Modern deep neural networks, while strong in many domains [29], have notmastered comparable language-basedgeneralization challenges, afactconjectured tounderlie their sample inefficiencyand inflexibility [26,25,8].
In this paper, we present a theory of stochastic optimal control (SOC) tailored to infinite-dimensional spaces, aiming to extend diffusion-based algorithms to function spaces. Specifically, we demonstrate how Doob's