Restarted contractive operators to learn at equilibrium

Bilevel optimization offers a methodology to learn hyperparameters in imaging inverse problems, yet its integration with automatic differentiation techniques remains challenging. On the one hand, inverse problems are typically solved by iterating arbitrarily many times some elementary scheme which maps any point to the minimizer of an energy functional, known as equilibrium point. On the other hand, introducing parameters to be learned in the energy functional yield architectures very reminiscent of Neural Networks (NN) known as Unrolled NN and thus suggests the use of Automatic Differentiation (AD) techniques. Yet, applying AD requires for the NN to be of relatively small depth, thus making necessary to truncate an unrolled scheme to a finite number of iterations. First, we show that, at the minimizer, the optimal gradient descent step computed in the Deep Equilibrium (DEQ) framework admits an approximation, known as Jacobian Free Backpropagation (JFB), that is much easier to compute and can be made arbitrarily good by controlling Lipschitz properties of the truncated unrolled scheme. Second, we introduce an algorithm that combines a restart strategy with JFB computed by AD and we show that the learned steps can be made arbitrarily close to the optimal DEQ framework. Third, we complement the theoretical analysis by applying the proposed method to a variety of problems in imaging that progressively depart from the theoretical framework. In particular we show that this method is effective for training weights in weighted norms; stepsizes and regularization levels of Plug-and-Play schemes; and a DRUNet denoiser embedded in Forward-Backward iterates.