Probabilistic bisection converges almost as quickly as stochastic approximation. Frazier, P. I., Henderson, S. G., & Waeber, R. *Mathematics of Operations Research*, 44(2):651–667, 2019. Arxiv abstract bibtex The probabilistic bisection algorithm (PBA) solves a class of stochastic root-finding problems in one dimension by successively updating a prior belief on the location of the root based on noisy responses to queries at chosen points. The responses indicate the direction of the root from the queried point, and are incorrect with a fixed probability. The fixed-probability assumption is problematic in applications, and so we extend the PBA to apply when this assumption is relaxed. The extension involves the use of a power-one test at each queried point. We explore the convergence behavior of the extended PBA, showing that it converges at a rate arbitrarily close to, but slower than, the canonical ``square root'' rate of stochastic approximation.

@article{frahenwae16,
abstract = {The probabilistic bisection algorithm (PBA) solves a class of
stochastic root-finding problems in one dimension by successively updating a
prior belief on the location of the root based on noisy responses to
queries at chosen points. The responses indicate
the direction of the root from the queried point, and are incorrect
with a fixed probability. The fixed-probability assumption is
problematic in applications, and so we extend the PBA to apply when
this assumption is relaxed. The extension involves the use of a
power-one test at each queried
point. We explore the convergence behavior of the extended PBA, showing that it
converges at a rate arbitrarily close to, but slower than, the canonical ``square root''
rate of stochastic approximation.},
author = {P. I. Frazier and S. G. Henderson and R. Waeber},
date-added = {2016-08-26 18:10:53 +0000},
date-modified = {2020-02-24 13:10:58 +1300},
journal = {Mathematics of Operations Research},
number = {2},
pages = {651--667},
title = {Probabilistic bisection converges almost as quickly as stochastic approximation},
url_arxiv = {http://arxiv.org/abs/1612.03964},
volume = {44},
year = {2019}}

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We explore the convergence behavior of the extended PBA, showing that it converges at a rate arbitrarily close to, but slower than, the canonical ``square root'' rate of stochastic approximation.","author":[{"firstnames":["P.","I."],"propositions":[],"lastnames":["Frazier"],"suffixes":[]},{"firstnames":["S.","G."],"propositions":[],"lastnames":["Henderson"],"suffixes":[]},{"firstnames":["R."],"propositions":[],"lastnames":["Waeber"],"suffixes":[]}],"date-added":"2016-08-26 18:10:53 +0000","date-modified":"2020-02-24 13:10:58 +1300","journal":"Mathematics of Operations Research","number":"2","pages":"651–667","title":"Probabilistic bisection converges almost as quickly as stochastic approximation","url_arxiv":"http://arxiv.org/abs/1612.03964","volume":"44","year":"2019","bibtex":"@article{frahenwae16,\n\tabstract = {The probabilistic bisection algorithm (PBA) solves a class of\nstochastic root-finding problems in one dimension by successively updating a\nprior belief on the location of the root based on noisy responses to \nqueries at chosen points. The responses indicate\nthe direction of the root from the queried point, and are incorrect\nwith a fixed probability. The fixed-probability assumption is\nproblematic in applications, and so we extend the PBA to apply when\nthis assumption is relaxed. The extension involves the use of a\npower-one test at each queried\npoint. We explore the convergence behavior of the extended PBA, showing that it\nconverges at a rate arbitrarily close to, but slower than, the canonical ``square root''\nrate of stochastic approximation.},\n\tauthor = {P. I. Frazier and S. G. Henderson and R. Waeber},\n\tdate-added = {2016-08-26 18:10:53 +0000},\n\tdate-modified = {2020-02-24 13:10:58 +1300},\n\tjournal = {Mathematics of Operations Research},\n\tnumber = {2},\n\tpages = {651--667},\n\ttitle = {Probabilistic bisection converges almost as quickly as stochastic approximation},\n\turl_arxiv = {http://arxiv.org/abs/1612.03964},\n\tvolume = {44},\n\tyear = {2019}}\n\n","author_short":["Frazier, P. 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