# Minimax agnostic risk for Lipschitz functions

For $L>0$, let $F_L$ be the class of all $L$-Lipschitz functions on $[0,1]$. Let $D$ be a joint distribution on $[0,1]\times\mathbb{R}$, from which we sample $n$ iid copies $(X_i,Y_i)$. Given any $f:[0,1]\to\mathbb{R}$, $%its empirical risk is$ $$% R_n(f) = \frac1n\sum_{i=1}^n (f(X_i)-Y_i)^2$$ its true risk (w.r.t. $D$) is $$R(f) = \mathbb{E}_D[(f(X)-Y)^2].$$ Define the minimax agnostic excess risk by $$\Delta_n = \inf_{\hat f}\sup_D \mathbb{E}_D[ R(\hat f)-\inf_{f\in F_L}R(f)],$$ where the sup is over all distributions $D$ and the inf is over all estimators taking an $n$-point sample $(X_i,Y_i)$ to a function $\hat f:[0,1]\to\mathbb{R}$.

What is known about the behavior of $\Delta_n$? I was only able to find mimimax rates for additive -- rather than agnostic -- noise models.

where a rate of $n^{-2/3}$ is shown.