Dual Averaging on Compactly-Supported Distributions And Application to No-Regret Learning on a Continuum

Abstract: 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 $\ell^{(t)} : S \to \mathbb{R}<em>+$ is revealed, and the decision maker incurs expected loss $$\langle \ell^{(t)}, x^{(t)} \rangle = \mathbb{E}</em>{s \sim x^{(t)}} \ell<sup>{(t)}(s)$$. 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 $L<sup>2(S)$, 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).