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

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 $\widetilde{O}((αρ)^{-1})$, where $α$ is the teleportation parameter and $ρ$ is the $\ell_1$-regularization parameter. It is not known whether classical acceleration methods can improve $1/α$ to $1/\sqrtα$ while preserving the $1/ρ$ 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 $(ρ\sqrtα)^{-1}\log(α/\varepsilon)$ term plus a boundary overhead $\sqrt{vol(\mathcal{B})}/(ρα^{3/2})$. We also provide graph-structural sufficient conditions that imply such confinement.