# Low-variance Black-box Gradient Estimates for the Plackett-Luce Distribution

arXiv: https://arxiv.org/abs/1911.10036 · HTML: https://mlsanity.com/paper/1911.10036/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2019.

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

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- [Neural Topological Ordering for Computation Graphs](https://mlsanity.com/paper/2207.05899/) (2022)
- [Leveraging Recursive Gumbel-Max Trick for Approximate Inference in Combinatorial Spaces](https://mlsanity.com/paper/2110.15072/) (2021)
- [Large (Vision) Language Models are Unsupervised In-Context Learners](https://mlsanity.com/paper/2504.02349/) (2025)
- [Probabilistic Permutation Graph Search: Black-Box Optimization for Fairness in Ranking](https://mlsanity.com/paper/2204.13765/) (2022)
- [Learning Multiscale Non-stationary Causal Structures](https://mlsanity.com/paper/2208.14989/) (2022)
- [Fast Slate Policy Optimization: Going Beyond Plackett-Luce](https://mlsanity.com/paper/2308.01566/) (2023)
- [ReCache: Learning Budget-Aware Caching Schedules for Diffusion Models via REINFORCE](https://mlsanity.com/paper/2606.06060/) (2026)
- [PACER: Acyclic Causal Discovery from Large-Scale Interventional Data](https://mlsanity.com/paper/2605.15353/) (2026)
- [Scoring Is Not Enough: Addressing Gaps in Utility-fairness Trade-offs for Ranking](https://mlsanity.com/paper/2606.26369/) (2026)
- [Rank-Conditioned Sample Reuse for the Plackett--Luce Best-of-K Objective](https://mlsanity.com/paper/2607.11146/) (2026)

## Abstract (arXiv)

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.