Strong Antithetic Variance Reduction Inequalities

Antithetic variates constitute a well-known variance reduction technique for Monte Carlo sampling methods and related applications. However, many standard antithetic variance bounds do not provide quantitative estimates of the theoretical gains enjoyed by these methods. As a step towards remedying this, in this work, we derive stronger antithetic variance reduction inequalities under anti-Lipschitz and strongly isotonic (as opposed to merely monotonic) assumptions in the univariate and multivariate settings, respectively, which quantify the magnitude of variance reduction. Over the course of our analysis, we develop the concept of an antithetic index and illustrate some of its properties. Furthermore, we show how our stronger antithetic variance reduction inequalities can be used to provide better theoretical guarantees when using antithetic variates in various applications. These applications include approximating integrals, function approximation, concentration inequalities, as well as stochastic optimization. Our arguments utilize and develop ideas from correlation inequalities and first-order optimization theory among other tools.