An Adaptive Accelerated Proximal Gradient Method and its Homotopy Continuation for Sparse Optimization

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Qihang Lin, Lin Xiao ;
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(1):73-81, 2014.

Abstract

We first propose an adaptive accelerated proximal gradient(APG) method for minimizing strongly convex composite functions with unknown convexity parameters. This method incorporates a restarting scheme to automatically estimate the strong convexity parameter and achieves a nearly optimal iteration complexity. Then we consider the ℓ1-regularized least-squares (ℓ1-LS) problem in the high-dimensional setting. Although such an objective function is not strongly convex, it has restricted strong convexity over sparse vectors. We exploit this property by combining the adaptive APG method with a homotopy continuation scheme, which generates a sparse solution path towards optimality. This method obtains a global linear rate of convergence and its overall iteration complexity has a weaker dependency on the restricted condition number than previous work.

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