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arXiv · 2106.00749

Higher-order Derivatives of Weighted Finite-state Machines

Ran Zmigrod·Tim Vieira·Ryan Cotterell
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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 $\mathcal{O}(A^2 N^4)$ 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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Ran Zmigrod, Tim Vieira, Ryan Cotterell. 2021-06-01. Higher-order Derivatives of Weighted Finite-state Machines. https://arxiv.org/abs/2106.00749

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