We obtain improved distributed algorithms in the CONGEST message-passing setting for problems on power graphs of an input graph $G$. This includes Coloring, Maximal Independent Set, and related problems. We develop a general deterministic technique that transforms R-round algorithms for $G$ with certain properties into $O(R \cdot \Delta^{k/2 - 1})$-round algorithms for $G^k$. This improves the previously-known running time for such transformation, which was $O(R \cdot \Delta^{k - 1})$. Consequently, for problems that can be solved by algorithms with the required properties and within polylogarithmic number of rounds, we obtain {quadratic} improvement for $G^k$ and {exponential} improvement for $G^2$. We also obtain significant improvements for problems with larger number of rounds in $G$.
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In this study, we have developed an incremental machine learning (ML) method that efficiently obtains the optimal model when a small number of instances or features are added or removed. This problem holds practical importance in model selection, such as cross-validation (CV) and feature selection. Among the class of ML methods known as linear estimators, there exists an efficient model update framework called the low-rank update that can effectively handle changes in a small number of rows and columns within the data matrix. However, for ML methods beyond linear estimators, there is currently no comprehensive framework available to obtain knowledge about the updated solution within a specific computational complexity. In light of this, our study introduces a method called the Generalized Low-Rank Update (GLRU) which extends the low-rank update framework of linear estimators to ML methods formulated as a certain class of regularized empirical risk minimization, including commonly used methods such as SVM and logistic regression. The proposed GLRU method not only expands the range of its applicability but also provides information about the updated solutions with a computational complexity proportional to the amount of dataset changes. To demonstrate the effectiveness of the GLRU method, we conduct experiments showcasing its efficiency in performing cross-validation and feature selection compared to other baseline methods.
We show that any one-round algorithm that computes a minimum spanning tree (MST) in the unicast congested clique must use a link bandwidth of $\Omega(\log^3 n)$ bits in the worst case. Consequently, computing an MST under the standard assumption of $O(\log n)$-size messages requires at least $2$ rounds. This is the first round complexity lower bound in the unicast congested clique for a problem where the output size is small, i.e., $O(n\log n)$ bits. Our lower bound holds as long as every edge of the MST is output by an incident node. To the best of our knowledge, all prior lower bounds for the unicast congested clique either considered problems with large output sizes (e.g., triangle enumeration) or required every node to learn the entire output.
We describe a simple parallel-friendly lightweight graph reordering algorithm for COO graphs (edge lists). Our ``Batched Order By Attachment'' (BOBA) algorithm is linear in the number of edges in terms of reads and linear in the number of vertices for writes through to main memory. It is highly parallelizable on GPUs\@. We show that, compared to a randomized baseline, the ordering produced gives improved locality of reference in sparse matrix-vector multiplication (SpMV) as well as other graph algorithms. Moreover, it can substantially speed up the conversion from a COO representation to the compressed format CSR, a very common workflow. Thus, it can give \emph{end-to-end} speedups even in SpMV\@. Unlike other lightweight approaches, this reordering does not rely on explicitly knowing the degrees of the vertices, and indeed its runtime is comparable to that of computing degrees. Instead, it uses the structure and edge distribution inherent in the input edge list, making it a candidate for default use in a pragmatic graph creation pipeline. This algorithm is suitable for road-type networks as well as scale-free. It improves cache locality on both CPUs and GPUs, achieving hit rates similar to the heavyweight techniques (e.g., for SpMV, 7--52\% and 11--67\% in the L1 and L2 caches, respectively). Compared to randomly labeled graphs, BOBA-reordered graphs achieve end-to-end speedups of up to 3.45. The reordering time is approximately one order of magnitude faster than existing lightweight techniques and up to 2.5 orders of magnitude faster than heavyweight techniques.
Value at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, tools from Extreme Value Theory are usually employed, considering that the tail data follow a Generalized Pareto distribution (GPD). Using the existing relations from the parameters of the baseline distribution and the limit GPD's parameters, we define highly informative priors that incorporate all the information available for the whole set of observations. We show how to perform Metropolis-Hastings (MH) algorithm to estimate VaR and CVaR employing the highly informative priors, in the case of exponential, stable and Gamma distributions. Afterwards, we perform a thorough simulation study to compare the accuracy and precision provided by three different methods. Finally, data from a real example is analyzed to show the practical application of the methods.
For population studies or for the training of complex machine learning models, it is often required to gather data from different actors. In these applications, summation is an important primitive: for computing means, counts or mini-batch gradients. In many cases, the data is privacy-sensitive and therefore cannot be collected on a central server. Hence the summation needs to be performed in a distributed and privacy-preserving way. Existing solutions for distributed summation with computational privacy guarantees make trust or connection assumptions - e.g., the existence of a trusted server or peer-to-peer connections between clients - that might not be fulfilled in real world settings. Motivated by these challenges, we propose Secure Summation via Subset Sums (S5), a method for distributed summation that works in the presence of a malicious server and only two honest clients, and without the need for peer-to-peer connections between clients. S5 adds zero-sum noise to clients' messages and shuffles them before sending them to the aggregating server. Our main contribution is a proof that this scheme yields a computational privacy guarantee based on the multidimensional subset sum problem. Our analysis of this problem may be of independent interest for other privacy and cryptography applications.
In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result in [Auza and Lee, 2021] which shows an $\Omega(n)$ lower bound for zero-error randomized algorithms. %To our knowledge, it is the only regime of this problem where we have upper and lower bounds tight up to constants. These questions have been extensively studied in the past few years, especially due to the problem's connections to symmetric submodular function minimization. We also describe a simple polynomial time deterministic algorithm that makes $O(\frac{n\log n}{\log\log n})$ queries on undirected unweighted graphs and returns a maximal spanning forest, thereby (slightly) improving upon the state-of-the-art.
Graph neural networks generalize conventional neural networks to graph-structured data and have received widespread attention due to their impressive representation ability. In spite of the remarkable achievements, the performance of Euclidean models in graph-related learning is still bounded and limited by the representation ability of Euclidean geometry, especially for datasets with highly non-Euclidean latent anatomy. Recently, hyperbolic space has gained increasing popularity in processing graph data with tree-like structure and power-law distribution, owing to its exponential growth property. In this survey, we comprehensively revisit the technical details of the current hyperbolic graph neural networks, unifying them into a general framework and summarizing the variants of each component. More importantly, we present various HGNN-related applications. Last, we also identify several challenges, which potentially serve as guidelines for further flourishing the achievements of graph learning in hyperbolic spaces.
The aim of this work is to develop a fully-distributed algorithmic framework for training graph convolutional networks (GCNs). The proposed method is able to exploit the meaningful relational structure of the input data, which are collected by a set of agents that communicate over a sparse network topology. After formulating the centralized GCN training problem, we first show how to make inference in a distributed scenario where the underlying data graph is split among different agents. Then, we propose a distributed gradient descent procedure to solve the GCN training problem. The resulting model distributes computation along three lines: during inference, during back-propagation, and during optimization. Convergence to stationary solutions of the GCN training problem is also established under mild conditions. Finally, we propose an optimization criterion to design the communication topology between agents in order to match with the graph describing data relationships. A wide set of numerical results validate our proposal. To the best of our knowledge, this is the first work combining graph convolutional neural networks with distributed optimization.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
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