Abstract: It is well known that regression methods designed for clean data will lead to erroneous results if directly applied to corrupted data. Despite the recent methodological and algorithmic advances in Gaussian graphical model estimation, how to achieve efficient and scalable estimation under contaminated covariates is unclear. Here a new methodology called convex conditioned innovative scalable efficient estimation (COCOISEE) for Gaussian graphical models under both additive and multiplicative measurement errors is developed. It combines the strengths of the innovative scalable efficient estimation in the Gaussian graphical model and the nearest positive semidefinite matrix projection, thus enjoying stepwise convexity and scalability. Comprehensive theoretical guarantees are provided and the effectiveness of the proposed methodology is demonstrated through numerical studies.