Efficient Discovery of Association Rules and Frequent Itemsets through Sampling with Tight Performance Guarantees

The tasks of extracting (top-$K$) Frequent Itemsets (FI's) and Association Rules (AR's) are fundamental primitives in data mining and database applications. Exact algorithms for these problems exist and are widely used, but their running time is hindered by the need of scanning the entire dataset, possibly multiple times. High quality approximations of FI's and AR's are sufficient for most practical uses, and a number of recent works explored the application of sampling for fast discovery of approximate solutions to the problems. However, these works do not provide satisfactory performance guarantees on the quality of the approximation, due to the difficulty of bounding the probability of under- or over-sampling any one of an unknown number of frequent itemsets. In this work we circumvent this issue by applying the statistical concept of \emph{Vapnik-Chervonenkis (VC) dimension} to develop a novel technique for providing tight bounds on the sample size that guarantees approximation within user-specified parameters. Our technique applies both to absolute and to relative approximations of (top-$K$) FI's and AR's. The resulting sample size is linearly dependent on the VC-dimension of a range space associated with the dataset to be mined. The main theoretical contribution of this work is a proof that the VC-dimension of this range space is upper bounded by an easy-to-compute characteristic quantity of the dataset which we call \emph{d-index}, and is the maximum integer $d$ such that the dataset contains at least $d$ transactions of length at least $d$ such that no one of them is a superset of or equal to another. We show that this bound is strict for a large class of datasets.Comment: 19 pages, 7 figures. A shorter version of this paper appeared in the proceedings of ECML PKDD 201

Finding the True Frequent Itemsets

Frequent Itemsets (FIs) mining is a fundamental primitive in data mining. It requires to identify all itemsets appearing in at least a fraction $\theta$ of a transactional dataset $\mathcal{D}$. Often though, the ultimate goal of mining $\mathcal{D}$ is not an analysis of the dataset \emph{per se}, but the understanding of the underlying process that generated it. Specifically, in many applications $\mathcal{D}$ is a collection of samples obtained from an unknown probability distribution $\pi$ on transactions, and by extracting the FIs in $\mathcal{D}$ one attempts to infer itemsets that are frequently (i.e., with probability at least $\theta$) generated by $\pi$, which we call the True Frequent Itemsets (TFIs). Due to the inherently stochastic nature of the generative process, the set of FIs is only a rough approximation of the set of TFIs, as it often contains a huge number of \emph{false positives}, i.e., spurious itemsets that are not among the TFIs. In this work we design and analyze an algorithm to identify a threshold $\hat{\theta}$ such that the collection of itemsets with frequency at least $\hat{\theta}$ in $\mathcal{D}$ contains only TFIs with probability at least $1-\delta$, for some user-specified $\delta$. Our method uses results from statistical learning theory involving the (empirical) VC-dimension of the problem at hand. This allows us to identify almost all the TFIs without including any false positive. We also experimentally compare our method with the direct mining of $\mathcal{D}$ at frequency $\theta$ and with techniques based on widely-used standard bounds (i.e., the Chernoff bounds) of the binomial distribution, and show that our algorithm outperforms these methods and achieves even better results than what is guaranteed by the theoretical analysis.Comment: 13 pages, Extended version of work appeared in SIAM International Conference on Data Mining, 201

Flexible constrained sampling with guarantees for pattern mining

Pattern sampling has been proposed as a potential solution to the infamous pattern explosion. Instead of enumerating all patterns that satisfy the constraints, individual patterns are sampled proportional to a given quality measure. Several sampling algorithms have been proposed, but each of them has its limitations when it comes to 1) flexibility in terms of quality measures and constraints that can be used, and/or 2) guarantees with respect to sampling accuracy. We therefore present Flexics, the first flexible pattern sampler that supports a broad class of quality measures and constraints, while providing strong guarantees regarding sampling accuracy. To achieve this, we leverage the perspective on pattern mining as a constraint satisfaction problem and build upon the latest advances in sampling solutions in SAT as well as existing pattern mining algorithms. Furthermore, the proposed algorithm is applicable to a variety of pattern languages, which allows us to introduce and tackle the novel task of sampling sets of patterns. We introduce and empirically evaluate two variants of Flexics: 1) a generic variant that addresses the well-known itemset sampling task and the novel pattern set sampling task as well as a wide range of expressive constraints within these tasks, and 2) a specialized variant that exploits existing frequent itemset techniques to achieve substantial speed-ups. Experiments show that Flexics is both accurate and efficient, making it a useful tool for pattern-based data exploration.Comment: Accepted for publication in Data Mining & Knowledge Discovery journal (ECML/PKDD 2017 journal track

MCRapper: Monte-Carlo Rademacher Averages for Poset Families and Approximate Pattern Mining

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

Frequent Subgraph Mining via Sampling with Rigorous Guarantees

Frequent subgraph mining is a fundamental task in the analysis of collections of graphs that aims at finding all the subgraphs that appear with more than a user-specified frequency in the dataset. While several exact approaches have been proposed to solve the task, it remains computationally challenging on large graph datasets due to the complexity of the subgraph isomorphism problem inherent in the task and the huge number of candidate patterns even for fairly small subgraphs. In this thesis, we study two statistical learning measures of complexity, VC-dimension and Rademacher averages, for subgraphs, and derive efficiently computable bounds for both. We then show how such bounds can be applied to devise efficient sampling-based approaches for rigorously approximating the solutions of the frequent subgraph mining problem, providing sample sizes which are much tighter than what would be obtained by a straightforward application of Chernoff and union bounds. We also show that our bounds can be used for true frequent subgraph mining, which requires to identify subgraphs generated with probability above a given threshold using samples from an unknown generative process. Moreover, we carried out an extensive experimental evaluation of our methods on real datasets, which shows that our bounds lead to efficiently computable and high-quality approximations for both applications.Frequent subgraph mining is a fundamental task in the analysis of collections of graphs that aims at finding all the subgraphs that appear with more than a user-specified frequency in the dataset. While several exact approaches have been proposed to solve the task, it remains computationally challenging on large graph datasets due to the complexity of the subgraph isomorphism problem inherent in the task and the huge number of candidate patterns even for fairly small subgraphs. In this thesis, we study two statistical learning measures of complexity, VC-dimension and Rademacher averages, for subgraphs, and derive efficiently computable bounds for both. We then show how such bounds can be applied to devise efficient sampling-based approaches for rigorously approximating the solutions of the frequent subgraph mining problem, providing sample sizes which are much tighter than what would be obtained by a straightforward application of Chernoff and union bounds. We also show that our bounds can be used for true frequent subgraph mining, which requires to identify subgraphs generated with probability above a given threshold using samples from an unknown generative process. Moreover, we carried out an extensive experimental evaluation of our methods on real datasets, which shows that our bounds lead to efficiently computable and high-quality approximations for both applications