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.

翻译：受应急响应与实验设计应用的启发，我们考虑定义在欧几里得空间紧支集概率测度上的光滑随机优化问题。以影响函数作为变分对象，我们为概率空间构造了一种确定性Frank-Wolfe（dFW）递归算法——这特别得益于一个引理，该引理给出了无限维Frank-Wolfe子问题的“闭式”解。dFW的每次迭代均表示为当前迭代点与狄拉克测度的凸组合，该狄拉克测度集中于当前迭代点处影响函数的最小值点。针对仅能通过蒙特卡洛观测获取目标函数及影响函数的常见应用场景，我们构建了随机Frank-Wolfe（sFW）变体算法，该算法通过日益精确的影响函数估计的最小值序列，生成随机概率测度序列。我们证明：对于光滑凸目标函数，sFW的最优性间隙序列几乎必然且依期望具有$O(k^{-1})$的迭代复杂度；对于光滑非凸目标函数，其Frank-Wolfe间隙具有$O(k^{-1/2})$的复杂度。进一步，我们证明易于实现的固定步长、固定样本量的sFW变体算法具有指数收敛到$\varepsilon$最优解的特性。最后，我们建立了生成随机测度序列处观测目标值的中心极限定理。为增强直观理解，文中提供了多个包含精确影响函数计算的示例。