Complexity of Classical Acceleration for $\ell_1$-Regularized PageRank

2026-02-24

AI Summary

This work investigates how to leverage acceleration techniques to mitigate the dependence of $\ell_1$-regularized PageRank on the teleportation parameter $\alpha$ while preserving locality. By analyzing the behavior of the FISTA algorithm applied to a slightly over-regularized objective, we derive verifiable conditions that confine suboptimally activated nodes within a boundary set, thereby controlling non-locality during acceleration. Integrating graph-structural analysis, a degree-weighted workload model, and boundary-set theory, we establish a composite complexity bound that accounts for both acceleration gains and boundary-related overhead. Experiments on synthetic and real-world graphs confirm the existence of acceleration and deceleration regimes. Theoretically, under the proposed conditions, the algorithm achieves an accelerated convergence rate dominated by $(\rho\sqrt{\alpha})^{-1}\log(\alpha/\varepsilon)$.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the degree-weighted work required to compute $\ell_1$-regularized PageRank using the standard one-gradient-per-iteration accelerated proximal-gradient method (FISTA). For non-accelerated local methods, the best known worst-case work scales as $\widetilde{O} ((\alpha\rho)^{-1})$, where $\alpha$ is the teleportation parameter and $\rho$ is the $\ell_1$-regularization parameter. A natural question is whether FISTA can improve the dependence on $\alpha$ from $1/\alpha$ to $1/\sqrt{\alpha}$ while preserving the $1/\rho$ locality scaling. The challenge is that acceleration can break locality by transiently activating nodes that are zero at optimality, thereby increasing the cost of gradient evaluations. We analyze FISTA on a slightly over-regularized objective and show that, under a checkable confinement condition, all spurious activations remain inside a boundary set $\mathcal{B}$. This yields a bound consisting of an accelerated $(\rho\sqrt{\alpha})^{-1}\log(\alpha/\varepsilon)$ term plus a boundary overhead $\sqrt{vol(\mathcal{B})}/(\rho\alpha^{3/2})$. We provide graph-structural conditions that imply such confinement. Experiments on synthetic and real graphs show the resulting speedup and slowdown regimes under the degree-weighted work model.

Problem

• accelerated optimization
• PageRank
• locality
• sparsity
• graph algorithms

Innovation

• accelerated proximal gradient
• ℓ₁-regularized PageRank
• locality preservation
• confinement condition
• degree-weighted complexity