Generalized Probabilistic Bisection for Stochastic Root-Finding

TL;DR

This paper extends the Probabilistic Bisection Algorithm to handle unknown and location-dependent sampling distributions, incorporating Bayesian inference and various sampling strategies to improve stochastic root-finding efficiency, with applications in financial derivative pricing.

Contribution

It introduces a generalized PBA framework that learns oracle accuracy and employs diverse sampling methods, including randomized quantile sampling, for more effective stochastic root-finding.

Findings

Randomized quantile sampling outperforms other strategies.

The method effectively estimates root locations in noisy environments.

Application demonstrated in Bermudan option pricing.

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.