The Lazy Online Subgradient Algorithm is Universal on Strongly Convex Domains

Daron Anderson, Douglas J. Leith

Keywords: optimisation, online learning, FTRL, subgradient, gradient descent, sequential, machine learning, strongly convex, logarithmic, pseudoregret, expected regret, differential geometry, curvature

TL;DR: If the domain is strongly convex, Online Lazy Gradient Descent gets $O(\log N)$ regret against iid opponents and $O(\sqrt N)$ against all opponents.

Abstract: We study Online Lazy Gradient Descent for optimisation on a strongly convex domain. The algorithm is known to achieve $O(\sqrt N)$ regret against adversarial opponents; here we show it is universal in the sense that it also achieves $O(\log N)$ expected regret against i.i.d opponents. This improves upon the more complex meta-algorithm of Huang et al \cite{FTLBall} that only gets $O(\sqrt {N \log N})$ and $ O(\log N)$ bounds. In addition we show that, unlike for the simplex, order bounds for pseudo-regret and expected regret are equivalent for strongly convex domains.

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