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Abstract We propose a very fast approximate Markov chain Monte Carlo sampling framework that is applicable to a large class of sparse Bayesian inference problems. The computational cost per iteration in several regression models is of order O(n(s+J)), where n is the sample size, s is the underlying sparsity of the model, and J is the size of a randomly selected subset of regressors. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin dynamics are employed. The algorithm is an extension of the asynchronous Gibbs sampler of Johnson et al. [(2013). Analyzing Hogwild parallel Gaussian Gibbs sampling. In Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS’13) (Vol. 2, pp. 2715–2723)], but can be viewed from a statistical perspective as a form of Bayesian iterated sure independent screening [Fan, J., Samworth, R., & Wu, Y. (2009). Ultrahigh dimensional feature selection: Beyond the linear model. Journal of Machine Learning Research, 10, 2013–2038]. We show that in high-dimensional linear regression problems, the Markov chain generated by the proposed algorithm admits an invariant distribution that recovers correctly the main signal with high probability under some statistical assumptions. Furthermore, we show that its mixing time is at most linear in the number of regressors. We illustrate the algorithm with several models.
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Max Markov Chain
In this paper, we introduce Max Markov Chain (MMC), a novel model for sequential data with sparse correlations among the state variables.It may also be viewed as a special class of approximate models for High-order Markov Chains (HMCs).MMC is desirable for domains where the sparse correlations are long-term and vary in their temporal stretches.Although generally intractable, parameter optimization for MMC can be solved analytically.However, based on this result,we derive an approximate solution that is highly efficient empirically.When compared with HMC and approximate HMC models, MMCcombines better sample efficiency, model parsimony, and an outstanding computational advantage.Such a quality allows MMC to scale to large domainswhere the competing models would struggle to perform.We compare MMC with several baselines with synthetic and real-world datasets to demonstrate MMC as a valuable alternative for stochastic modeling.
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- Award ID(s):
- 2047186
- PAR ID:
- 10505645
- Publisher / Repository:
- International Joint Conferences on Artificial Intelligence Organization
- Date Published:
- ISBN:
- 978-1-956792-03-4
- Page Range / eLocation ID:
- 5758 to 5767
- Format(s):
- Medium: X
- Location:
- Macau, SAR China
- Sponsoring Org:
- National Science Foundation
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