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.

本文提出了一种使用控制变量的 Plackett-Luce 分布来进行离散潜在变量学习模型的随机梯度下降的方法， 能够在非可微、离散和连续数据的因果关系学习任务中胜过其他对比松弛优化方法。

Plackett-Luce 分布的低方差黑盒梯度估计

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

We consider the problem of statistical inference for ranking data,
specifically rank aggregation, under the assumption that samples are incomplete
in the sense of not comprising all choice alternatives. In contrast to most
existing methods, we explicitly model the process of turning a full ranking
into an incomplete one, which we call the coarsening process. To this end, we
propose the concept of rank-dependent coarsening, which assumes that incomplete
rankings are produced by projecting a full ranking to a random subset of ranks.
For a concrete instantiation of our model, in which full rankings are drawn
from a Plackett-Luce distribution and observations take the form of pairwise
preferences, we study the performance of various rank aggregation methods. In
addition to predictive accuracy in the finite sample setting, we address the
theoretical question of consistency, by which we mean the ability to recover a
target ranking when the sample size goes to infinity, despite a potential bias
in the observations caused by the (unknown) coarsening.

本论文探讨了不完整排序数据以及基于 Plackett-Luce 分布的完整排序数据下的各种排序方法的表现，提出了考虑等级相关粗化的概念，并研究了当样本大小趋近于无穷大时，是否存在偏差导致恢复目标排序的一致性问题。