“[A]llain Gersten, Hopfen, und Wasser”— 1516 Reinheitsgebot We presentBavarian, a collection of sampling-based algorithms for approximating the Betweenness Centrality (BC) of all vertices in a graph. Our algorithms use Monte-Carlo Empirical Rademacher Averages (MCERAs), a concept from statistical learning theory, to efficiently compute tight bounds on the maximum deviation of the estimates from the exact values. The MCERAs provide a sample-dependent approximation guarantee much stronger than the state-of-the-art, thanks to its use of variance-aware probabilistic tail bounds. The flexibility of the MCERAs allows us to introduce a unifying framework that can be instantiated with existing sampling-based estimators of BC, thus allowing a fair comparison between them, decoupled from the sample-complexity results with which they were originally introduced. Additionally, we prove novel sample-complexity results showing that, for all estimators, the sample size sufficient to achieve a desired approximation guarantee depends on the vertex-diameter of the graph, an easy-to-bound characteristic quantity. We also show progressive-sampling algorithms and extensions to other centrality measures, such as percolation centrality. Our extensive experimental evaluation ofBavarianshows the improvement over the state-of-the-art made possible by the MCERAs (2–4× reduction in the error bound), and it allows us to assess the different trade-offs between sample size and accuracy guarantees offered by the different estimators.

“I’m an MC still as honest” – Eminem, Rap God We present MCRapper , an algorithm for efficient computation of Monte-Carlo Empirical Rademacher Averages (MCERA) for families of functions exhibiting poset (e.g., lattice) structure, such as those that arise in many pattern mining tasks. The MCERA allows us to compute upper bounds to the maximum deviation of sample means from their expectations, thus it can be used to find both (1) statistically-significant functions (i.e., patterns) when the available data is seen as a sample from an unknown distribution, and (2) approximations of collections of high-expectation functions (e.g., frequent patterns) when the available data is a small sample from a large dataset. This flexibility offered by MCRapper is a big advantage over previously proposed solutions, which could only achieve one of the two. MCRapper uses upper bounds to the discrepancy of the functions to efficiently explore and prune the search space, a technique borrowed from pattern mining itself. To show the practical use of MCRapper , we employ it to develop an algorithm TFP-R for the task of True Frequent Pattern (TFP) mining, by appropriately computing approximations of the negative and positive borders of the collection of patterns of interest, which allow an effective pruning of the pattern space and the computation of strong bounds to the supremum deviation. TFP-R gives guarantees on the probability of including any false positives (precision) and exhibits higher statistical power (recall) than existing methods offering the same guarantees. We evaluate MCRapper and TFP-R and show that they outperform the state-of-the-art for their respective tasks.

We present Bavarian, a collection of sampling-based algorithms for approximating the Betweenness Centrality (BC) of all vertices in a graph. Our algorithms use Monte-Carlo Empirical Rademacher Averages (MCERAs), a concept from statistical learning theory, to efficiently compute tight bounds on the maximum deviation of the estimates from the exact values. The MCERAs provide a sample-dependent approximation guarantee much stronger than the state of the art, thanks to its use of variance-aware probabilistic tail bounds. The flexibility of the MCERA allows us to introduce a unifying framework that can be instantiated with existing sampling-based estimators of BC, thus allowing a fair comparison between them, decoupled from the sample-complexity results with which they were originally introduced. Additionally, we prove novel sample-complexity results showing that, for all estimators, the sample size sufficient to achieve a desired approximation guarantee depends on the vertex-diameter of the graph, an easy-to-bound characteristic quantity. We also show progressive-sampling algorithms and extensions to other centrality measures, such as percolation centrality. Our extensive experimental evaluation of Bavarian shows the improvement over the state-of-the art made possible by the MCERA, and it allows us to assess the different trade-offs between sample size and accuracy guarantee offered by the different estimators.

We present MCRapper, an algorithm for efficient computation of Monte-Carlo Empirical Rademacher Averages (MCERA) for families of functions exhibiting poset (e.g., lattice) structure, such as those that arise in many pattern mining tasks. The MCERA allows us to compute upper bounds to the maximum deviation of sample means from their expectations, thus it can be used to find both statistically-significant functions (i.e., patterns) when the available data is seen as a sample from an unknown distribution, and approximations of collections of high-expectation functions (e.g., frequent patterns) when the available data is a small sample from a large dataset. This feature is a strong improvement over previously proposed solutions that could only achieve one of the two. MCRapper uses upper bounds to the discrepancy of the functions to efficiently explore and prune the search space, a technique borrowed from pattern mining itself. To show the practical use of MCRapper, we employ it to develop an algorithm TFP-R for the task of True Frequent Pattern (TFP) mining. TFP-R gives guarantees on the probability of including any false positives (precision) and exhibits higher statistical power (recall) than existing methods offering the same guarantees. We evaluate MCRapper and TFP-R and show that they outperform the state-of-the-art for their respective tasks.

We define the k-Nearest Representatives (k-NR) classifier, a distance-based classifier similar to the k-nearest neighbors classifier with comparable accuracy in practice, and stronger generalization bounds. Uniform convergence is shown through Rademacher complexity, and generalizability is controlled through regularization. Finite-sample risk bound are also given. Compared to the k-NN, the k-NR requires less memory to store and classification queries may be made more efficiently. Training is also efficient, being polynomial in all parameters, and is accomplished via a simple empirical risk minimization process.