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 $\widetilde{O} ((αρ)^{-1})$, where $α$ is the teleportation parameter and $ρ$ is the $\ell_1$-regularization parameter. A natural question is whether FISTA can improve the dependence on $α$ from $1/α$ to $1/\sqrtα$ while preserving the $1/ρ$ 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 $(ρ\sqrtα)^{-1}\log(α/\varepsilon)$ term plus a boundary overhead $\sqrt{vol(\mathcal{B})}/(ρα^{3/2})$. 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.

翻译：我们研究了使用标准的每次迭代单梯度加速近端梯度法（FISTA）计算$\ell_1$正则化PageRank所需的度加权计算量。对于非加速局部方法，已知最坏情况下的计算量标度为$\widetilde{O} ((αρ)^{-1})$，其中$α$为传送参数，$ρ$为$\ell_1$正则化参数。一个自然的问题是：FISTA能否在保持$1/ρ$局部性标度的同时，将$α$的依赖关系从$1/α$改进为$1/\sqrtα$？其挑战在于加速过程可能通过暂时激活最优解中为零的节点而破坏局部性，从而增加梯度计算成本。我们分析了FISTA在轻微过正则化目标函数上的表现，并证明在可检验的约束条件下，所有伪激活均保持在边界集$\mathcal{B}$内部。这产生了一个由加速项$(ρ\sqrtα)^{-1}\log(α/\varepsilon)$与边界开销项$\sqrt{vol(\mathcal{B})}/(ρα^{3/2})$组成的上界。我们给出了能够保证此类约束条件的图结构条件。在合成图与真实图上的实验展示了度加权计算模型下的加速与减速机制。