🤖 AI Summary
This work addresses smooth stochastic optimization of probability measures over compact subsets of Euclidean space, motivated by applications in emergency response and experimental design. We extend the Frank–Wolfe (FW) algorithm to the infinite-dimensional space of probability measures—its first such generalization—and prove that subproblem solutions are necessarily Dirac measures concentrated at extreme points. We propose a stochastic FW algorithm with Monte Carlo sampling (sFW) and establish rigorous convergence guarantees: for convex objectives, the optimization gap is (O(1/k)) almost surely and in expectation; for nonconvex objectives, the FW gap is (O(1/sqrt{k})); and with fixed step size and sample size, exponential convergence to an (varepsilon)-optimal solution is achieved. Additionally, we derive a central limit theorem for the objective value sequence. Our core contribution lies in systematically lifting the classical finite-dimensional FW framework to the infinite-dimensional space of probability measures, yielding a theoretically grounded and practically implementable stochastic optimization method.
📝 Abstract
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. The dFW recursion is made especially possible by a lemma that identifies the solution to the infinite-dimensional Frank–Wolfe subproblem as a Dirac measure concentrating on the minimum of the influence function at the incumbent iterate. Each iterate in the dFW recursion is thus expressed through a “particle update,” as a convex combination of the incumbent iterate and a Dirac measure. 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 the sFW optimality gap sequence exhibits [Formula: see text] iteration complexity almost surely and in expectation for smooth convex objectives, and [Formula: see text] (in the Frank–Wolfe gap) for smooth nonconvex objectives. Furthermore, we show that an easy-to-implement fixed-step, fixed-sample version of the sFW method exhibits exponential convergence to [Formula: see text]-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. Funding: This work was partially supported by the National Science Foundation [Grants CMMI-2035086, DMS-2230023, and OAC-2410950] and by the Office of Naval Research [Grants 13000991 and N000141712295].