  We consider an online learning problem on a continuum. A decision maker is
given a compact feasible set $S$, and is faced with the following sequential
problem: at iteration~$t$, the decision maker chooses a distribution $x^{(t)}
\in \Delta(S)$, then a loss function $\ell^{(t)} : S \to \mathbb{R}_+$ is
revealed, and the decision maker incurs expected loss $\langle \ell^{(t)},
x^{(t)} \rangle = \mathbb{E}_{s \sim x^{(t)}} \ell^{(t)}(s)$. We view the
problem as an online convex optimization problem on the space $\Delta(S)$ of
Lebesgue-continnuous distributions on $S$. We prove a general regret bound for
the Dual Averaging method on $L^2(S)$, then prove that dual averaging with
$\omega$-potentials (a class of strongly convex regularizers) achieves
sublinear regret when $S$ is uniformly fat (a condition weaker than convexity).
