Approximation Algorithms for Minimizing Empirical Error by Axis-Parallel Hyperplanes

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Many learning situations involve separation of labeled training instances by hyperplanes. Consistent separation is of theoretical interest, but the real goal is rather to minimize the number of errors using a bounded number of hyperplanes. Exact minimization of empirical error in a high-dimensional grid induced into the feature space by axis-parallel hyperplanes is NP-hard. We develop two approximation schemes with performance guarantees, a greedy set covering scheme for producing a consistently labeled grid, and integer programming rounding scheme for finding the minimum error grid with bounded number of hyperplanes.