Online Learning with Local Permutations and Delayed Feedback
This work addresses a bottleneck in online learning for scenarios with delayed feedback, offering a novel theoretical improvement but is incremental in the broader context of optimization algorithms.
The paper tackles the problem of online learning with delayed feedback by introducing a setting where the learner can permute loss functions locally, proving that this improves regret bounds from O(√(τT)) to O(√T(1+√(τ²/M))) when M ≥ τ, and providing experiments to validate the results.
We propose an Online Learning with Local Permutations (OLLP) setting, in which the learner is allowed to slightly permute the \emph{order} of the loss functions generated by an adversary. On one hand, this models natural situations where the exact order of the learner's responses is not crucial, and on the other hand, might allow better learning and regret performance, by mitigating highly adversarial loss sequences. Also, with random permutations, this can be seen as a setting interpolating between adversarial and stochastic losses. In this paper, we consider the applicability of this setting to convex online learning with delayed feedback, in which the feedback on the prediction made in round $t$ arrives with some delay $τ$. With such delayed feedback, the best possible regret bound is well-known to be $O(\sqrt{τT})$. We prove that by being able to permute losses by a distance of at most $M$ (for $M\geq τ$), the regret can be improved to $O(\sqrt{T}(1+\sqrt{τ^2/M}))$, using a Mirror-Descent based algorithm which can be applied for both Euclidean and non-Euclidean geometries. We also prove a lower bound, showing that for $M<τ/3$, it is impossible to improve the standard $O(\sqrt{τT})$ regret bound by more than constant factors. Finally, we provide some experiments validating the performance of our algorithm.