Generalized Probabilistic Bisection for Stochastic Root-Finding
This work addresses a practical challenge in stochastic root-finding, particularly for pricing Bermudan financial derivatives, but appears incremental as it builds on existing methods with new adaptations.
The authors tackled the problem of root finding with noisy responses by generalizing the Probabilistic Bisection Algorithm to handle unknown, location-dependent sampling distributions, achieving efficiency through strategies like randomized quantile sampling.
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.