Fixed-Size Active Statistical Inference

Variance reduction techniques improve the efficiency of drawing conclusions from data. Recent proposals that incorporate machine learning predictions into estimators based on weighted sampling use the Poisson Sampling (PS) algorithm, which independently flips a biased coin at each unit and includes all heads in the sample. A fundamental flaw with PS is that it has a random sample-size, which presents practical challenges and introduces variance into the estimator. We show that a fixed-size alternative, known in survey-sampling as Conditional Poisson Sampling (CPS), is a drop-in replacement for PS on general multivariate Z-estimation problems, which includes convex M-estimation considered in prior work. Notably, we show that the asymptotic variance of CPS is no worse than that of PS, and can be substantially better in realistic scenarios where predictions are imperfect. We illustrate these advantages on both synthetic and real datasets. In light of our contributions, there is no need to suffer a variable sample-size.