Monte Carlo gradient estimation

September 30, 2020 — May 2, 2023

Figure 1

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.

1 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.

1.1 Rao-Blackwellization

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.

2 Parametric

I can imagine that our observed rv \({\mathsf{x}}\in \mathbb{R}\) is generated via lookups from its iCDF $F(;) $ 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.

TBC

3 “Measure-valued”

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

4 Tooling

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.

5 Reparameterization trick

See reparameterization trick.

6 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.

7 References

Ahn, Korattikara, and Welling. 2012. Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring.” In Proceedings of the 29th International Coference on International Conference on Machine Learning. ICML’12.
Arya, Schauer, Schäfer, et al. 2022. Automatic Differentiation of Programs with Discrete Randomness.” In.
Blundell, Cornebise, Kavukcuoglu, et al. 2015. Weight Uncertainty in Neural Networks.” In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37. ICML’15.
Casella, and Robert. 1996. Rao-Blackwellisation of Sampling Schemes.” Biometrika.
Fu. 2005. Stochastic Gradient Estimation.”
Glasserman, and Ho. 1991. Gradient Estimation Via Perturbation Analysis.
Grathwohl, Swersky, Hashemi, et al. 2021. Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.”
Hyvärinen. 2005. Estimation of Non-Normalized Statistical Models by Score Matching.” The Journal of Machine Learning Research.
Liu, Regier, Tripuraneni, et al. 2019. Rao-Blackwellized Stochastic Gradients for Discrete Distributions.”
Mohamed, Rosca, Figurnov, et al. 2020. Monte Carlo Gradient Estimation in Machine Learning.” Journal of Machine Learning Research.
Oktay, McGreivy, Aduol, et al. 2020. Randomized Automatic Differentiation.” arXiv:2007.10412 [Cs, Stat].
Ranganath, Gerrish, and Blei. 2013. Black Box Variational Inference.” arXiv:1401.0118 [Cs, Stat].
Richter, Boustati, Nüsken, et al. 2020. VarGrad: A Low-Variance Gradient Estimator for Variational Inference.”
Rosca, Figurnov, Mohamed, et al. 2019. “Measure–Valued Derivatives for Approximate Bayesian Inference.” In NeurIPS Workshop on Approximate Bayesian Inference.
Rubinstein, Reuven Y, and Kroese. 2004. The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning.
Rubinstein, Reuven Y., and Kroese. 2016. Simulation and the Monte Carlo Method. Wiley series in probability and statistics.
Schulman, Heess, Weber, et al. 2015. Gradient Estimation Using Stochastic Computation Graphs.” In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2. NIPS’15.
Shi, Sun, and Zhu. 2018. A Spectral Approach to Gradient Estimation for Implicit Distributions.” In.
Stoker. 1986. Consistent Estimation of Scaled Coefficients.” Econometrica.
van Krieken, Tomczak, and Teije. 2021. Storchastic: A Framework for General Stochastic Automatic Differentiation.” In arXiv:2104.00428 [Cs, Stat].
Walder, Roussel, Nock, et al. 2019. New Tricks for Estimating Gradients of Expectations.” arXiv:1901.11311 [Cs, Stat].