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Off-policy evaluation often refers to two related tasks: estimating theexpected return of a policy and estimating its value function (or otherfunctions of interest, such as density ratios). While recent works onmarginalized importance sampling (MIS) show that the former can enjoy provableguarantees under realizable function approximation, the latter is only known tobe feasible under much stronger assumptions such as prohibitively expressivediscriminators. In this work, we provide guarantees for off-policy functionestimation under only realizability, by imposing proper regularization on theMIS objectives. Compared to commonly used regularization in MIS, ourregularizer is much more flexible and can account for an arbitraryuser-specified distribution, under which the learned function will be close tothe groundtruth. We provide exact characterization of the optimal dual solutionthat needs to be realized by the discriminator class, which determines thedata-coverage assumption in the case of value-function learning. As anothersurprising observation, the regularizer can be altered to relax thedata-coverage requirement, and completely eliminate it in the ideal case withstrong side information.

Beyond the Return: Off-policy Function Estimation under User-specified Error-measuring Distributions