Maksym Neyra-Nesterenko

Maksym Neyra-Nesterenko, Ben Adcock

Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for solving inverse problems. However, it is known that current data-driven approaches face several key issues, notably hallucinations, instabilities and unpredictable generalization, with potential impact in critical tasks such as medical imaging. This raises the key question of whether or not one can construct deep neural networks for inverse problems with explicit stability and accuracy guarantees. In this work, we present a novel construction of accurate, stable and efficient neural networks for inverse problems with general analysis-sparse models, termed NESTANets. To construct the network, we first unroll NESTA, an accelerated first-order method for convex optimization. The slow convergence of this method leads to deep networks with low efficiency. Therefore, to obtain shallow, and consequently more efficient, networks we combine NESTA with a novel restart scheme. We then use compressed sensing techniques to demonstrate accuracy and stability. We showcase this approach in the case of Fourier imaging, and verify its stability and performance via a series of numerical experiments. The key impact of this work is demonstrating the construction of efficient neural networks based on unrolling with guaranteed stability and accuracy.

Ben Adcock, Matthew J. Colbrook, Maksym Neyra-Nesterenko

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.