  Empirical risk minimization (ERM) is a fundamental learning rule for
statistical learning problems where the data is generated according to some
unknown distribution $\mathsf{P}$ and returns a hypothesis $f$ chosen from a
fixed class $\mathcal{F}$ with small loss $\ell$. In the parametric setting,
depending upon $(\ell, \mathcal{F},\mathsf{P})$ ERM can have slow
$(1/\sqrt{n})$ or fast $(1/n)$ rates of convergence of the excess risk as a
function of the sample size $n$. There exist several results that give
sufficient conditions for fast rates in terms of joint properties of $\ell$,
$\mathcal{F}$, and $\mathsf{P}$, such as the margin condition and the Bernstein
condition. In the non-statistical prediction with expert advice setting, there
is an analogous slow and fast rate phenomenon, and it is entirely characterized
in terms of the mixability of the loss $\ell$ (there being no role there for
$\mathcal{F}$ or $\mathsf{P}$). The notion of stochastic mixability builds a
bridge between these two models of learning, reducing to classical mixability
in a special case. The present paper presents a direct proof of fast rates for
ERM in terms of stochastic mixability of $(\ell,\mathcal{F}, \mathsf{P})$, and
in so doing provides new insight into the fast-rates phenomenon. The proof
exploits an old result of Kemperman on the solution to the general moment
problem. We also show a partial converse that suggests a characterization of
fast rates for ERM in terms of stochastic mixability is possible.
