Bilevel optimization is a method used in machine learning for tasks like hyperparameter optimization. New algorithms have been developed to solve these problems faster. Two new algorithms were introduced in this study: one uses momentum-based recursive iterations, and the other uses gradient estimations in nested loops to reduce variance. Both algorithms have been shown to be significantly faster than existing methods, with a complexity of $\mathcal{\widetilde O}(\epsilon^{-1.5})$. Experimental results confirm the superior performance of these algorithms in hyperparameter applications.