On the optimality of antithetic randomization for cross-validation

Abstract

In the classical normal means problem, independent train--test folds can be constructed by perturbing the data with normal randomization. Averaging over $K$ such folds yields a cross-validation estimator whose bias depends on the marginal distribution of the randomization variables, while its variance depends on their joint distribution. This raises the questions: which joint law is optimal, and how to construct the corresponding randomization scheme? We show that: (i) for smooth estimators, antithetic randomization with pairwise correlation $\rho=-1/(K-1)$ is necessary and sufficient for the reducible variance due to randomization to remain bounded as the bias vanishes; (ii) a general construction yields a class of antithetic schemes, within which the jointly normal scheme is minimax optimal; and (iii) for non-smooth estimators with finitely many jump discontinuities, antithetic randomization improves the asymptotic rate of the reducible variance, while a simple control variate restores bounded variance when the discontinuities are known.