Iterative Methods via Locally Evolving Set Process

Given the damping factor $\alpha$ and precision tolerance $\epsilon$, \citet{andersen2006local} introduced Approximate Personalized PageRank (APPR), the \textit{de facto local method} for approximating the PPR vector, with runtime bounded by $\Theta(1/(\alpha\epsilon))$ independent of the graph size. Recently, Fountoulakis \& Yang asked whether faster local algorithms could be developed using $\tilde{\mathcal{O}}(1/(\sqrt{\alpha}\epsilon))$ operations. By noticing that APPR is a local variant of Gauss-Seidel, this paper explores the question of *whether standard iterative solvers can be effectively localized*. We propose to use the *locally evolving set process*, a novel framework to characterize the algorithm locality, and demonstrate that many standard solvers can be effectively localized. Let $\overline{\operatorname{vol}}{ (\mathcal S_t)}$ and $\overline{\gamma_t}$ be the running average of volume and the residual ratio of active nodes $\textstyle \mathcal{S_t}$ during the process. We show $\overline{\operatorname{vol}}{ (\mathcal S_t)}/\overline{\gamma_t} \leq 1/\epsilon$ and prove APPR admits a new runtime bound $\tilde{\mathcal{O}}(\overline{\operatorname{vol}}(\mathcal S_t)/(\alpha\overline{\gamma_t}))$ mirroring the actual performance. Furthermore, when the geometric mean of residual reduction is $\Theta(\sqrt{\alpha})$, then there exists $c \in (0,2)$ such that the local Chebyshev method has runtime $\tilde{\mathcal{O}}(\overline{\operatorname{vol}}(\mathcal{S_t})/(\sqrt{\alpha}(2-c)))$ without the monotonicity assumption. Numerical results confirm the efficiency of this novel framework and show up to a hundredfold speedup over corresponding standard solvers on real-world graphs.