Monte Carlo Samplers

MC-based acquisition functions rely on the reparameterization trick, which transforms some set of $\epsilon$ from some base distribution into a target distribution. For example, when drawing posterior samples from a Gaussian process, the classical parameterization is $\mu(x) + L(x) \epsilon$, where $\epsilon$ are i.i.d standard normal, $\mu$ is the mean of the posterior, and $L(x)$ is a root decomposition of the covariance matrix such that $L(x)L(x)^T = \Sigma(x)$.

Exactly how base samples are generated when using the reparameterization trick can have substantial effects on the convergence of gradients estimated from these samples. Because of this, BoTorch implements a generic module capable of flexible sampling from any type of probabilistic model.

A MCSampler is a Module that provides base samples from a Posterior object. These samplers may then in turn be used in conjunction with MC-based acquisition functions. BoTorch includes two types of MC samplers for sampling isotropic normal deviates: a vanilla, normal sampler (IIDNormalSampler) and randomized quasi-Monte Carlo sampler (SobolQMCNormalSampler).

For most use cases, we recommend using SobolQMCNormalSampler, as it tends to produce more accurate (i.e. lower variance) gradient estimates with much fewer samples relative to the IIDNormalSampler. To experiment with alternative sampling procedures, please see the source code for SobolQMCNormalSampler as an example.