Establish convergence guarantees for stability safeguards

Prove convergence of the data-driven reset and adaptive-projection safeguards for structure-aware online Age of Information optimization when the fixed stability assumptions are violated or the stability boundary is unknown.

Background

The numerical study introduces two safeguards for operation near or beyond an unknown stability boundary: resetting the update rate to the smallest feasible value and adaptively projecting the gradient step onto a contracted stability interval. These safeguards are intended to address situations in which the fixed truncation and stability conditions required by Propositions 5 and 9 cannot be verified or do not hold.

The experiments indicate that the safeguards can restore convergence in an example outside the theoretical stability hypothesis. However, the paper does not establish a convergence theorem for either safeguarded variant, leaving their formal convergence analysis unresolved.