Title: Accelerated Evolving Set Processes for Local PageRank Computation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Personalized PageRank

Local methods

Sublinear methods
Abstract: This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by $\min\{\tilde{\mathcal{O}}(R^2/\epsilon^2), \tilde{\mathcal{O}}(m)\}$ to obtain an $\epsilon$-approximation of the PPR vector, where $m$ denotes the number of edges in the graph and $R$ is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only $\tilde{\mathcal{O}}(1/\sqrt{\alpha})$ such linear systems, where $\alpha$ is the damping factor. When $1/\epsilon^2\ll m$, this implies the existence of an algorithm that computes an $\epsilon$-approximation of the PPR vector with an overall time complexity of $\tilde{\mathcal{O}}(R^2 / (\sqrt{\alpha}\epsilon^2))$, independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel framework called AESP (Accelerated Evolving Set Process) to approximate the Personalized PageRank (PPR) vectors in a faster way using local computations. AESP has a complexity of O(min{ R^2/(sqrt(α)ϵ^2) , m/sqrt(α) }), where R is a constant depending on the evolving set process and m, α, ϵ are the number of edges, the damping factor, the precision, respectively. Interestingly, when 1/ϵ^2 ≪ m the algorithm computes an ϵ-approximation of the PPR vector with an time  complexity independent of the underlying graph size. The authors show that AESP can accelerate any existing local solver, e.g. APPR, by (1) reformulating the PPR computation as a convex quadratic problem, (2) applying proximal point acceleration with an evolving set process to avoid accessing the whole graph and to solve easier subproblems, (3) using the local solver for each subproblem, (4) using momentum [39] to update the estimate with less iterations (from 1/α to 1/sqrt(α)).\n\nSTRENGTHS AND WEAKNESSES:\nPros: The paper answers an open conjecture [19, 52] about achievable time complexity for the PPR computation. The problem is well introduced and the presentation is rigorous. Theoretical analysis with mathematical proofs are provided. The evaluation on real-world large graphs has been conducted, with a comparison with baseline algorithms (APPR and LocGD from [52]). Experiments include ablations (different initialization methods, different α values) and are consistent with the theory. They show acceleration in error reduction during initial operations and overall speedup.\n\nCons: The constant R depends not only on graph parameters, and R is not explicitly bounded. The authors claim that given the empirical results, it still seems to be small and show in Fig.3 a value of at most 1.4. However these values are shown for alpha = 0.1 and 0.01. Even for these two values of damping factor, there is an increase in R for lower alpha. This means that since R(a) then this should be also incorporated in the claimed complexity. How does R grow with smaller alpha? \nMore generally, a study of the algorithm over the range of the damping factor is missing. For example, the authors assume that α < 0.5, but is the algorithm also robust to high α values?\n\nAnother issue that troubles me is that the complexity has a denominator in ϵ^2 instead of ϵ. Certainly there is a benefit to go from 1/α to 1/(sqrt(α) but there is penalty in having the squared precision in the denominator. This is not discussed sufficiently.\n\nFurthermore, the comparison against APPR, LocGD and FISTA give interesting results, however, there are other state-of-the-art methods for approximating PageRank, e.g. it is missing comparison against Ref.[37] although extensively discussed in the Related Works section.\n\nQUESTIONS:\nIf one compares with O(1/(αϵ)) from APPR where are the wins and losses? How does the ϵ^2 in the proposed method here affect performance compared to APPR and other methods with just  ϵ involved?\n\nHow does the R in the complexity scale with dumping factor? \n\nHow does the method perform against other methods from the literature e.g. Ref.[37] ?\n\nLIMITATIONS:\nThe limitations of the current algorithmic proposal have not been sufficiently discussed, although to be fair there are some interesting ablation studies. Furthermore, some cases in large graphs where the method underperforms compared to baselines have been identified, but I was not convinced from the explanations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Accelerated Evolving Set Process (AESP), a novel framework designed to accelerate the Approximate Personalized PageRank (APPR) algorithm. The core of AESP lies in its formulation based on inexact accelerated proximal point iterations. The authors provide a rigorous theoretical analysis, establishing sufficient conditions for convergence and detailing the method's time complexity. A key theoretical finding is that AESP achieves faster convergence rates than traditional APPR in the early stages. Extensive experiments on large-scale graphs, with node counts ranging from 6 million to 240 million, empirically validate this theoretical advantage, demonstrating that AESP-based methods significantly outperform baseline approaches in terms of early-stage convergence speed.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nSignificant Theoretical Contributions: The paper provides a rigorous theoretical foundation for the proposed method. A key strength is the inclusion of not only a detailed time complexity analysis but also formal convergence guarantees. Generality and Extensibility: The authors demonstrate the versatility of the AESP framework by analyzing its application to several variational forms of the PPR problem. This analysis effectively shows how the proposed framework can be extended beyond the standard PPR setting, broadening its potential impact. Practical and Meaningful Problem: The work addresses the important and practical challenge of accelerating a general formulation of the Personalized PageRank problem. Developing faster algorithms for this fundamental task is a meaningful contribution to the field of large-scale graph analysis.\n\nWeaknesses:\nClarity and Presentation: The presentation of the proposed framework is overly complex, which significantly hinders readability and makes the core ideas difficult to follow. Choice of Baselines in Experiments: The persuasiveness of the experimental results is somewhat limited by the choice of baseline methods. The selected baselines (e.g., APPR, APPR-opt) could be considered dated. A more robust evaluation against more recent PPR approximation algorithms would be necessary to convincingly demonstrate the superiority of the proposed AESP framework.