Constant gossip steps for sublinear network regret

Determine whether performing a constant number of gossip rounds per iteration (q(t) = O(1)) in the DECO decentralized parameter-free coin-betting algorithms suffices to guarantee sublinear network regret R_T^{net}(u) as T grows, under the standard assumptions used in the paper (doubly stochastic gossip matrix W and excellent coin-betting potentials with associated betting functions). Equivalently, ascertain whether the disagreement term in the network regret decomposition can be kept sublinear without increasing the number of gossip steps with time.

Background

The paper decomposes the network regret into the average local regret—shown to be sublinear and independent of topology—and a non-negative disagreement term that captures the cost of agents making different decisions. The disagreement term depends on the spectral properties of the gossip matrix W and the number of gossip steps performed per round.

To ensure sublinear overall network regret, the analysis establishes sufficient conditions that require the number of gossip steps q(t) to grow with time (e.g., linearly) so that the disagreement term is also sublinear. The authors highlight an unresolved theoretical question regarding whether this communication burden can be reduced: specifically, whether a constant number of gossip steps per round is enough to maintain sublinear network regret. They also conjecture in favor of this possibility.