Probabilistic Bisection Converges Almost as Quickly as Stochastic Approximation

TL;DR

This paper extends the probabilistic bisection algorithm to handle variable error probabilities using power-one tests, demonstrating it converges nearly as fast as stochastic approximation methods in noisy root-finding tasks.

Contribution

The authors develop an extension of PBA that relaxes the fixed error probability assumption, employing power-one tests to maintain convergence properties.

Findings

Convergence rate is close to the square root rate of stochastic approximation.

Extended PBA remains effective under relaxed error probability assumptions.

The method converges almost as quickly as traditional stochastic approximation.

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.