Abstract
In this work, we study the online convex optimization problem with curved losses and delayed feedback.When losses are strongly convex, existing approaches obtain regret bounds of order $d_{\max} \ln T$, where $d_{\max}$ is the maximum delay and $T$ is the time horizon. However, in many cases, this guarantee can be much worse than $\sqrt{d_{\mathrm{tot}}}$ as obtained by a delayed version of online gradient descent, where $d_{\mathrm{tot}}$ is the total delay.We bridge this gap by proposing a variant of follow-the-regularized-leader that obtains regret of order $\min\\{\sigma_{\max}\ln T, \sqrt{d_{\mathrm{tot}}}\\}$, where $\sigma_{\max}$ is the maximum number of missing observations.We then consider exp-concave losses and extend the Online Newton Step algorithm to handle delays with an adaptive learning rate tuning, achieving regret $\min\\{d_{\max} n\ln T, \sqrt{d_{\mathrm{tot}}}\\}$ where $n$ is the dimension.To our knowledge, this is the first algorithm to achieve such a regret bound for exp-concave losses.We further consider the problem of unconstrained online linear regression and achieve a similar guarantee by designing a variant of the Vovk-Azoury-Warmuth forecaster with a clipping trick.Finally, we implement our algorithms and conduct experiments under various types of delay and losses, showing an improved performance over existing methods.