Classi er Selection using the Predicate Depth

Typically, one approaches a supervised machine learning problem by writing down an objective function and nding a hypothesis that minimizes it. This is equivalent to nding the Maximum A Posteriori (MAP) hypothesis for a Boltzmann distribution. However, MAP is not a robust statistic. We present an alternative approach by de ning a median of the distribution, which we show is both more robust, and has good generalization guarantees. We present algorithms to approximate this median. One contribution of this work is an e cient method for approximating the Tukey median. The Tukey median, which is often used for data visualization and outlier detection, is a special case of the family of medians we de ne: however, computing it exactly is exponentially slow in the dimension. Our algorithm approximates such medians in polynomial time while making weaker assumptions than those required by previous work.