  Stochastic bilevel optimization tackles challenges involving nested
optimization structures. Its fast-growing scale nowadays necessitates efficient
distributed algorithms. In conventional distributed bilevel methods, each
worker must transmit full-dimensional stochastic gradients to the server every
iteration, leading to significant communication overhead and thus hindering
efficiency and scalability. To resolve this issue, we introduce the first
family of distributed bilevel algorithms with communication compression. The
primary challenge in algorithmic development is mitigating bias in
hypergradient estimation caused by the nested structure. We first propose
C-SOBA, a simple yet effective approach with unbiased compression and provable
linear speedup convergence. However, it relies on strong assumptions on bounded
gradients. To address this limitation, we explore the use of moving average,
error feedback, and multi-step compression in bilevel optimization, resulting
in a series of advanced algorithms with relaxed assumptions and improved
convergence properties. Numerical experiments show that our compressed bilevel
algorithms can achieve $10\times$ reduction in communication overhead without
severe performance degradation.
