  Bilevel optimization refers to scenarios whereby the optimal solution of a
lower-level energy function serves as input features to an upper-level
objective of interest. These optimal features typically depend on tunable
parameters of the lower-level energy in such a way that the entire bilevel
pipeline can be trained end-to-end. Although not generally presented as such,
this paper demonstrates how a variety of graph learning techniques can be
recast as special cases of bilevel optimization or simplifications thereof. In
brief, building on prior work we first derive a more flexible class of energy
functions that, when paired with various descent steps (e.g., gradient descent,
proximal methods, momentum, etc.), form graph neural network (GNN)
message-passing layers; critically, we also carefully unpack where any residual
approximation error lies with respect to the underlying constituent
message-passing functions. We then probe several simplifications of this
framework to derive close connections with non-GNN-based graph learning
approaches, including knowledge graph embeddings, various forms of label
propagation, and efficient graph-regularized MLP models. And finally, we
present supporting empirical results that demonstrate the versatility of the
proposed bilevel lens, which we refer to as BloomGML, referencing that BiLevel
Optimization Offers More Graph Machine Learning. Our code is available at
https://github.com/amberyzheng/BloomGML. Let graph ML bloom.
