  In recent years, Bi-Level Optimization (BLO) techniques have received
extensive attentions from both learning and vision communities. A variety of
BLO models in complex and practical tasks are of non-convex follower structure
in nature (a.k.a., without Lower-Level Convexity, LLC for short). However, this
challenging class of BLOs is lack of developments on both efficient solution
strategies and solid theoretical guarantees. In this work, we propose a new
algorithmic framework, named Initialization Auxiliary and Pessimistic
Trajectory Truncated Gradient Method (IAPTT-GM), to partially address the above
issues. In particular, by introducing an auxiliary as initialization to guide
the optimization dynamics and designing a pessimistic trajectory truncation
operation, we construct a reliable approximate version of the original BLO in
the absence of LLC hypothesis. Our theoretical investigations establish the
convergence of solutions returned by IAPTT-GM towards those of the original BLO
without LLC. As an additional bonus, we also theoretically justify the quality
of our IAPTT-GM embedded with Nesterov's accelerated dynamics under LLC. The
experimental results confirm both the convergence of our algorithm without LLC,
and the theoretical findings under LLC.
