# Probabilistic Bisection Converges Almost as Quickly as Stochastic Approximation

arXiv: https://arxiv.org/abs/1612.03964 · HTML: https://mlsanity.com/paper/1612.03964/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2016.

## Built on by (most cited first, 5 of 5)

- [Computing Systemic Risk Measures with Graph Neural Networks](https://mlsanity.com/paper/2410.07222/) (2024)
- [A Probabilistic Approach for Model Alignment with Human Comparisons](https://mlsanity.com/paper/2403.10771/) (2024)
- [Characterizing the Accuracy-Communication-Privacy Trade-off in Distributed Stochastic Convex Optimization](https://mlsanity.com/paper/2501.03222/) (2025)
- [Detecting Statements in Text: A Domain-Agnostic Few-Shot Solution](https://mlsanity.com/paper/2405.05705/) (2024)
- [On the Optimality of Tracking Fisher Information in Adaptive Testing with Stochastic Binary Responses](https://mlsanity.com/paper/2510.07862/) (2025)

## Abstract (arXiv)

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.