Boosting Algorithms for Maximizing the Soft Margin

We present a novel boosting algorithm, called SoftBoost, designed for sets of binary labeled examples that are not necessarily separable by convex combinations of base hypotheses. Our algorithm achieves robustness by capping the distributions on the examples. Our update of the distribution is motivated by minimizing a relative entropy subject to the capping constraints and constraints on the edges of the obtained base hypotheses. The capping constraints imply a soft margin in the dual optimization problem. Our algorithm produces a convex combination of hypotheses whose soft margin is within δ of its maximum. We employ relative entropy projection methods to prove an O(ln N/δ2) iteration bound for our algorithm, where N is number of examples. We compare our algorithm with other approaches including LPBoost, Brown-Boost, and SmoothBoost. We show that there exist cases where the number of iterations required by LPBoost grows linearly in N instead of the logarithmic growth for SoftBoost. In simulation studies we show that our algorithm converges about as fast as LPBoost, faster than BrownBoost, and much faster than SmoothBoost. In a benchmark comparison we illustrate the competitiveness of our approach.