@inproceedings{zmigrod-etal-2021-higher,
title = "Higher-order Derivatives of Weighted Finite-state Machines",
author = "Zmigrod, Ran and
Vieira, Tim and
Cotterell, Ryan",
booktitle = "Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)",
month = aug,
year = "2021",
address = "Online",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2021.acl-short.32",
doi = "10.18653/v1/2021.acl-short.32",
pages = "240--248",
abstract = "Weighted finite-state machines are a fundamental building block of NLP systems. They have withstood the test of time{---}from their early use in noisy channel models in the 1990s up to modern-day neurally parameterized conditional random fields. This work examines the computation of higher-order derivatives with respect to the normalization constant for weighted finite-state machines. We provide a general algorithm for evaluating derivatives of all orders, which has not been previously described in the literature. In the case of second-order derivatives, our scheme runs in the optimal O(A{\^{}}2 N{\^{}}4) time where A is the alphabet size and N is the number of states. Our algorithm is significantly faster than prior algorithms. Additionally, our approach leads to a significantly faster algorithm for computing second-order expectations, such as covariance matrices and gradients of first-order expectations.",
}

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%0 Conference Proceedings
%T Higher-order Derivatives of Weighted Finite-state Machines
%A Zmigrod, Ran
%A Vieira, Tim
%A Cotterell, Ryan
%S Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)
%D 2021
%8 August
%I Association for Computational Linguistics
%C Online
%F zmigrod-etal-2021-higher
%X Weighted finite-state machines are a fundamental building block of NLP systems. They have withstood the test of time—from their early use in noisy channel models in the 1990s up to modern-day neurally parameterized conditional random fields. This work examines the computation of higher-order derivatives with respect to the normalization constant for weighted finite-state machines. We provide a general algorithm for evaluating derivatives of all orders, which has not been previously described in the literature. In the case of second-order derivatives, our scheme runs in the optimal O(A\² N\⁴) time where A is the alphabet size and N is the number of states. Our algorithm is significantly faster than prior algorithms. Additionally, our approach leads to a significantly faster algorithm for computing second-order expectations, such as covariance matrices and gradients of first-order expectations.
%R 10.18653/v1/2021.acl-short.32
%U https://aclanthology.org/2021.acl-short.32
%U https://doi.org/10.18653/v1/2021.acl-short.32
%P 240-248

##### Markdown (Informal)

[Higher-order Derivatives of Weighted Finite-state Machines](https://aclanthology.org/2021.acl-short.32) (Zmigrod et al., ACL-IJCNLP 2021)

##### ACL

- Ran Zmigrod, Tim Vieira, and Ryan Cotterell. 2021. Higher-order Derivatives of Weighted Finite-state Machines. In
*Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)*, pages 240–248, Online. Association for Computational Linguistics.