Learning Bounds for Importance Weighting

This paper presents an analysis of importance weighting for learning from finite samples and gives a series of theoretical and algorithmic results. We point out simple cases where importance weighting can fail, which suggests the need for an analysis of the properties of this technique. We then give both upper and lower bounds for generalization with bounded importance weights and, more significantly, give learning guarantees for the more common case of unbounded importance weights under the weak assumption that the second moment is bounded, aconditionrelatedtotheRényidivergenceofthetrainingand test distributions. These results are based on a series of novel and general boundswederiveforunbounded loss functions, which are of independent interest. We use these bounds to guide the definition of an alternative reweighting algorithm andreporttheresults of experiments demonstrating its benefits. Finally, we analyze the properties of normalized importance weights which are also commonly used. 1