A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning

Aug 3, 2026

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The Gumbel-max trick enables sampling from a categorical distribution using unnormalized (log-) probabilities. Recent machine learning work extends this idea to support tasks such as drawing multiple samples, sampling from structured domains, and gradient estimation for backpropagation in neural network optimization. This survey provides background on the original trick, a structured overview to help with algorithm selection, and a comprehensive mapping of related literature using Gumbel-based methods. It also highlights common design choices and outlines future perspectives.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

AccessFree to download

Introduction

Motivation for discrete stochastic modeling

Overview of the Gumbel-max trick

Sampling from categorical distributions

Extensions and applications

Multiple samples, structured domains, and gradient estimation

Survey scope and literature mapping

Design choices and future perspective

A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning Citation for published version (APA): Huijben, I. A. M. , Kool, W. , Paulus, M. B. , & van Sloun, R. J. G. (2023) . A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(2), 1353-1371 . Article 9729603. [https://doi.org/10.1109/TPAMI.2022.3157042](https://doi.org/10.1109/TPAMI.2022.3157042) Document license: TAVERNE DOI: 10.1109/TPAMI.2022.3157042 Document status and date: Published: 01/02/2023 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: [www.tue.nl/taverne](www.tue.nl/taverne) Take down policy If you believe that this document breaches copyright please contact us at: [openaccess@tue.nl](openaccess@tue.nl) providing details and we will investigate your claim. Download date: 02. Aug. 2026 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 45, NO. 2, FEBRUARY 2023 1353 A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning Iris A. M. Huijben , Student Member, IEEE, Wouter Kool , Max B. Paulus, and Ruud J. G. van Sloun , Member, IEEE Abstract—The Gumbel-max trick is a method to draw a sample from a categorical distribution, given by its unnormalized (log-) probabilities. Over the past years, the machine learning community has proposed several extensions of this trick to facilitate, e.g. , drawing multiple samples, sampling from structured domains, or gradient estimation for error backpropagation in neural network optimization. The goal of this survey article is to present background about the Gumbel-max trick, and to provide a structured overview of its extensions to ease algorithm selection. Moreover, it presents a comprehensive outline of (machine learning) literature in which Gumbel-based algorithms have been leveraged, reviews commonly-made design choices, and sketches a future perspective. Index Terms—Gumbel-max trick, sampling, gradient estimation, gumbel-softmax, categorical distribution, structured models ~~ ~~ Ç ~~ ~~ 1 INTRODUCTION THE world around us is discrete in many aspects. Think about decision making, e.g., in trafﬁc (Should I decelerate, accelerate or maintain a constant speed?), for product selection (Given that I liked the trousers from shop X last time, which new t

FAQs

What problem does the Gumbel-max trick solve in machine learning?

It provides a way to draw a sample from a categorical distribution using unnormalized (log-) probabilities, enabling discrete stochastic sampling within learning pipelines.

What kinds of extensions to the Gumbel-max trick are covered in the survey?

The survey discusses extensions for drawing multiple samples, sampling from structured domains, and gradient estimation to support error backpropagation in neural network optimization.

How does the survey help readers choose among Gumbel-based algorithms?

It offers background on the original trick and a structured overview of its extensions, along with a review of commonly made design choices and a mapping of relevant literature that uses Gumbel-based methods.

Aug 3, 2026

Upload by Logic