  In this paper we consider linearly constrained optimization problems and
propose a loopless projection stochastic approximation (LPSA) algorithm. It
performs the projection with probability $p_n$ at the $n$-th iteration to
ensure feasibility. Considering a specific family of the probability $p_n$ and
step size $\eta_n$, we analyze our algorithm from an asymptotic and continuous
perspective. Using a novel jump diffusion approximation, we show that the
trajectories connecting those properly rescaled last iterates weakly converge
to the solution of specific stochastic differential equations (SDEs). By
analyzing SDEs, we identify the asymptotic behaviors of LPSA for different
choices of $(p_n, \eta_n)$. We find that the algorithm presents an intriguing
asymptotic bias-variance trade-off and yields phase transition phenomenons,
according to the relative magnitude of $p_n$ w.r.t. $\eta_n$. This finding
provides insights on selecting appropriate ${(p_n, \eta_n)}_{n \geq 1}$ to
minimize the projection cost. Additionally, we propose the Debiased LPSA
(DLPSA) as a practical application of our jump diffusion approximation result.
DLPSA is shown to effectively reduce projection complexity compared to vanilla
LPSA.
