6,650 research outputs found
A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem
We consider solving the -regularized least-squares (-LS)
problem in the context of sparse recovery, for applications such as compressed
sensing. The standard proximal gradient method, also known as iterative
soft-thresholding when applied to this problem, has low computational cost per
iteration but a rather slow convergence rate. Nevertheless, when the solution
is sparse, it often exhibits fast linear convergence in the final stage. We
exploit the local linear convergence using a homotopy continuation strategy,
i.e., we solve the -LS problem for a sequence of decreasing values of
the regularization parameter, and use an approximate solution at the end of
each stage to warm start the next stage. Although similar strategies have been
studied in the literature, there have been no theoretical analysis of their
global iteration complexity. This paper shows that under suitable assumptions
for sparse recovery, the proposed homotopy strategy ensures that all iterates
along the homotopy solution path are sparse. Therefore the objective function
is effectively strongly convex along the solution path, and geometric
convergence at each stage can be established. As a result, the overall
iteration complexity of our method is for finding an
-optimal solution, which can be interpreted as global geometric rate
of convergence. We also present empirical results to support our theoretical
analysis
A Proximal Stochastic Gradient Method with Progressive Variance Reduction
We consider the problem of minimizing the sum of two convex functions: one is
the average of a large number of smooth component functions, and the other is a
general convex function that admits a simple proximal mapping. We assume the
whole objective function is strongly convex. Such problems often arise in
machine learning, known as regularized empirical risk minimization. We propose
and analyze a new proximal stochastic gradient method, which uses a multi-stage
scheme to progressively reduce the variance of the stochastic gradient. While
each iteration of this algorithm has similar cost as the classical stochastic
gradient method (or incremental gradient method), we show that the expected
objective value converges to the optimum at a geometric rate. The overall
complexity of this method is much lower than both the proximal full gradient
method and the standard proximal stochastic gradient method
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
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