Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation

Question

Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding $$ |\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)| $$ for a given function $f$. Here $P \otimes Q$ denotes the product distribution of $P$ and $Q$, while $P, Q$ denotes their joint distribution (or equivalently a coupling).

Question: Can we upper bound such a mean difference using the Wasserstein distance between $P$ and $Q$, or the Wasserstein distance between $P \otimes Q$ and $P, Q$?

If needed, we are allowed to impose regularity conditions, such as Lipschitz continuity, on $f$.

Answer 1

By the dual representation of the Wasserstein metric $W_1$, $$ |\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)|\le\text{Lip}(f)\,W_1(P \otimes Q,(P, Q)), $$ where $\text{Lip}(f)$ is the Lipschitz constant of $f$.

On the other hand, the closeness of $P$ to $Q$ in any metric will not by itself provide any interesting upper bound on $|\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)|$; e.g., take $P=Q$.