  Implicit equilibrium models, i.e., deep neural networks (DNNs) defined by
implicit equations, have been becoming more and more attractive recently. In
this paper, we investigate an emerging question: can an implicit equilibrium
model's equilibrium point be regarded as the solution of an optimization
problem? To this end, we first decompose DNNs into a new class of unit layer
that is the proximal operator of an implicit convex function while keeping its
output unchanged. Then, the equilibrium model of the unit layer can be derived,
named Optimization Induced Equilibrium Networks (OptEq), which can be easily
extended to deep layers. The equilibrium point of OptEq can be theoretically
connected to the solution of its corresponding convex optimization problem with
explicit objectives. Based on this, we can flexibly introduce prior properties
to the equilibrium points: 1) modifying the underlying convex problems
explicitly so as to change the architectures of OptEq; and 2) merging the
information into the fixed point iteration, which guarantees to choose the
desired equilibrium point when the fixed point set is non-singleton. We show
that deep OptEq outperforms previous implicit models even with fewer
parameters. This work establishes the first step towards the
optimization-guided design of deep models.
