Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization

TL;DR

This work establishes fundamental limits on the convergence rates of gradient-based algorithms for time-varying quadratic objectives by combining the internal model principle with root-locus analysis. It proves a universal lower bound $\rho_{TV}=\left(\frac{\kappa-1}{\kappa+1}\right)^{1/n}$ linking problem conditioning $\kappa$ and temporal-model degree $n$, for minimal-order, non-accelerated methods. The authors derive explicit root-locus-based controllers that achieve this bound in low-degree models ($n=1,2,3$), and demonstrate the results via numerical simulations on a representative quadratic problem. These findings reveal a fundamental tradeoff: greater temporal complexity (larger $n$) inherently slows convergence, even under optimal controller design, highlighting intrinsic limits for time-varying online optimization. The work also points to future directions in incorporating additional dynamics and extending beyond quadratics to broaden applicability.

Abstract

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 $κ$ and time variation whose model has degree $n$, we show that the worst-case convergence rate of any minimal-order gradient-based algorithm is $ρ_\text{TV} = (\frac{κ-1}{κ+1})^{1/n}$. 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.

Figures (5)

• Figure 1: Structure of the gradient-basing optimization algorithms, as a block-diagram in the frequency domain. See \ref{['eq:tf_00']}.
• Figure 2: Root locus with controller $-\alpha/(z-1)$ (left) and $\alpha/(z+1)$ (right). The locus (blue) starts at the open-loop poles ($\times$). The pole locations at gains $\lambda=\mu$ and $\lambda=L$ are shown ($\bullet$). For all $\lambda\in[\mu,L]$, the root locus is entirely contained in the $\rho$ circle (gray).
• Figure 3: Root locus with controller for a single frequency $\theta = \pi/4$. The locus (blue) starts at the open-loop poles ($\times$) and ends at the open-loop zeros ($\circ$). The pole locations at gains $\lambda=\mu$ and $\lambda=L$ are shown ($\bullet$). For all $\lambda\in[\mu,L]$, the root locus is entirely contained in the $\rho$ circle (gray).
• Figure 4: Root locus with controller for both a constant and a single frequency $\theta = \pi/4$. The locus (blue) starts at the open-loop poles ($\times$) and ends at the open-loop zeros ($\circ$). The pole locations at gains $\lambda=\mu$ and $\lambda=L$ are shown ($\bullet$). For all $\lambda\in[\mu,L]$, the root locus is entirely contained in the $\rho$ circle (gray).
• Figure 5: Numerical simulation of the optimal worst-case controller from \ref{['sec:design-n3']} to minimize the quadratic objective in \ref{['eq:objective']}. (Top) Trajectories of each component in the time-varying linear term $b_k$. (Middle) Trajectories of each component in the time-varying optimizer $x_k^\star$. (Bottom) Gradient norm $\|A x_k + b_k\|$ and the bound on the worst-case rate $\rho^k$ (dashed). See \ref{['sec:simulation']} for details.

Theorems & Definitions (11)

• Remark 1: Quadratic objective functions
• Example 1: Gradient descent
• Definition 1: Asymptotic tracking
• Remark 2
• Definition 2: Optimization filters of minimal order
• Lemma 1: Fundamental structure of tracking filters
• proof
• Lemma 2: Characterization of the convergence rate
• proof
• Theorem 1: Bound on worst-case convergence rate