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.

Probabilistic bisection with spatial metamodels

“…In analogy to Waeber (2013); Rodriguez and Ludkovski (2017), we utilize the following three test functions h i (x) defined for x ∈ (0, 1):…”

“…Table 1 shows the results for the linear test function (31). To allow a direct comparison to the non-spatial G-PBA, the last few rows present the performance of the best local G-PBA schemes as identified in Rodriguez and Ludkovski (2017):…”

“…However, in the more general and practical case, including the SRFP in (2), p(x) is unknown and location-dependent and hence must be itself estimated. The Generalized PBA (G-PBA) that we developed in Rodriguez and Ludkovski (2017) extends the classical PBA by using the observed data to construct a point estimate,p(x), for p(x), as well as to learn the root location X * in parallel. The proposed estimatorsp(x n+1 ) under the aforementioned G-PBA paradigm were constructed locally at x n+1 (i.e., without using information from previous locations x 1:n ).…”

“…Since the true Bayesian posterior g n is not attainable due to unknown p(·), we notationally distinguish between the approximate knowledge state f n and the true g n (3). For assimilating information, we mimic the exact Bayesian updating from Waeber et al (2011) and use the batched knowledge state transition introduced in Rodriguez and Ludkovski (2017). Thus we take…”

“…Approach (i) utilizes fixed replication amount a ≥ 1 and selects the new x n+1 using an information-theoretic criterion. In analogy to the Information Directed Sampling (IDS) policy used in the G-PBA context (Rodriguez and Ludkovski, 2017), we consider a criterion based on the batched expected Kullback-Leibler (KL) divergence E[D(f n+1 ; f n )] between f n and the updated knowledge state f n+1 = Ψ(f n , x, B n+1 ;p n+1 , a),…”