[Paper Review] On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems

Risheng Liu, Shichao Cheng+4 more|arXiv (Cornell University)|Aug 16, 2018

Sparse and Compressive Sensing Techniques36 references3 citations

TL;DR

This paper proposes Flexible Iterative Modularization Algorithm (FIMA), a provably globally convergent framework for learning-based iterative methods in nonconvex inverse problems. By integrating learnable modules with theoretical convergence guarantees under the Kurdyka-Łojasiewicz (KŁ) property, FIMA ensures convergence to critical points while enabling data-adaptive optimization, outperforming classical and deep learning-based methods in real-world applications.

ABSTRACT

Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA.

Motivation & Objective

• To address the lack of theoretical convergence guarantees in learning-based iterative methods for nonconvex inverse problems.
• To overcome the limitations of fixed, handcrafted update rules in classical solvers that fail to exploit data-specific structures.
• To develop a generic, modular framework that allows learnable components while preserving global convergence.
• To provide a unified theoretical analysis for convergence of iterative methods in nonconvex settings with both smooth and nonsmooth terms.
• To demonstrate the practical superiority of the proposed method on real-world inverse problems in vision and learning.

Proposed method

• Proposes FIMA, a flexible iterative modularization framework that decomposes the optimization problem into sequential, learnable subproblems.
• Introduces scheduling policies for modular components that ensure convergence under the Kurdyka-Łojasiewicz (KŁ) condition.
• Employs a block-iterative update strategy where each variable is updated in sequence with proximal-like steps, using learnable parameters.
• Derives convergence guarantees by analyzing the descent of the objective function and showing that the sequence of iterates is Cauchy.
• Uses the KŁ property to establish convergence rates and prove that the sequence converges globally to critical points.
• Applies the framework to nonconvex problems where both the data-fidelity term f and prior term g are possibly nonconvex, with f continuously differentiable and g nonsmooth.

Experimental results

Research questions

• RQ1 Can a learning-based iterative method be designed with provable global convergence for nonconvex inverse problems?
• RQ2 How can learnable modules be integrated into iterative solvers without sacrificing theoretical convergence guarantees?
• RQ3 What theoretical conditions ensure convergence of such hybrid learning-optimization schemes in nonconvex settings?
• RQ4 Can the proposed framework improve performance on real-world inverse problems compared to classical or end-to-end deep learning methods?
• RQ5 What convergence rates can be established under the Kurdyka-Łojasiewicz (KŁ) property for the proposed method?

Key findings

• FIMA ensures global convergence of the iterates to critical points of the objective function under the Kurdyka-Łojasiewicz (KŁ) property.
• The sequence of iterates generated by FIMA is Cauchy, implying convergence to a stationary point.
• The method establishes convergence rates that match those of classical proximal schemes under the same KŁ assumptions.
• Empirical results show FIMA outperforms both classical iterative solvers and end-to-end deep learning baselines on real inverse problems.
• The convergence analysis holds even when both f and g are nonconvex, provided f is continuously differentiable and g is nonsmooth.
• The framework allows for flexible, data-adaptive module scheduling, enabling better performance than fixed-structure solvers.