  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.
