  Variational inference is a technique that approximates a target distribution
by optimizing within the parameter space of variational families. On the other
hand, Wasserstein gradient flows describe optimization within the space of
probability measures where they do not necessarily admit a parametric density
function. In this paper, we bridge the gap between these two methods. We
demonstrate that, under certain conditions, the Bures-Wasserstein gradient flow
can be recast as the Euclidean gradient flow where its forward Euler scheme is
the standard black-box variational inference algorithm. Specifically, the
vector field of the gradient flow is generated via the path-derivative gradient
estimator. We also offer an alternative perspective on the path-derivative
gradient, framing it as a distillation procedure to the Wasserstein gradient
flow. Distillations can be extended to encompass $f$-divergences and
non-Gaussian variational families. This extension yields a new gradient
estimator for $f$-divergences, readily implementable using contemporary machine
learning libraries like PyTorch or TensorFlow.
