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Iterative Methods via Locally Evolving Set Process

Baojian Zhou, Yifan Sun, Reza Babanezhad Harikandeh, Xingzhi Guo, Deqing Yang, Yanghua Xiao

TL;DR

This paper proposes to use thelocally evolving set process, a novel framework to characterize the algorithm locality, and demonstrates that many standard solvers can be effectively localized and shows up to a hundredfold speedup over corresponding standard solvers on real-world graphs.

Abstract

Given the damping factor $α$ and precision tolerance $ε$, \citet{andersen2006local} introduced Approximate Personalized PageRank (APPR), the \textit{de facto local method} for approximating the PPR vector, with runtime bounded by $Θ(1/(αε))$ independent of the graph size. Recently, \citet{fountoulakis2022open} asked whether faster local algorithms could be developed using $\tilde{O}(1/(\sqrtαε))$ operations. By noticing that APPR is a local variant of Gauss-Seidel, this paper explores the question of \textit{whether standard iterative solvers can be effectively localized}. We propose to use the \textit{locally evolving set process}, a novel framework to characterize the algorithm locality, and demonstrate that many standard solvers can be effectively localized. Let $\overline{\operatorname{vol}}{ (S_t)}$ and $\overlineγ_{t}$ be the running average of volume and the residual ratio of active nodes $\textstyle S_{t}$ during the process. We show $\overline{\operatorname{vol}}{ (S_t)}/\overlineγ_{t} \leq 1/ε$ and prove APPR admits a new runtime bound $\tilde{O}(\overline{\operatorname{vol}}(S_t)/(α\overlineγ_{t}))$ mirroring the actual performance. Furthermore, when the geometric mean of residual reduction is $Θ(\sqrtα)$, then there exists $c \in (0,2)$ such that the local Chebyshev method has runtime $\tilde{O}(\overline{\operatorname{vol}}(S_{t})/(\sqrtα(2-c)))$ 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.

Iterative Methods via Locally Evolving Set Process

TL;DR

This paper proposes to use thelocally evolving set process, a novel framework to characterize the algorithm locality, and demonstrates that many standard solvers can be effectively localized and shows up to a hundredfold speedup over corresponding standard solvers on real-world graphs.

Abstract

Given the damping factor and precision tolerance , \citet{andersen2006local} introduced Approximate Personalized PageRank (APPR), the \textit{de facto local method} for approximating the PPR vector, with runtime bounded by independent of the graph size. Recently, \citet{fountoulakis2022open} asked whether faster local algorithms could be developed using operations. By noticing that APPR is a local variant of Gauss-Seidel, this paper explores the question of \textit{whether standard iterative solvers can be effectively localized}. We propose to use the \textit{locally evolving set process}, a novel framework to characterize the algorithm locality, and demonstrate that many standard solvers can be effectively localized. Let and be the running average of volume and the residual ratio of active nodes during the process. We show and prove APPR admits a new runtime bound mirroring the actual performance. Furthermore, when the geometric mean of residual reduction is , then there exists such that the local Chebyshev method has runtime 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.

Paper Structure

This paper contains 42 sections, 27 theorems, 209 equations, 15 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Revisiting Anderson's APPR and its local runtime bound
  4. Problem reformulation
  5. Local Methods via Evolving Set Process
  6. Locally evolving set process
  7. APPR via locally evolving set process
  8. Faster local variant of GS-SOR
  9. Parallelizable local gradient descent
  10. Accelerated Local Iterative Methods
  11. Local Chebyshev method
  12. Generalization and Open Problems
  13. Experiments
  14. Limitations and Conclusion
  15. Notations and Proof of Lemma \ref{['lemma:anderson-local-lemma']}
  16. ...and 27 more sections

Key Result

Lemma 2.1

Given $\alpha \in (0,1)$ and the precision ${\epsilon} \leq 1/d_s$ for node $s \in {\mathcal{V}}$ with ${\bm{p}} \leftarrow \bm 0, {\bm{z}} \leftarrow {\bm{e}}_s$ at the initial, $\textsc{APPR}(\alpha,{\epsilon},s,{\mathcal{G}})$ defined in algo:appr returns an estimate ${\bm{p}}$ of ${\bm{\pi}}$. T Furthermore, the estimate $\hat{{\bm{\pi}}}:={\bm{p}}$ satisfies $\|{\bm{D}}^{-1}(\hat{{\bm{\pi}}}

Figures (15)

  • Figure 1: Runtime of APPR in the locally evolving set process on the com-dblp graph with $s=0, \alpha = 0.1$, and ${\epsilon} = 1/m$. The red region of the left figure is ${\mathcal{T}}_{\textsc{APPR}}$. The right two figures show active ratios and $\overline{\operatorname{vol}}({\mathcal{S}}_T)/\overline{\gamma}_T \leq 1/{\epsilon}$.
  • Figure 2: Comparison of runtime between APPR and LocSOR (left) and runtime bounds (right) as a function of ${\epsilon}$. We used the same setting as in Fig. \ref{['fig:evolving-processing-appr-dataset-com-dblp']}.
  • Figure 3: Comparison of runtime between APPR and ASPR. The setting is the same as in Fig. \ref{['fig:evolving-processing-appr-dataset-com-dblp']}. Left ${\epsilon} = 10^{-4}$ while $\tfrac{1}{n}$ for right.
  • Figure 4: The speedup of local solvers as a function of ${\epsilon}$. The vertical line is ${\epsilon} =1/n$.
  • Figure 5: Estimation error as a function of operations required. (${\epsilon} = 10^{-4}/n$)
  • ...and 10 more figures

Theorems & Definitions (58)

  • Lemma 2.1: Runtime bound of APPR andersen2006local
  • Definition 3.1: Locally evolving set process
  • Lemma 3.2: New local evolving-based bound for APPR
  • Theorem 3.3: Runtime bound of LocSOR $(\omega = 1)$
  • Corollary 3.4
  • Theorem 3.5: Runtime bound of LocGD
  • Lemma 4.1
  • Theorem 4.2: Runtime bound of LocCH
  • proof
  • Definition B.1: Local variant of GS-SOR
  • ...and 48 more