Fast Rates in Online Convex Optimization by Exploiting the Curvature of
  Feasible Sets
Taira Tsuchiya Shinji Ito
In this paper, we explore online convex optimization (OCO) and introduce a
new analysis that provides fast rates by exploiting the curvature of feasible
sets. In online linear optimization, it is known that if the average gradient
of loss functions is larger than a certain value, the curvature of feasible
sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a
logarithmic regret. This paper reveals that algorithms adaptive to the
curvature of loss functions can also leverage the curvature of feasible sets.
We first prove that if an optimal decision is on the boundary of a feasible set
and the gradient of an underlying loss function is non-zero, then the algorithm
achieves a regret upper bound of $O(\rho \log T)$ in stochastic environments.
Here, $\rho > 0$ is the radius of the smallest sphere that includes the optimal
decision and encloses the feasible set. Our approach, unlike existing ones, can
work directly with convex loss functions, exploiting the curvature of loss
functions simultaneously, and can achieve the logarithmic regret only with a
local property of feasible sets. Additionally, it achieves an $O(\sqrt{T})$
regret even in adversarial environments where FTL suffers an $\Omega(T)$
regret, and attains an $O(\rho \log T + \sqrt{C \rho \log T})$ regret bound in
corrupted stochastic environments with corruption level $C$. Furthermore, by
extending our analysis, we establish a regret upper bound of
$O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$ for
$q$-uniformly convex feasible sets, where uniformly convex sets include
strongly convex sets and $\ell_p$-balls for $p \in [1,\infty)$. This bound
bridges the gap between the $O(\log T)$ regret bound for strongly convex sets
($q=2$) and the $O(\sqrt{T})$ regret bound for non-curved sets ($q\to\infty$).