Monte Carlo gradient estimation
September 30, 2020 — May 2, 2023
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.
- Shakir Mohamed, Log Derivative Trick
- Syed Ashar Javed, REINFORCE vs Reparameterization Trick
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
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.