Tightness and improvement of the general convergence-rate bound

Determine whether the general upper bound on the exponential convergence rate for heterogeneous and time-varying non-Bayesian social learning is tight, and, if it is not, derive an improved bound; additionally, characterize when the dominated- and dominating-path bounds are tight or can be surpassed.

Background

The paper derives a general upper bound based on the informativeness of a hypothetical agent receiving the most informative signal at each time, together with more specialized lower and upper bounds based on dominated and dominating paths.

The authors state that the general bound is obtained through a straightforward inequality and that its exactness has not been established. They also note that the path-based bounds can behave differently in special network configurations, while exact rates are currently available only under restrictive conditions.

References

One can wonder whether this bound is tight, we were not able to prove it. To arrive at Theorem \ref{thm:upperbound}, we applied a straightforward upper bounding on the equation (i) obtained in Theorem \ref{thm:linearrep}, therefore, we make the conjecture that this bound might be improved.

— Non-Bayesian Social Learning with Heterogeneous and Time-Varying Aggregation Rules  (2610.13611 - Oukaci et al., 9 Oct 2026) in Section 5, Discussion and Future Work, paragraph “Tightness of the convergence rate bounds”

As we have shown in Section~\ref{subsec:CES}, from our setting we conjecture that any network should use the harmonic rule with weights decaying to 0 (having the best lower bound) , but this is only proved for the CES family case and under homogeneity among agents. Would this remain true in a more realistic and complex setup, such as heterogeneous families of aggregation functions?

— Non-Bayesian Social Learning with Heterogeneous and Time-Varying Aggregation Rules  (2610.13611 - Oukaci et al., 9 Oct 2026) in Section 5, Discussion and Future Work, paragraph “Optimal configuration under heterogeneous time varying rules”