A New Simple Stochastic Gradient Descent Type Algorithm With Lower Computational Complexity for Bilevel Optimization

Abstract: Bilevel optimization has been widely used in many machine learning applications such as hyperparameter optimization and meta learning. Recently, many simple stochastic gradient descent(SGD) type algorithms(without using momentum and variance techniques) have been proposed to solve the bilevel optimization problems. However, all the existing simple SGD type algorithms estimate the hypergradient via stochastic estimation of Neumann series. In the paper, we propose to estimate the hypergradient via SGD-based Estimation(i.e., solving the linear system with SGD). By using warm start initialization strategy, a new simple SGD type algorithm SSGD based on SGD-based Estimation is proposed. We provide the convergence rate guarantee for SSGD and show that SSGD outperforms the best known computational complexity achieved by the existing simple SGD type algorithms. Our experiments validate our theoretical results and demonstrate the efficiency of our proposed algorithm SSGD in hyperparameter optimization applications.