Differentiable Architecture Search (DARTS) is a recent neural architecture search (NAS) method based on a differentiable relaxation. Due to its success, numerous variants analyzing and improving parts of the DARTS framework have recently been proposed. By considering the problem as a constrained bilevel optimization, we present and analyze DARTS-PRIME, a variant including improvements to architectural weight update scheduling and regularization towards discretization. We propose a dynamic schedule based on per-minibatch network information to make architecture updates more informed, as well as proximity regularization to promote well-separated discretization. Our results in multiple domains show that DARTS-PRIME improves both performance and reliability, comparable to state-of-the-art in differentiable NAS.