  Plug-and-Play Priors (PnP) and Regularization by Denoising (RED) are
widely-used frameworks for solving imaging inverse problems by computing
fixed-points of operators combining physical measurement models and learned
image priors. While traditional PnP/RED formulations have focused on priors
specified using image denoisers, there is a growing interest in learning
PnP/RED priors that are end-to-end optimal. The recent Deep Equilibrium Models
(DEQ) framework has enabled memory-efficient end-to-end learning of PnP/RED
priors by implicitly differentiating through the fixed-point equations without
storing intermediate activation values. However, the dependence of the
computational/memory complexity of the measurement models in PnP/RED on the
total number of measurements leaves DEQ impractical for many imaging
applications. We propose ODER as a new strategy for improving the efficiency of
DEQ through stochastic approximations of the measurement models. We
theoretically analyze ODER giving insights into its convergence and ability to
approximate the traditional DEQ approach. Our numerical results suggest the
potential improvements in training/testing complexity due to ODER on three
distinct imaging applications.
