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

TL;DR

This paper investigates the computational complexity of accelerated methods for $\, ext{l}_1$-regularized PageRank, revealing limitations of FISTA and proposing conditions for effective confinement of activations.

Contribution

It demonstrates that standard FISTA can be asymptotically worse than ISTA for this problem and provides conditions under which accelerated methods perform efficiently.

Findings

FISTA can be asymptotically worse than ISTA for certain instances.

Under confinement conditions, FISTA's complexity is bounded by a logarithmic term plus a boundary overhead.

Graph-structural conditions can ensure effective confinement and improved performance.

Abstract

We study the degree-weighted work required to compute $\ell_1$-regularized PageRank using the standard accelerated proximal-gradient method (FISTA). For non-accelerated methods (ISTA), the best known worst-case work is $\widetilde{O}((\alpha\rho)^{-1})$, where $\alpha$ is the teleportation parameter and $\rho$ is the $\ell_1$-regularization parameter. It is not known whether classical acceleration methods can improve $1/\alpha$ to $1/\sqrt{\alpha}$ while preserving the $1/\rho$ locality scaling, or whether they can be asymptotically worse. For FISTA, we show a negative result by constructing a family of instances for which standard FISTA is asymptotically worse than ISTA. On the positive side, we analyze FISTA on a slightly over-regularized objective and show that, under a 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 also provide graph-structural sufficient conditions that imply such confinement.