Barber Rina Foygel, Sidky Emil Y
Department of Statistics, University of Chicago, Chicago, IL 60637, USA.
Department of Radiology, University of Chicago, Chicago, IL 60637, USA.
J Mach Learn Res. 2024;25.
The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form . ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions and , and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective. In this work, our new theoretical results provide convergence guarantees under a restricted strong convexity assumption without requiring smoothness or differentiability, while still allowing differentiable terms to be treated approximately if needed. We validate these theoretical results empirically, with a simulated example where both and are nondifferentiable-and thus outside the scope of existing theory-as well as a simulated CT image reconstruction problem.
交替方向乘子法（ADMM）算法是处理形如 的复杂优化问题的强大且灵活的工具。ADMM在一系列具有挑战性的场景中展现出稳健的经验性能，包括目标函数 和 的非光滑性和非凸性，并且为计算机断层扫描（CT）成像的图像重建逆问题提供了一种简单自然的方法。从理论角度来看，非凸情形下现有的收敛结果通常假设目标函数中至少有一个分量函数是光滑的。在这项工作中，我们的新理论结果在受限强凸性假设下提供了收敛保证，无需光滑性或可微性，同时在需要时仍允许对可微项进行近似处理。我们通过一个模拟示例对这些理论结果进行了实证验证，在该示例中 和 都是不可微的——因此超出了现有理论的范围——以及一个模拟CT图像重建问题。
