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A Quasi-Newton Approach to Nonsmooth Convex Optimization

A Quasi-Newton Approach to Nonsmooth Convex Optimization

This video was recorded at 25th International Conference on Machine Learning (ICML), Helsinki 2008. We extend the well-known BFGS quasi-Newton method and its limited-memory 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 apply the resulting sub(L)BFGS algorithm to L2-regularized risk minimization with binary hinge loss, and its direction-finding component to L1-regularized risk minimization with logistic loss. In both settings our generic algorithms perform comparable to or better than their counterparts in specialized state-of-the-art solvers.

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