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We consider the problem of correctly identifying the \textit{mode} of adiscrete distribution $\mathcal{P}$ with sufficiently high probability byobserving a sequence of i.i.d. samples drawn from $\mathcal{P}$. This problemreduces to the estimation of a single parameter when $\mathcal{P}$ has asupport set of size $K = 2$. After noting that this special case is tackledvery well by prior-posterior-ratio (PPR) martingale confidence sequences\citep{waudby-ramdas-ppr}, we propose a generalisation to mode estimation, inwhich $\mathcal{P}$ 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 ismore efficient than the "one-versus-rest" alternative. We then prove that ourresulting stopping rule, denoted PPR-1v1, is asymptotically optimal (as themistake probability is taken to $0$). PPR-1v1 is parameter-free andcomputationally light, and incurs significantly fewer samples than competitorseven in the non-asymptotic regime. We demonstrate its gains in two practicalapplications of sampling: election forecasting and verification of smartcontracts in blockchains.

PAC Mode Estimation using PPR Martingale Confidence Sequences