Antithetic variates revisited again

We revisit the method of antithetic variates, perhaps the simplest yet profound variance reduction technique among many others, with the aim of comprehensively improving the method in its general formulation in the seminal works A new Monte Carlo technique: Antithetic variates [Hammersley and Morton, Math. Proc. Cambridge Philos. Soc. 52 (1956), 449–475] and Antithetic variates revisited [Fishman and Huang, Commun. ACM 26 (1983), 964–971]. The achieved advancement under this general formulation contrasts with the conventional approach based on equally weighted two variates with negative correlation, commonly referred to as the method of antithetic variates in various contexts. In pursuit of the aim, we investigate its second-order structure in depth and introduce an adaptive algorithm designed to optimize the weights among multiple variates throughout the primary Monte Carlo estimation process. In order to effectively demonstrate the theoretical advancements, we present numerical results throughout the presentation, vividly showcasing the potential effectiveness of the proposed approach and adaptive algorithm in the face of varying weights.