Discovering causal relationships from observational data is a crucial problem and it has applications in many research areas. The PC algorithm is the state-of-the-art constraint based method for causal discovery. However, runtime of the PC algorithm, in the worst-case, is exponential to the number of nodes (variables), and thus it is inefficient when being applied to high dimensional data, e.g. gene expression datasets. On another note, the advancement of computer hardware in the last decade has resulted in the widespread availability of multi-core personal computers. There is a significant motivation for designing a parallelised PC algorithm that is suitable for personal computers and does not require end users’ parallel computing knowledge beyond their competency in using the PC algorithm. In this paper, we develop parallel-PC, a fast and memory efficient PC algorithm using the parallel computing technique. We apply our method to a range of synthetic and real-world high dimensional datasets. Experimental results on a dataset from the DREAM 5 challenge show that the original PC algorithm could not produce any results after running more than 24 hours; meanwhile, our parallel-PC algorithm managed to finish within around 12 hours with a 4-core CPU computer, and less than 6 hours with a 8-core CPU computer. Furthermore, we integrate parallel-PC into a causal inference method for inferring miRNA-mRNA regulatory relationships. The experimental results show that parallel-PC helps improve both the efficiency and accuracy of the causal inference algorithm.
Meanwhile, the other constraint based methods @cite_26 infer causal relationships by performing CI tests and searching for specific causal structures. For instance, Cooper @cite_56 proposed the LCD method to learn only a small portion of edges rather than the complete network. Specifically, the LCD algorithm searches for the CCC structures which consist of three pairs of correlated variables: ( @math , @math ), ( @math , @math ) and ( @math , @math ). If @math and @math become independent when conditioned on @math , we may infer that one of the following causal relations exists between @math , @math , and @math : @math ; @math ; and @math . @cite_28 proposed the similar method that focuses on the CCU structures. The CCU method searches for two correlated pairs @math and @math and an independent pair @math . When conditioning on @math , if @math becomes dependent then the causal relationships @math are concluded. Compared to the PC algorithm, such specific structure finding methods are more efficient, but the causal discoveries are incomplete.
@cite_28 @cite_26 @cite_56
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Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, Let @math be an @math real random matrix whose elements are independent and identically distributed standard normal random variables, and let @math be the 2-norm condition number of @math . We prove that, for any @math , @math , and @math , @math satisfies @math where @math and @math @math are universal positive constants independent of @math , @math , and @math . Moreover, for any @math and @math , @math A similar pair of results for complex Gaussian random matrices is also established.