16 research outputs found
Learning Model-Based Sparsity via Projected Gradient Descent
Several convex formulation methods have been proposed previously for
statistical estimation with structured sparsity as the prior. These methods
often require a carefully tuned regularization parameter, often a cumbersome or
heuristic exercise. Furthermore, the estimate that these methods produce might
not belong to the desired sparsity model, albeit accurately approximating the
true parameter. Therefore, greedy-type algorithms could often be more desirable
in estimating structured-sparse parameters. So far, these greedy methods have
mostly focused on linear statistical models. In this paper we study the
projected gradient descent with non-convex structured-sparse parameter model as
the constraint set. Should the cost function have a Stable Model-Restricted
Hessian the algorithm produces an approximation for the desired minimizer. As
an example we elaborate on application of the main results to estimation in
Generalized Linear Model
Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization
Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure
for finding sparse solutions of underdetermined linear systems. This method has
been shown to have strong theoretical guarantee and impressive numerical
performance. In this paper, we generalize HTP from compressive sensing to a
generic problem setup of sparsity-constrained convex optimization. The proposed
algorithm iterates between a standard gradient descent step and a hard
thresholding step with or without debiasing. We prove that our method enjoys
the strong guarantees analogous to HTP in terms of rate of convergence and
parameter estimation accuracy. Numerical evidences show that our method is
superior to the state-of-the-art greedy selection methods in sparse logistic
regression and sparse precision matrix estimation tasks
Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators on Domains Given by Linear Minimization Oracles
The majority of First Order methods for large-scale convex-concave saddle
point problems and variational inequalities with monotone operators are
proximal algorithms which at every iteration need to minimize over problem's
domain X the sum of a linear form and a strongly convex function. To make such
an algorithm practical, X should be proximal-friendly -- admit a strongly
convex function with easy to minimize linear perturbations. As a byproduct, X
admits a computationally cheap Linear Minimization Oracle (LMO) capable to
minimize over X linear forms. There are, however, important situations where a
cheap LMO indeed is available, but X is not proximal-friendly, which motivates
search for algorithms based solely on LMO's. For smooth convex minimization,
there exists a classical LMO-based algorithm -- Conditional Gradient. In
contrast, known to us LMO-based techniques for other problems with convex
structure (nonsmooth convex minimization, convex-concave saddle point problems,
even as simple as bilinear ones, and variational inequalities with monotone
operators, even as simple as affine) are quite recent and utilize common
approach based on Fenchel-type representations of the associated
objectives/vector fields. The goal of this paper is to develop an alternative
(and seemingly much simpler) LMO-based decomposition techniques for bilinear
saddle point problems and for variational inequalities with affine monotone
operators