Randomized Stochastic Variance-Reduced Methods for Multi-Task Stochastic Bilevel Optimization
This work solves efficiency and scalability issues in SBO for machine learning applications, particularly in multi-task settings, representing an incremental advancement with specific gains.
The paper tackles non-convex stochastic bilevel optimization (SBO) problems, addressing high sample complexities and limitations in handling multiple lower-level problems, by proposing randomized stochastic methods that achieve sample complexities of O(1/ε^3) for single lower-level problems and O(m/ε^3) for multi-task SBO, matching or improving upon existing bounds.
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 $O(1/ε^3)$ for finding an $ε$-stationary point under Lipschitz continuous conditions of stochastic oracles, 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 (multi-task SBO) by processing a constant number of lower problems at each iteration, and establish its sample complexity no worse than $O(m/ε^3)$, which could be a better complexity than that of simply processing all $m$ lower problems at each iteration. Lastly, we establish even faster convergence results for gradient-dominant functions. To the best of our knowledge, this is the first work considering multi-task SBO and developing state-of-the-art sample complexity results.