Dual Averaging on Compactly-Supported Distributions And Application to No-Regret Learning on a Continuum
This addresses the problem of no-regret learning in continuous spaces for decision-makers, offering an incremental improvement by extending results to non-convex settings.
The paper tackles online learning on a continuum with compact feasible sets, proving that dual averaging with ω-potentials achieves sublinear regret under a uniformly fat condition, which is weaker than convexity.
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 $x^{(t)} \in Δ(S)$, 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 $Δ(S)$ 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 $ω$-potentials (a class of strongly convex regularizers) achieves sublinear regret when $S$ is uniformly fat (a condition weaker than convexity).