We consider the problem of correctly identifying the mode of a discrete distribution $mathcal{P}$ with sufficiently high probability by observing a sequence of i.i.d. samples drawn according to $mathcal{P}$. This problem reduces to the estimation of a single parameter when $mathcal{P}$ has a support set of size $K = 2$. Noting the efficiency of prior-posterior-ratio (PPR) martingale confidence sequences for handling this special case, we propose a generalisation to mode estimation, in which $mathcal{P}$ may take $K geq 2$ values. We observe that the one-versus-one principle yields a more efficient generalisation than the one-versus-rest alternative. Our resulting stopping rule, denoted PPR-ME, is optimal in its sample complexity up to a logarithmic factor. Moreover, PPR-ME empirically outperforms several other competing approaches for mode estimation. We demonstrate the gains offered by PPR-ME in two practical applications: (1) sample-based forecasting of the winner in indirect election systems, and (2) efficient verification of smart contracts in permissionless blockchains.
تحميل البحث