Learnability of Parameter-Bounded Bayes Nets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911976653848576 |
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| author | Bhattacharyya, Arnab Choo, Davin Gayen, Sutanu Myrisiotis, Dimitrios |
| author_facet | Bhattacharyya, Arnab Choo, Davin Gayen, Sutanu Myrisiotis, Dimitrios |
| contents | Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations. In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution $\mathbb{P}$, that is defined as the marginal distribution of a Bayes net, it is $\mathsf{NP}$-hard to decide whether there is a parameter-bounded Bayes net that represents $\mathbb{P}$. They called this problem LEARN. In this work, we extend the $\mathsf{NP}$-hardness result of LEARN and prove the $\mathsf{NP}$-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net. We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution $\mathbb{P}$, that is represented by some parameter-bounded Bayes net, generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learnability of Parameter-Bounded Bayes Nets Bhattacharyya, Arnab Choo, Davin Gayen, Sutanu Myrisiotis, Dimitrios Machine Learning Computational Complexity Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations. In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution $\mathbb{P}$, that is defined as the marginal distribution of a Bayes net, it is $\mathsf{NP}$-hard to decide whether there is a parameter-bounded Bayes net that represents $\mathbb{P}$. They called this problem LEARN. In this work, we extend the $\mathsf{NP}$-hardness result of LEARN and prove the $\mathsf{NP}$-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net. We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution $\mathbb{P}$, that is represented by some parameter-bounded Bayes net, generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020). |
| title | Learnability of Parameter-Bounded Bayes Nets |
| topic | Machine Learning Computational Complexity |
| url | https://arxiv.org/abs/2407.00927 |