Monte Carlo gradient estimation

September 30, 2020 — May 22, 2024

Bayes
calculus
density
estimator distribution
Monte Carlo
probabilistic algorithms
probability
risk
uncertainty
Figure 1

Taking gradients through integrals/expectations using randomness, i.e. can I estimate this?

\[ g[f] := \frac{\partial}{\partial \theta} \mathbb{E}_{\mathsf{x}\sim p(\mathsf{x};\theta)}[f(\mathsf{x})] \]

A concept 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 applications.

1 Score function estimator

A.k.a. REINFORCE (all-caps, for some reason?) A generic method that works on lots of things, including discrete variables; notoriously high variance if done naïvely. Credited to (Williams 1992), but surely it must be older than that?

\[ \hat{g}_{\text{REINFORCE}}(f) = f(\mathsf{x})\frac{\partial}{\partial \theta} \mathbb{E}_{\mathsf{x}\sim p(\mathsf{x};\theta)} \log p(\mathsf{x};\theta) \]

I am pretty sure this was called a “score function estimator” in my statistics degreee.

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 Reparameterization trick

Define some base distribution \(\mathsf{e}\sim p(\mathsf{e})\) such that \(f(\mathsf{x};\theta) \simeq f(T(\mathsf{e});\theta)\) for some transform \(T\). Then \[\begin{aligned} \hat{g}_{\text{reparam}}(f) &= \frac{\partial}{\partial \theta} \mathbb{E}_{\mathsf{e}\sim p(\mathsf{e})}[f(\mathsf{x})] \\ &= \mathbb{E}_{\mathsf{e}\sim p(\mathsf{e})}\left[\frac{\partial f}{\partial T}\frac{\partial T}{\partial \theta}(\mathsf{e};\theta)\right]. \end{aligned}\]

Less general but better-behaved than the score-function/REINFORCE estimator.

See reparameterization trick for more about that.

3 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

4 “Measure-valued”

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

5 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.

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, Choi, Wu, et al. 2018. Backpropagation Through the Void: Optimizing Control Variates for Black-Box Gradient Estimation.” In Proceedings of ICLR.
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.” In.
Mnih, and Gregor. 2014. Neural Variational Inference and Learning in Belief Networks.” In Proceedings of The 31st International Conference on Machine Learning. ICML’14.
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].
Prillo, and Eisenschlos. 2020. SoftSort: A Continuous Relaxation for the Argsort Operator.”
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.
Tucker, Mnih, Maddison, et al. 2017. REBAR: Low-Variance, Unbiased Gradient Estimates for Discrete Latent Variable Models.” In Proceedings of the 31st International Conference on Neural Information Processing Systems. NIPS’17.
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].
Williams. 1992. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning.” Machine Learning.