For this post, $\Gamma$ is an expectation operator, and we have two distributions $S$ and $T$. $(x_S,y_S)\sim S$ and $(x_t,y_t)\sim T$.

Define $\Gamma^*_f = \arg \min_{\Gamma} \Gamma l(f(x_t),y_s)$ and for now, let $l(w,z)=(w-z)^2$

I'm trying to interpret $\Gamma_f^* d(y_s,y_t)$ as it comes up in a bound I came up with recently in my research. I just don't see any natural way of doing so. Does anyone have any insights? Or at least a nice idea for a simple case to begin with? I've tried looking at linear classes of functions for $f$, but nothing has come of it.