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In this paper we consider linearly constrained optimization problems andpropose a loopless projection stochastic approximation (LPSA) algorithm. Itperforms the projection with probability $p_n$ at the $n$-th iteration toensure feasibility. Considering a specific family of the probability $p_n$ andstep size $\eta_n$, we analyze our algorithm from an asymptotic and continuousperspective. Using a novel jump diffusion approximation, we show that thetrajectories connecting those properly rescaled last iterates weakly convergeto the solution of specific stochastic differential equations (SDEs). Byanalyzing SDEs, we identify the asymptotic behaviors of LPSA for differentchoices of $(p_n, \eta_n)$. We find that the algorithm presents an intriguingasymptotic bias-variance trade-off and yields phase transition phenomenons,according to the relative magnitude of $p_n$ w.r.t. $\eta_n$. This findingprovides insights on selecting appropriate ${(p_n, \eta_n)}_{n \geq 1}$ tominimize 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 vanillaLPSA.

Asymptotic Behaviors and Phase Transitions in Projected Stochastic Approximation: A Jump Diffusion Approach