Learning with Comparison Feedback: Online Estimation of Sample Statistics

TL;DR

This paper introduces algorithms for online estimation of sample statistics like median, CDF, and mean using comparison feedback in an adversarial setting, with theoretical guarantees and high-dimensional extensions.

Contribution

It develops robust algorithms for online sample statistic estimation under adversarial comparison feedback, with nearly matching lower bounds and high-dimensional generalizations.

Findings

Robust algorithms for median, CDF, and mean estimation.

Nearly matching lower bounds established for these estimation tasks.

Extensions to high-dimensional data scenarios.

Abstract

We study an online version of the noisy binary search problem where feedback is generated by a non-stochastic adversary rather than perturbed by random noise. We reframe this as maintaining an accurate estimate for the median of an adversarial sequence of integers, $x_1, x_2, \dots$, in a model where each number $x_t$ can only be accessed through a single threshold query of the form ${1(x_t \leq q_t)}$. In this online comparison feedback model, we explore estimation of general sample statistics, providing robust algorithms for median, CDF, and mean estimation with nearly matching lower bounds. We conclude with several high-dimensional generalizations.