Best-of-All-Worlds Bounds for Online Learning with Feedback Graphs
This work provides a unified algorithm for online learning with feedback graphs, addressing a theoretical challenge in sequential decision-making, but it is incremental as it builds on prior entropy-based methods.
The paper tackles the problem of online learning with feedback graphs by developing an algorithm that achieves simultaneous regret bounds for adversarial losses, stochastic losses, and corrupted stochastic losses, with bounds scaling with the clique covering number of the graph, such as O(√(θ(G)T)) for adversarial cases.
We study the online learning with feedback graphs framework introduced by Mannor and Shamir (2011), in which the feedback received by the online learner is specified by a graph $G$ over the available actions. We develop an algorithm that simultaneously achieves regret bounds of the form: $\smash{\mathcal{O}(\sqrt{θ(G) T})}$ with adversarial losses; $\mathcal{O}(θ(G)\operatorname{polylog}{T})$ with stochastic losses; and $\mathcal{O}(θ(G)\operatorname{polylog}{T} + \smash{\sqrt{θ(G) C})}$ with stochastic losses subject to $C$ adversarial corruptions. Here, $θ(G)$ is the clique covering number of the graph $G$. Our algorithm is an instantiation of Follow-the-Regularized-Leader with a novel regularization that can be seen as a product of a Tsallis entropy component (inspired by Zimmert and Seldin (2019)) and a Shannon entropy component (analyzed in the corrupted stochastic case by Amir et al. (2020)), thus subtly interpolating between the two forms of entropies. One of our key technical contributions is in establishing the convexity of this regularizer and controlling its inverse Hessian, despite its complex product structure.