  In recent years, a variety of gradient-based first-order methods have been
developed to solve bi-level optimization problems for learning applications.
However, theoretical guarantees of these existing approaches heavily rely on
the simplification that for each fixed upper-level variable, the lower-level
solution must be a singleton (a.k.a., Lower-Level Singleton, LLS). In this
work, we first design a counter-example to illustrate the invalidation of such
LLS condition. Then by formulating BLPs from the view point of optimistic
bi-level and aggregating hierarchical objective information, we establish
Bi-level Descent Aggregation (BDA), a flexible and modularized algorithmic
framework for generic bi-level optimization. Theoretically, we derive a new
methodology to prove the convergence of BDA without the LLS condition. Our
investigations also demonstrate that BDA is indeed compatible to a verify of
particular first-order computation modules. Additionally, as an interesting
byproduct, we also improve these conventional first-order bi-level schemes
(under the LLS simplification). Particularly, we establish their convergences
with weaker assumptions. Extensive experiments justify our theoretical results
and demonstrate the superiority of the proposed BDA for different tasks,
including hyper-parameter optimization and meta learning.
