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Register a new userApproximate Graph Propagation
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Added by
Hanzhi Wang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Efficient computation of node proximity queries such as transition probabilities, Personalized PageRank, and Katz are of fundamental importance in various graph mining and learning tasks. In particular, several recent works leverage fast node proximity computation to improve the scalability of Graph Neural Networks (GNN). However, prior studies on proximity computation and GNN feature propagation are on a case-by-case basis, with each paper focusing on a particular proximity measure. In this paper, we propose Approximate Graph Propagation (AGP), a unified randomized algorithm that computes various proximity queries and GNN feature propagation, including transition probabilities, Personalized PageRank, heat kernel PageRank, Katz, SGC, GDC, and APPNP. Our algorithm provides a theoretical bounded error guarantee and runs in almost optimal time complexity. We conduct an extensive experimental study to demonstrate AGPs effectiveness in two concrete applications: local clustering with heat kernel PageRank and node classification with GNNs. Most notably, we present an empirical study on a billion-edge graph Papers100M, the largest publicly available GNN dataset so far. The results show that AGP can significantly improve various existing GNN models scalability without sacrificing prediction accuracy.
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The problem of matching a query string to a directed graph, whose vertices are labeled by strings, has application in different fields, from data mining to computational biology. Several variants of the problem have been considered, depending on the fact that the match is exact or approximate and, in this latter case, which edit operations are considered and where are allowed. In this paper we present results on the complexity of the approximate matching problem, where edit operations are symbol substitutions and are allowed only on the graph labels or both on the graph labels and the query string. We introduce a variant of the problem that asks whether there exists a path in a graph that represents a query string with any number of edit operations and we show that is is NP-complete, even when labels have length one and in the case the alphabet is binary. Moreover, when it is parameterized by the length of the input string and graph labels have length one, we show that the problem is fixed-parameter tractable and it is unlikely to admit a polynomial kernel. The NP-completeness of this problem leads to the inapproximability (within any factor) of the approximate matching when edit operations are allowed only on the graph labels. Moreover, we show that the variants of approximate string matching to graph we consider are not fixed-parameter tractable, when the parameter is the number of edit operations, even for graphs that have distance one from a DAG. The reduction for this latter result allows us to prove the inapproximability of the variant where edit operations can be applied both on the query string and on graph labels.
Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [1983] (and further improved by Bondarenko, Prymak, and Radchenko [2013]). We revisit their proof idea that is based on expander graphs, and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function $h$ that is close to an approximately linear function $f$, while querying the value of $f$ on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, by this improving over a previous algorithm of Chierichetti, Das, Dasgupta and Kumar [2015] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on a Hadamard transform.
Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. We provide efficient constructions in both of these models, but our core results are for MPC. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds. Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use to construct distance sketches are an MPC construction of the hopsets of Elkin and Neiman (2016). This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting.
Approximate inference techniques are the cornerstone of probabilistic methods based on Gaussian process priors. Despite this, most work approximately optimizes standard divergence measures such as the Kullback-Leibler (KL) divergence, which lack the basic desiderata for the task at hand, while chiefly offering merely technical convenience. We develop a new approximate inference method for Gaussian process models which overcomes the technical challenges arising from abandoning these convenient divergences. Our method---dubbed Quantile Propagation (QP)---is similar to expectation propagation (EP) but minimizes the $L_2$ Wasserstein distance (WD) instead of the KL divergence. The WD exhibits all the required properties of a distance metric, while respecting the geometry of the underlying sample space. We show that QP matches quantile functions rather than moments as in EP and has the same mean update but a smaller variance update than EP, thereby alleviating EPs tendency to over-estimate posterior variances. Crucially, despite the significant complexity of dealing with the WD, QP has the same favorable locality property as EP, and thereby admits an efficient algorithm. Experiments on classification and Poisson regression show that QP outperforms both EP and variational Bayes.
We consider the problem of evaluating certain types of functional aggregation queries on relational data subject to additive inequalities. Such aggregation queries, with a smallish number of additive inequalities, arise naturally/commonly in many applications, particularly in learning applications. We give a relatively complete categorization of the computational complexity of such problems. We first show that the problem is NP-hard, even in the case of one additive inequality. Thus we turn to approximating the query. Our main result is an efficient algorithm for approximating, with arbitrarily small relative error, many natural aggregation queries with one additive inequality. We give examples of natural queries that can be efficiently solved using this algorithm. In contrast, we show that the situation with two additive inequalities is quite different, by showing that it is NP-hard to evaluate simple aggregation queries, with two additive inequalities, with any bounded relative error.
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