Probabilistic Bisection with Spatial Metamodels

TL;DR

This paper introduces a novel spatial probabilistic bisection algorithm that constructs statistical surrogates for unknown, location-dependent oracle responses, improving root-finding in noisy, complex environments.

Contribution

It proposes a new G-PBA framework using spatial surrogates like Gaussian Processes and logistic regression, with adaptive sampling policies for better root estimation.

Findings

Spatial PBA outperforms previous models in synthetic tests.

The method effectively handles complex, stochastic root-finding problems.

Active learning strategies improve sampling efficiency.

Abstract

Probabilistic Bisection Algorithm performs root finding based on knowledge acquired from noisy oracle responses. We consider the generalized PBA setting (G-PBA) where the statistical distribution of the oracle is unknown and location-dependent, so that model inference and Bayesian knowledge updating must be performed simultaneously. To this end, we propose to leverage the spatial structure of a typical oracle by constructing a statistical surrogate for the underlying logistic regression step. We investigate several non-parametric surrogates, including Binomial Gaussian Processes (B-GP), Polynomial, Kernel, and Spline Logistic Regression. In parallel, we develop sampling policies that adaptively balance learning the oracle distribution and learning the root. One of our proposals mimics active learning with B-GPs and provides a novel look-ahead predictive variance formula. The resulting gains of our Spatial PBA algorithm relative to earlier G-PBA models are illustrated with synthetic examples and a challenging stochastic root finding problem from Bermudan option pricing.