PAC Mode Estimation using PPR Martingale Confidence Sequences

We consider the problem of correctly identifying the textitmode of a discrete distribution P with sufficiently high probability by observing a sequence of i.i.d. samples drawn from P. This problem reduces to the estimation of a single parameter when 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 citepwaudby-ramdas-ppr, we propose a generalisation to mode estimation, in which P may take K ≥ 2 values. To begin, we show that the "one-versus-one" principle to generalise from K = 2 to K ≥ 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.