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

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 ∈ ∆(S), then a loss function ` : S → R+ is revealed, and the decision maker incurs expected loss 〈 `, x 〉 = Es∼x(t) ` (s). We view the problem as an online convex optimization problem on the space ∆(S) of Lebesgue-continnuous distributions on S. We prove a general regret bound for the Dual Averaging method on L(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).