Generalized Probabilistic Bisection for Stochastic Root Finding

Generalized Probabilistic Bisection for Stochastic Root-Finding

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 ...

Adaptive Designs for Stochastic Root-Finding

The Robbins-Monro procedure (1951) for stochastic root-finding is a nonparametric approach. Wu (1985, 1986) has shown that the convergence of the sequential procedure can be greatly improved if we know the distribution of the response. Wu’s approach assumes a parametric model and therefore its convergence rate slows down when the assumed model is very different from the true model. This article...

Coupled bisection for root ordering

We consider the problem of solving multiple ‘‘coupled’’ root-finding problems at once, in that we can evaluate every function at the same point simultaneously. Using a dynamic programming formulation, we show that a sequential bisection algorithm is a close-to-optimalmethod for finding a rankingwith respect to the zeros of these functions. We show the ranking can be found in linear time, prove ...

Some Issues in Multivariate Stochastic Root Finding

The stochastic root finding problem (SRFP) involves finding points in a region where a function attains a prespecified target value, using only a consistent estimator of the function. Due to the properties that the SRFP contexts entail, the development of good solutions to SRFPs has proven difficult, at least in the multi-dimensional setting. This paper discusses certain key issues, insights an...

The Bisection Method: Which Root?