  Motivated by applications in emergency response and experimental design, we
consider smooth stochastic optimization problems over probability measures
supported on compact subsets of the Euclidean space. With the influence
function as the variational object, we construct a deterministic Frank-Wolfe
(dFW) recursion for probability spaces, made especially possible by a lemma
that identifies a ``closed-form'' solution to the infinite-dimensional
Frank-Wolfe sub-problem. Each iterate in dFW is expressed as a convex
combination of the incumbent iterate and a Dirac measure concentrating on the
minimum of the influence function at the incumbent iterate. To address common
application contexts that have access only to Monte Carlo observations of the
objective and influence function, we construct a stochastic Frank-Wolfe (sFW)
variation that generates a random sequence of probability measures constructed
using minima of increasingly accurate estimates of the influence function. We
demonstrate that sFW's optimality gap sequence exhibits $O(k^{-1})$ iteration
complexity almost surely and in expectation for smooth convex objectives, and
$O(k^{-1/2})$ (in Frank-Wolfe gap) for smooth non-convex objectives.
Furthermore, we show that an easy-to-implement fixed-step, fixed-sample version
of (sFW) exhibits exponential convergence to $\varepsilon$-optimality. We end
with a central limit theorem on the observed objective values at the sequence
of generated random measures. To further intuition, we include several
illustrative examples with exact influence function calculations.
