PAC Mode Estimation using PPR Martingale Confidence Sequences

TL;DR

This paper introduces a new, efficient method for identifying the most common value in a discrete distribution using confidence sequences, improving sample efficiency in practical sampling tasks like election forecasting.

Contribution

It generalizes PPR martingale confidence sequences from binary to multi-class mode estimation, providing an asymptotically optimal, parameter-free, and computationally efficient stopping rule.

Findings

PPR-1v1 outperforms competitors in sample efficiency.

The method is asymptotically optimal as error probability approaches zero.

Effective in practical applications like election forecasting and blockchain verification.

Abstract

We consider the problem of correctly identifying the \textit{mode} of a discrete distribution $\mathcal{P}$ with sufficiently high probability by observing a sequence of i.i.d. samples drawn from $\mathcal{P}$. This problem reduces to the estimation of a single parameter when $\mathcal{P}$ has a support set of size $K = 2$. After noting that this special case is tackled very well by prior-posterior-ratio (PPR) martingale confidence sequences \citep{waudby-ramdas-ppr}, we propose a generalisation to mode estimation, in which $\mathcal{P}$ may take $K \geq 2$ values. To begin, we show that the "one-versus-one" principle to generalise from $K = 2$ to $K \geq 2$ classes is more efficient than the "one-versus-rest" alternative. We then prove that our resulting stopping rule, denoted PPR-1v1, is asymptotically optimal (as the mistake probability is taken to $0$). PPR-1v1 is parameter-free and computationally light, and incurs significantly fewer samples than competitors even in the non-asymptotic regime. We demonstrate its gains in two practical applications of sampling: election forecasting and verification of smart contracts in blockchains.