Reparameterization methods for MC gradient estimation

Tutorials

• Shakir Mohamed, Machine Learning Trick of the Day (4): Reparameterisation Tricks:

Suppose we want the gradient of an expectation of a smooth function \(f\): \[ \nabla_\theta \mathbb {E}_{p(z; \theta)}[f (z)]=\nabla_\theta \int p(z; \theta) f (z) d z \] […] This gradient is often difficult to compute because the integral is typically unknown and the parameters \(\theta\), with respect to which we are computing the gradient, are of the distribution \(p(z; \theta)\).

Now we suppose that we know some function \(g\) such that for some easy distribution \(p(\epsilon)\), \(z | \theta=g(\epsilon, \theta)\). Now we can try to estimate the gradient of the expectation by Monte Carlo:

\[ \nabla_\theta \mathbb {E}_{p(z; \theta)}[f (z)]=\mathbb {E}_{p (c)}\left[\nabla_\theta f(g(\epsilon, \theta))\right] \] Let’s derive this expression and explore the implications of it for our optimisation problem. One-liners give us a transformation from a distribution \(p(\epsilon)\) to another \(p (z)\), thus the differential area (mass of the distribution) is invariant under the change of variables. This property implies that: \[ p (z)=\left|\frac{d \epsilon}{d z}\right—p(\epsilon) \Longrightarrow—p (z) d z|=|p(\epsilon) d \epsilon| \] Re-expressing the troublesome stochastic optimisation problem using random variate reparameterisation, we find: \[ \begin {aligned} & \nabla_\theta \mathbb {E}_{p(z; \theta)}[f (z)]=\nabla_\theta \int p(z; \theta) f (z) d z \\ = & \nabla_\theta \int p(\epsilon) f (z) d \epsilon=\nabla_\theta \int p(\epsilon) f(g(\epsilon, \theta)) d \epsilon \\ = & \nabla_\theta \mathbb {E}_{p (c)}[f(g(\epsilon, \theta))]=\mathbb {E}_{p (e)}\left[\nabla_\theta f(g(\epsilon, \theta))\right] \end {aligned} \]

• Yuge Shi’s variational inference tutorial is a tour of cunning reparameterisation gradient tricks. Written for her paper Shi et al. (2019). She punts some details to Mohamed et al. (2020) which in turn tells me that this adventure continues at reparameterization gradients and Monte Carlo gradient estimation. Figurnov, Mohamed, and Mnih (2018), Devroye (2006) and Jankowiak and Obermeyer (2018).

Normalizing flows

Cunning reparameterization maps with desirable properties for nonparametric inference. See normalizing flows.

General measure transport

See transport maps.

Tooling

Storchastic.

Incoming

Universal representation theorems? Probably many, here are some I saw: Perekrestenko, Müller, and Bölcskei (2020); Perekrestenko, Eberhard, and Bölcskei (2021).

References