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

Monte Carlo optimization approach for decentralized estimation networks under communication constraints

We consider designing decentralized estimation schemes over bandwidth limited communication links with a particular interest in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We take two classes of in–network processing strategies into account which yield graph representations through modeling the sensor platforms as the vertices and the communication links by edges as well as a tractable Bayesian risk that comprises the cost of transmissions and penalty for the estimation errors. This approach captures a broad range of possibilities for “online” processing of observations as well as the constraints imposed and enables a rigorous design setting in the form of a constrained optimization problem. Similar schemes as well as the structures exhibited by the solutions to the design problem has been studied previously in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization schemes involve integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both classes of in–network processing strategies and their optimization. The proposed Monte Carlo optimization procedures operate in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk

Monte Carlo optimization approach for decentralized estimation networks under communication constraints

We consider designing decentralized estimation schemes over bandwidth limited communication links with a particular interest in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We take two classes of in–network processing strategies into account which yield graph representations through modeling the sensor platforms as the vertices and the communication links by edges as well as a tractable Bayesian risk that comprises the cost of transmissions and penalty for the estimation errors. This approach captures a broad range of possibilities for “online” processing of observations as well as the constraints imposed and enables a rigorous design setting in the form of a constrained optimization problem. Similar schemes as well as the structures exhibited by the solutions to the design problem has been studied previously in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization schemes involve integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both classes of in–network processing strategies and their optimization. The proposed Monte Carlo optimization procedures operate in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk

Techniques for improving clustering and association rules mining from very large transactional databases

Clustering and association rules mining are two core data mining tasks that have been actively studied by data mining community for nearly two decades. Though many clustering and association rules mining algorithms have been developed, no algorithm is better than others on all aspects, such as accuracy, efficiency, scalability, adaptability and memory usage. While more efficient and effective algorithms need to be developed for handling the large-scale and complex stored datasets, emerging applications where data takes the form of streams pose new challenges for the data mining community. The existing techniques and algorithms for static stored databases cannot be applied to the data streams directly. They need to be extended or modified, or new methods need to be developed to process the data streams.In this thesis, algorithms have been developed for improving efficiency and accuracy of clustering and association rules mining on very large, high dimensional, high cardinality, sparse transactional databases and data streams.A new similarity measure suitable for clustering transactional data is defined and an incremental clustering algorithm, INCLUS, is proposed using this similarity measure. The algorithm only scans the database once and produces clusters based on the user’s expectations of similarities between transactions in a cluster, which is controlled by the user input parameters, a similarity threshold and a support threshold. Intensive testing has been performed to evaluate the effectiveness, efficiency, scalability and order insensitiveness of the algorithm.To extend INCLUS for transactional data streams, an equal-width time window model and an elastic time window model are proposed that allow mining of clustering changes in evolving data streams. The minimal width of the window is determined by the minimum clustering granularity for a particular application. Two algorithms, CluStream_EQ and CluStream_EL, based on the equal-width window model and the elastic window model respectively, are developed by incorporating these models into INCLUS. Each algorithm consists of an online micro-clustering component and an offline macro-clustering component. The online component writes summary statistics of a data stream to the disk, and the offline components uses those summaries and other user input to discover changes in a data stream. The effectiveness and scalability of the algorithms are evaluated by experiments.This thesis also looks into sampling techniques that can improve efficiency of mining association rules in a very large transactional database. The sample size is derived based on the binomial distribution and central limit theorem. The sample size used is smaller than that based on Chernoff Bounds, but still provides the same approximation guarantees. The accuracy of the proposed sampling approach is theoretically analyzed and its effectiveness is experimentally evaluated on both dense and sparse datasets.Applications of stratified sampling for association rules mining is also explored in this thesis. The database is first partitioned into strata based on the length of transactions, and simple random sampling is then performed on each stratum. The total sample size is determined by a formula derived in this thesis and the sample size for each stratum is proportionate to the size of the stratum. The accuracy of transaction size based stratified sampling is experimentally compared with that of random sampling.The thesis concludes with a summary of significant contributions and some pointers for further work