Optimization Guarantees of Unfolded ISTA and ADMM Networks With Smooth Soft-Thresholding

TL;DR

This paper provides theoretical optimization guarantees for unfolded ISTA and ADMM networks with smooth soft-thresholding, showing conditions for near-zero training loss and comparing their sample complexity to standard neural networks.

Contribution

It introduces a PL* condition-based analysis for finite-layer unfolded networks, deriving spectral norm bounds and sample complexity thresholds, and compares these with fully-connected networks.

Findings

Unfolded networks satisfy PL* condition under certain width and sample size.

Threshold on training samples increases with network width.

Unfolded networks have higher sample complexity thresholds than FFNN, implying better error bounds.

Abstract

Solving linear inverse problems plays a crucial role in numerous applications. Algorithm unfolding based, model-aware data-driven approaches have gained significant attention for effectively addressing these problems. Learned iterative soft-thresholding algorithm (LISTA) and alternating direction method of multipliers compressive sensing network (ADMM-CSNet) are two widely used such approaches, based on ISTA and ADMM algorithms, respectively. In this work, we study optimization guarantees, i.e., achieving near-zero training loss with the increase in the number of learning epochs, for finite-layer unfolded networks such as LISTA and ADMM-CSNet with smooth soft-thresholding in an over-parameterized (OP) regime. We achieve this by leveraging a modified version of the Polyak-Lojasiewicz, denoted PL$^*$, condition. Satisfying the PL$^*$ condition within a specific region of the loss landscape ensures the existence of a global minimum and exponential convergence from initialization using gradient descent based methods. Hence, we provide conditions, in terms of the network width and the number of training samples, on these unfolded networks for the PL$^*$ condition to hold. We achieve this by deriving the Hessian spectral norm of these networks. Additionally, we show that the threshold on the number of training samples increases with the increase in the network width. Furthermore, we compare the threshold on training samples of unfolded networks with that of a standard fully-connected feed-forward network (FFNN) with smooth soft-thresholding non-linearity. We prove that unfolded networks have a higher threshold value than FFNN. Consequently, one can expect a better expected error for unfolded networks than FFNN.