Monte Carlo gradient estimation

Taking gradients through integrals using randomness. A thing with similar name but which is not the same is Stochastic Gradient MCMC which uses stochastic gradients to sample from a target posterior distribution. Some similar tools and concepts pop up in both uses.

Score function estimator

A.k.a. REINFORCE, all-caps, for some reason. Could do with a decent intro. TBD.

A very generic method that works on lots of things, including discrete variables; however, notoriously high variance if done naïvely.

For unifying overviews see (Mohamed et al. 2020; Schulman et al. 2015; van Krieken, Tomczak, and Teije 2021) and the Storchastic docs.


Rao-Blackwellization (Casella and Robert 1996) seems like a natural trick gradient estimators. How would it work? Liu et al. (2019) is a contemporary example; I have a vague feeling that I saw something similar in Reuven Y. Rubinstein and Kroese (2016). TODO: follow up.


I can imagine that our observed rv \({\mathsf{x}}\in \mathbb{R}\) is generated via lookups from its iCDF \(F(\cdot;\theta)\) with parameter \(\theta\): \[\mathsf{x} = F^{-1}(\mathsf{u};\theta) \] where \(\mathsf{u}\sim\operatorname{Uniform}(0,1)\). Each realization corresponds to a choice of \(u_i\sim \mathsf{u}\) independently. How can I get the derivative of such a map?

Maybe I generated my original variable not by the icdf method but by simulating some variable \({\mathsf{z}}\sim F(\cdot; \theta).\) In which case I may as well have generated those \(\mathsf{u}_i\) by taking \(\mathsf{u}_i=F(\mathsf{z}_i;\theta)\) for some \(\mathsf{z} \sim F(\cdot;\theta)\) and I am conceptually generating my RV by fixing \(z_i\sim\mathsf{z}_i\) and taking \(\phi := F^{-1}(F(z_i;\theta);\tau).\) So to find the effect of my perturbation what I actually need is

\[\begin{aligned} \left.\frac{\partial}{\partial \tau} F^{-1}(F(z;\theta);\tau)\right|_{\tau=\theta}\\ \end{aligned}\]

Does this do what we want? Kinda. So suppose that the parameters in question are something boring, such as the location parameter of a location-scale distribution, i.e. \(F(\cdot;\theta)=F(\cdot-\theta;0).\) Then \(F^{-1}(\cdot;\theta)=F^{-1}(\cdot;0)+\theta\) and thus

\[\begin{aligned} \left.\frac{\partial}{\partial \tau} F^{-1}(F(z;\theta);\tau)\right|_{\tau=\theta} &=\left.\frac{\partial}{\partial \tau} F^{-1}(F(z-\theta;0);0)+\tau\right|_{\tau=\theta}\\ &=\left.\frac{\partial}{\partial \tau}\left(z-\theta+\tau\right)\right|_{\tau=\theta}\\ &=1\\ \end{aligned}\]

OK grand that came out simple enough.



TBD (Mohamed et al. 2020; Rosca et al. 2019).


van Krieken, Tomczak, and Teije (2021) supplies us with a large library of pytorch tools for stochastic gradient estimation purposes, under the rubric Storchastic. (Source.). See also Deepmind’s mc_gradients.

Reparameterization trick

See reparameterization trick.

Optimising Monte Carlo

Let us say I need to differentiate through a monte carlo algorithm to alter its parameters while holding the PRNG fixed. See Tuning MC.


Ahn, Sungjin, Anoop Korattikara, and Max Welling. 2012. Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring.” In Proceedings of the 29th International Coference on International Conference on Machine Learning, 1771–78. ICML’12. Madison, WI, USA: Omnipress.
Arya, Gaurav, Moritz Schauer, Frank Schäfer, and Christopher Vincent Rackauckas. 2022. Automatic Differentiation of Programs with Discrete Randomness.” In.
Blundell, Charles, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. 2015. Weight Uncertainty in Neural Networks.” In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, 1613–22. ICML’15. Lille, France:
Casella, George, and Christian P. Robert. 1996. Rao-Blackwellisation of Sampling Schemes.” Biometrika 83 (1): 81–94.
Fu, Michael. 2005. Stochastic Gradient Estimation,” 32.
Glasserman, Paul, and Yu-Chi Ho. 1991. Gradient Estimation Via Perturbation Analysis. Springer Science & Business Media.
Grathwohl, Will, Kevin Swersky, Milad Hashemi, David Duvenaud, and Chris J. Maddison. 2021. Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.” arXiv.
Hyvärinen, Aapo. 2005. Estimation of Non-Normalized Statistical Models by Score Matching.” The Journal of Machine Learning Research 6 (December): 695–709.
Krieken, Emile van, Jakub M. Tomczak, and Annette ten Teije. 2021. Storchastic: A Framework for General Stochastic Automatic Differentiation.” In arXiv:2104.00428 [Cs, Stat].
Liu, Runjing, Jeffrey Regier, Nilesh Tripuraneni, Michael I. Jordan, and Jon McAuliffe. 2019. Rao-Blackwellized Stochastic Gradients for Discrete Distributions.” arXiv.
Mohamed, Shakir, Mihaela Rosca, Michael Figurnov, and Andriy Mnih. 2020. Monte Carlo Gradient Estimation in Machine Learning.” Journal of Machine Learning Research 21 (132): 1–62.
Oktay, Deniz, Nick McGreivy, Joshua Aduol, Alex Beatson, and Ryan P. Adams. 2020. Randomized Automatic Differentiation.” arXiv:2007.10412 [Cs, Stat], July.
Ranganath, Rajesh, Sean Gerrish, and David M. Blei. 2013. Black Box Variational Inference.” arXiv:1401.0118 [Cs, Stat], December.
Richter, Lorenz, Ayman Boustati, Nikolas Nüsken, Francisco J. R. Ruiz, and Ömer Deniz Akyildiz. 2020. VarGrad: A Low-Variance Gradient Estimator for Variational Inference.” arXiv.
Rosca, Mihaela, Michael Figurnov, Shakir Mohamed, and Andriy Mnih. 2019. “Measure–Valued Derivatives for Approximate Bayesian Inference.” In NeurIPS Workshop on Approximate Bayesian Inference.
Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. Simulation and the Monte Carlo Method. 3 edition. Wiley series in probability and statistics. Hoboken, New Jersey: Wiley.
Rubinstein, Reuven Y, and Dirk P Kroese. 2004. The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. New York, NY: Springer New York.
Schulman, John, Nicolas Heess, Theophane Weber, and Pieter Abbeel. 2015. Gradient Estimation Using Stochastic Computation Graphs.” In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2, 3528–36. NIPS’15. Cambridge, MA, USA: MIT Press.
Shi, Jiaxin, Shengyang Sun, and Jun Zhu. 2018. A Spectral Approach to Gradient Estimation for Implicit Distributions.” In. arXiv.
Stoker, Thomas M. 1986. Consistent Estimation of Scaled Coefficients.” Econometrica 54 (6): 1461–81.
Walder, Christian J., Paul Roussel, Richard Nock, Cheng Soon Ong, and Masashi Sugiyama. 2019. New Tricks for Estimating Gradients of Expectations.” arXiv:1901.11311 [Cs, Stat], June.

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