To strengthen the asymptotic convergence analysis of norm-PRR and establish non-asymptotic linear convergence rates under suitable geometric conditions, one approach could be to refine the analysis of the error terms and their impact on the convergence properties of the algorithm. By conducting a more detailed investigation of the error bounds and their relationship to the convergence rates, it may be possible to derive tighter bounds that guarantee non-asymptotic linear convergence under specific geometric conditions. Additionally, exploring the interplay between the step sizes, variance terms, and the geometry of the objective function could provide insights into how to optimize the algorithm for faster convergence rates in practical settings.