To illustrate this phenomenon we consider a simple example of stochastic optimization problem andapplySGDwithconstant stepsizetosolveit.

Neural transport augmented sampling, firstintroduced byParnoandMarzouk (2018),isageneral method for using normalizing flows to sample from a given densityπ. Thus, samples can be generated fromπ(θ)by running MCMC chain in theZ-space and pushing these samples onto theΘ-space usingT. Neural transport augmented samplers havebeen subsequently extended by Hoffman etal. In this paper, we proposed equivariant Stein variational gradient descent algorithm for sampling fromdensities thatareinvarianttosymmetry transformations. Another contributionofourworkis subsequently using this equivariant sampling method to efficiently train equivariant energy based models forprobabilistic modeling andinference.

Subsequently,wedefineequivariant energy based models to model invariant densities that are learned using contrastivedivergence.