# Efficient Computation of Expectations under Spanning Tree Distributions

arXiv: https://arxiv.org/abs/2008.12988 · HTML: https://mlsanity.com/paper/2008.12988/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Aug 2020.

## Built on by (most cited first, 11 of 11)

- [What Languages are Easy to Language-Model? A Perspective from Learning Probabilistic Regular Languages](https://mlsanity.com/paper/2406.04289/) (2024)
- [Differentiable Clustering with Perturbed Spanning Forests](https://mlsanity.com/paper/2305.16358/) (2023)
- [Measuring and Reducing Model Update Regression in Structured Prediction for NLP](https://mlsanity.com/paper/2202.02976/) (2022)
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- [Joint Learning of Energy-based Models and their Partition Function](https://mlsanity.com/paper/2501.18528/) (2025)
- [Unbiased and Efficient Sampling of Dependency Trees](https://mlsanity.com/paper/2205.12621/) (2022)
- [SynJax: Structured Probability Distributions for JAX](https://mlsanity.com/paper/2308.03291/) (2023)
- [Higher-order Derivatives of Weighted Finite-state Machines](https://mlsanity.com/paper/2106.00749/) (2021)
- [Efficient Sampling of Dependency Structures](https://mlsanity.com/paper/2109.06521/) (2021)
- [Syntactic Belief Update as the Driver of Garden Path Processing Difficulty](https://mlsanity.com/paper/2606.27206/) (2026)

## Abstract (arXiv)

We give a general framework for inference in spanning tree models. We propose unified algorithms for the important cases of first-order expectations and second-order expectations in edge-factored, non-projective spanning-tree models. Our algorithms exploit a fundamental connection between gradients and expectations, which allows us to derive efficient algorithms. These algorithms are easy to implement with or without automatic differentiation software. We motivate the development of our framework with several cautionary tales of previous research, which has developed numerous inefficient algorithms for computing expectations and their gradients. We demonstrate how our framework efficiently computes several quantities with known algorithms, including the expected attachment score, entropy, and generalized expectation criteria. As a bonus, we give algorithms for quantities that are missing in the literature, including the KL divergence. In all cases, our approach matches the efficiency of existing algorithms and, in several cases, reduces the runtime complexity by a factor of the sentence length. We validate the implementation of our framework through runtime experiments. We find our algorithms are up to 15 and 9 times faster than previous algorithms for computing the Shannon entropy and the gradient of the generalized expectation objective, respectively.