Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of aDoeblin curve—a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

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.