Efficient learning of pseudo-Boolean functions from limited training data

Pseudo-Boolean functions are generalizations of Boolean functions. We present a new method for learning pseudo-Boolean functions from limited training data. The objective of learning is to obtain a function f which is a good approximation of the target function f*. We define suitable criteria for the "goodness" of an approximating function. One criterion is to choose a function f that minimizes the "expected distance" with respect to a distance function d (over pairs of pseudo-Boolean functions) and the uniform distribution over all feasible pseudo-Boolean functions. We define two alternative "distance measures" over pairs of pseudo-Boolean functions, and show that they are are actually equivalent with respect to the criterion of minimal expected distance. We outline efficient algorithms for learning pseudo-Boolean functions according to these criteria. Other reasonable distance measures and "goodness" criteria are also discussed.