Title: Generalized Probabilistic Bisection for Stochastic Root-Finding,
Abstract: We consider numerical schemes for root finding of noisy responses through
generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical
context where the sampling distribution is unknown and location-dependent. As
in standard PBA, we rely on a knowledge state for the approximate posterior of
the root location. To implement the corresponding Bayesian updating, we also
carry out inference of oracle accuracy, namely learning the probability of
correct response. To this end we utilize batched querying in combination with a
variety of frequentist and Bayesian estimators based on majority vote, as well
as the underlying functional responses, if available. For guiding sampling
selection we investigate both Information Directed sampling, as well as
Quantile sampling. Our numerical experiments show that these strategies perform
quite differently; in particular we demonstrate the efficiency of randomized
quantile sampling which is reminiscent of Thompson sampling. Our work is
motivated by the root-finding sub-routine in pricing of Bermudan financial
derivatives, illustrated in the last section of the paper.