John Duchi, Yoram Singer.
Year: 2009, Volume: 10, Issue: 99, Pages: 2899−2934
We describe, analyze, and experiment with a framework for empirical loss minimization with regularization. Our algorithmic framework alternates between two phases. On each iteration we first perform an unconstrained gradient descent step. We then cast and solve an instantaneous optimization problem that trades off minimization of a regularization term while keeping close proximity to the result of the first phase. This view yields a simple yet effective algorithm that can be used for batch penalized risk minimization and online learning. Furthermore, the two phase approach enables sparse solutions when used in conjunction with regularization functions that promote sparsity, such as l1. We derive concrete and very simple algorithms for minimization of loss functions with l1, l2, l22, and l∞ regularization. We also show how to construct efficient algorithms for mixed-norm l1/lq regularization. We further extend the algorithms and give efficient implementations for very high-dimensional data with sparsity. We demonstrate the potential of the proposed framework in a series of experiments with synthetic and natural data sets.