ILAMM for Sparse Learning: Simultaneous Control of Algorithmic Complexity and Statistical Error
Abstract
We propose a computational framework named iterative local adaptive majorizeminimization (ILAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. ILAMM is a twostage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasilikelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family of nonconvex optimization problems. The iteration effects on statistical errors are clearly demonstrated via a contraction property. Our theory relies on a localized version of the sparse/restricted eigenvalue condition, which allows us to analyze a large family of loss and penalty functions and provide optimality guarantees under very weak assumptions (For example, ILAMM requires much weaker minimal signal strength than other procedures). Thorough numerical results are provided to support the obtained theory.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.01037
 Bibcode:
 2015arXiv150701037F
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 66 pages, 5 figures