Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization

This paper investigates the fundamental performance limits of gradient-based algorithms for time-varying optimization. Leveraging the internal model principle and root locus techniques, we show that temporal variabilities impose intrinsic limits on the achievable rate of convergence. For a problem with condition ratio $\kappa$ and time variation whose model has degree $n$, we show that the worst-case convergence rate of any minimal-order gradient-based algorithm is $\rho_\text{TV} = (\frac{\kappa-1}{\kappa+1})^{1/n}$. This bound reveals a fundamental tradeoff between problem conditioning, temporal complexity, and rate of convergence. We further construct explicit controllers that attain the bound for low-degree models of time variation.