A Quasi-Newton Approach to Nonsmooth Convex Optimization Problems in Machine Learning
Journal of Machine Learning Research(2010)
摘要
We extend the well-known BFGS quasi-Newton method and its memory-limited variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: the local quadratic model, the identification of a descent direction, and the Wolfe line search conditions. We prove that under some technical conditions, the resulting subBFGS algorithm is globally convergent in objective function value. We apply its memory-limited variant (subLBFGS) to L2-regularized risk minimization with the binary hinge loss. To extend our algorithm to the multiclass and multilabel settings, we develop a new, efficient, exact line search algorithm. We prove its worst-case time complexity bounds, and show that our line search can also be used to extend a recently developed bundle method to the multiclass and multilabel settings. We also apply the direction-finding component of our algorithm to L1-regularized risk minimization with logistic loss. In all these contexts our methods perform comparable to or better than specialized state-of-the-art solvers on a number of publicly available data sets. An open source implementation of our algorithms is freely available.
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关键词
bundle method,multilabel setting,nonsmooth convex optimization problems,machine learning,available data set,l1-regularized risk minimization,subbfgs algorithm,logistic loss,quasi-newton approach,l2-regularized risk minimization,exact line search algorithm,wolfe line search condition,line search,quasi newton method,time complexity,convex optimization,objective function
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