We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parameterized by an input-convex neural network (ICNN). While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle non-smooth problems which often leads to slow convergence. Moreover, the nested structure of the neural network complicates the application of standard non-smooth optimization techniques, such as proximal algorithms. To overcome these challenges, we reformulate the problem and eliminate the network's nested structure. By relating this reformulation to epigraphical projections of the activation functions, we transform the problem into a convex optimization problem that can be efficiently solved using a primal-dual algorithm. We also prove that this reformulation is equivalent to the original variational problem. Through experiments on several imaging tasks, we show that the proposed approach not only outperforms subgradient methods and even accelerated methods in the smooth setting, but also facilitates the training of the regularizer itself.

翻译：我们研究数据驱动的变分重建框架中的优化问题，其中正则化器由输入凸神经网络参数化。虽然基于梯度的方法常用于求解此类问题，但其难以有效处理非光滑问题，通常导致收敛缓慢。此外，神经网络中的嵌套结构使标准非光滑优化技术（如近端算法）的应用变得复杂。为克服这些挑战，我们重构了该问题并消除了网络的嵌套结构。通过将该重构与激活函数的上镜图投影相关联，我们将问题转化为一个凸优化问题，该问题可使用原对偶算法高效求解。我们还证明了该重构与原始变分问题等价。通过在多个成像任务上的实验，我们表明所提方法不仅在光滑设定下优于次梯度方法乃至加速方法，还能促进正则化器本身的训练。