-order method
based on an alternating linearization technique that exploits the problem's
special structure; in particular, the subproblems solved in each iteration have
closed-form solutions. Moreover, our algorithm obtains an -optimal
solution in iterations. Numerical experiments on both synthetic
and real data from gene association networks show that a practical version of
this algorithm outperforms other competitive algorithms
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