\n\nQUESTIONS:\nThank you for your work on this interesting problem. The paper is promising, particularly due to its strong theoretical contributions. However, I have a few key concerns regarding the clarity of the presentation and the experimental evaluation that prevent me from giving a higher score at this stage. My questions and suggestions below are intended to clarify these points. My evaluation score will be raised if you can address them convincingly in your rebuttal.\n1.\tOn the clarity of proposed AESP Framework: \nMy primary concern is with the presentation of the AESP framework in Section 2.2. I found the exposition overly complex and the notation difficult to follow, which obscures the core intuition behind the method. To address this, I strongly encourage the authors to Consider adding a simple, illustrative figure or a running example that walks the reader through one or two iterations of the algorithm. This would significantly improve comprehensibility. My score would increase substantially if the authors can provide a clearer, more accessible explanation of their framework in the rebuttal.\n2.\tOn the Robustness of the experimental evaluation:\nThe experimental section provides evidence for AESP's fast early-stage convergence. However, the choice of baseline methods seems somewhat outdated, which limits the persuasiveness of the results. To claim superiority, a comparison against the current state-of-the-art is essential. Could the authors please justify their choice of the current baselines? Are there specific reasons (e.g., implementation availability, direct comparability) why these were chosen over more recent methods from top conferences from the last 3-5 years?\n3.\tOn the characterization of early-stage convergence:\nThe paper's key empirical results emphasize that AESP's primary advantage is its faster early-stage convergence. This raises a question about the method's performance envelope. Could the authors elaborate on the behavior of AESP in the later stages of convergence? Specifically, does the performance advantage over baselines diminish or disappear as the approximation error tolerance becomes very small? Is there a crossover point after which traditional methods might become more efficient?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the time complexity of Personalized PageRank (PPR) computation. It introduces a nested evolving set process to improve the time complexity to $\\tilde{O}(\\min(R^2/\\epsilon^2, m))$, where $R$ is a constant defined by the process and $\\epsilon$ is the normalized additive error parameter. In particular, the time dependence on the damping factor $\\alpha$ is improved to $1/\\sqrt{\\alpha}$. Theoretical analysis and experimental results are provided to demonstrate the effectiveness of the proposed methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nS1. The problem of efficiently estimating Personalized PageRank (PPR) on large graphs has been widely studied and has numerous applications. Improving the time dependence on $1/\\alpha$ is of particular interest in recent work, so the paper is well motivated.\n\nS2. The paper is generally well written and easy to follow (though some issues should be addressed, as outlined below). The theoretical analysis appears sound. Experimental results and source code are also provided, which strengthens the paper’s contributions.\n\nS3. The paper demonstrates that the time dependence on $1/\\alpha$ can be improved using nested evolving sets. While the improvement is partial, the underlying idea and algorithmic structure could offer new insights to the community.\n\n\nWeaknesses:\n\nW1. The complexity results presented in the paper are not very interesting to me. The proposed algorithm has quadratic dependence on $1/\\epsilon$, where $\\epsilon$ is the degree-normalized additive error. In contrast, the basic Forward Push algorithm (FOCS 2006) has a time complexity of $O(1/(\\epsilon \\alpha))$ (please correct me during the rebuttal if there is any misunderstanding). In other words, while the paper improves the dependence on $\\alpha$ from $1/\\alpha$ to $1/\\sqrt{\\alpha}$, it introduces a worse dependence on $\\epsilon$. This trade-off weakens the overall contribution.\n\nW2. Although the writing is generally clear, some aspects of the presentation need refinement: First, it would be better to clearly define the problem (i.e., the approximation error requirement) at the beginning of the paper. Presenting the complexity bound before defining the problem makes the bound hard to interpret. Second, the use of the term ”$\\epsilon$-approximation” is potentially confusing. In most literature, this refers to relative error, while in the paper, $\\epsilon$ denotes degree-normalized additive error. Clarifying this distinction earlier would help avoid confusion.\n\nQUESTIONS:\nI’ve described my questions in the “Weaknesses” section above. Please refer to that for details.\n\nLIMITATIONS:\nYes, there is not much concern about the potential negative societal impact of this paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper proposes the Accelerated Evolving Set Process (AESP) framework that leverages the accelerated inexact proximal operator to improve the computation efficiency of personalized PageRank. Particularly, the proposed algorithm improves the time complexity of existing methods for $\epsilon$-approximate personalized PageRank computation from $O(vol(\mathcal{S}_t)/(\alpha \gamma_t))$ to $O(vol(\mathcal{S}_t)/(\sqrt{\alpha} \gamma_t))$.

STRENGTHS AND WEAKNESSES:
S1. The paper is theoretically grounded and technically grounded.

S2. The idea of utilizing the accelerated inexact proximal operator is novel.

S3. The methods are guaranteed to converge without any additional assumptions, and the time complexity achieves a provable improvement over existing works.

W1. The assumption on constant $R$ may not hold.

W2. The empirical studies that compare the proposed methods against the algorithms for personalized PageRank estimation on more datasets are needed. e.g., FORA [KDD'17], TopPPR [SIGMOD'19].

QUESTIONS:
1. How do the proposed methods compare to the sampling-based PPR estimation algorithms (e.g., FORA [KDD'2017]) in empirical efficiency?
2. What is the intuition underlying the proposed methods for the lower time complexity?

LIMITATIONS:
yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03