Efficient discovery of association rules and frequent itemsets through sampling with tight performance guarantees

• Authors: Matteo Riondato;Eli Upfal
• Venue: ECML PKDD'12 Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part I
• Year: 2012

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 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 characterization of the VC-dimension of this range space and a proof that it is upper bounded by an easy-to-compute characteristic quantity of the dataset which we call d-index, namely the maximum integer d such that the dataset contains at least d transactions of length at least d. We show that this bound is strict for a large class of datasets. The resulting sample size for an absolute (resp. relative) (ε, δ)-approximation of the collection of FI's is $O(\frac{1}{\varepsilon^2}(d+\log\frac{1}{\delta}))$ (resp. $O(\frac{2+\varepsilon}{\varepsilon^2(2-\varepsilon)\theta}(d\log\frac{2+\varepsilon}{(2-\varepsilon)\theta}+\log\frac{1}{\delta}))$) transactions, which is a significant improvement over previous known results. We present an extensive experimental evaluation of our technique on real and artificial datasets, demonstrating the practicality of our methods, and showing that they achieve even higher quality approximations than what is guaranteed by the analysis.