An optimal unrestricted learning procedure

We study learning problems involving arbitrary classes of functions $F$, distributions $X$ and targets $Y$. Because proper learning procedures, i.e., procedures that are only allowed to select functions in $F$, tend to perform poorly unless the problem satisfies some additional structural property (e.g., that $F$ is convex), we consider unrestricted learning procedures that are free to choose functions outside the given class. We present a new unrestricted procedure that is optimal in a very strong sense: the required sample complexity is essentially the best one can hope for, and the estimate holds for (almost) any problem, including heavy-tailed situations. Moreover, the sample complexity coincides with the what one would expect if $F$ were convex, even when $F$ is not. And if $F$ is convex, the procedure turns out to be proper. Thus, the unrestricted procedure is actually optimal in both realms, for convex classes as a proper procedure and for arbitrary classes as an unrestricted procedure.