Show HN: Stochastic Gradient in Hilbert Spaces (zenodo.org)

A new manuscript from Lightcap develops a fully rigorous, end-to-end theory of stochastic gradient methods in infinite-dimensional Hilbert spaces. After building the minimal functional-analytic and measure-theoretic toolkit, the work shows that common practitioner definitions of a “stochastic gradient” in function spaces coincide under mild assumptions, proves well-posedness of both discrete- and continuous-time dynamics, and makes the continuum link to gradient-flow PDEs precise. The core technical deliverable is a suite of non-asymptotic convergence guarantees with explicit constants (not just big-O) across regimes: convex and strongly convex objectives, PL/KL-type nonconvex landscapes, heavy-tailed noise, and composite (proximal) models. The paper also separates weak vs. strong convergence, constructs the necessary martingale machinery from first principles, resolves infinite-dimensional measurability issues, and uses spectral analysis of linearized dynamics to explain mode-by-mode (slow-direction) behavior via the operator spectrum. Beyond theory, the manuscript treats Gaussian/RKHS settings, Hilbert manifolds, and carefully delineates what breaks in general Banach spaces. From a numerical perspective it analyzes five practical discretizations, proving stability + consistency ⇒ convergence, provides pseudocode whose cost tracks mesh size, step-size and accuracy ε, and shows fully discrete schemes converge to the infinite-dimensional limit with explicit error bounds. Four case studies—quantum ground states, elasticity, optimal control, and Bayesian inverse problems—illustrate applications, and a curated list of open problems maps next steps for stochastic optimization in function spaces.