In this paper, we prove rigorous guarantees on the convergence performance of the EM algorithm and gradient EM algorithm. They first analyze the population level and they apply the results to the sample level. The researchers simulate the convergence rate with three models: Gaussian mixture models, the mixture of regressions, and regression with missing covariates. These examples showed the necessity of qualified initialization and the rate of convergence of the EM and gradient EM algorithms, which confirms that various aspects of the theoretical predictions. Overall, the goal of this paper is to develop some general tools for characterizing the suitably initialized sample-based EM algorithm, and their relation to MLE.