Randomized Stochastic Variance-Reduced Methods for Stochastic Bilevel Optimization

In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To address these limitations, this paper proposes fast randomized stochastic algorithms for non-convex SBO problems. First, we present a stochastic method for non-convex SBO with only one lower problem and establish its sample complexity of $O(1/\epsilon^3)$ for finding an $\epsilon$-stationary point under appropriate conditions, matching the lower bound for stochastic smooth non-convex optimization. Second, we present a randomized stochastic method for non-convex SBO with $m>1$ lower level problems by processing only one lower problem at each iteration, and establish its sample complexity no worse than $O(m/\epsilon^3)$, which could have a better complexity than simply processing all $m$ lower problems at each iteration. To the best of our knowledge, this is the first work considering SBO with many lower level problems and establishing state-of-the-art sample complexity.