We introduce a machine-learning framework to warm-start fixed-point optimization algorithms. Our architecture consists of a neural network mapping problem parameters to warm starts, followed by a predefined number of fixed-point iterations. We propose two loss functions designed to either minimize the fixed-point residual or the distance to a ground truth solution. In this way, the neural network predicts warm starts with the end-to-end goal of minimizing the downstream loss. An important feature of our architecture is its flexibility, in that it can predict a warm start for fixed-point algorithms run for any number of steps, without being limited to the number of steps it has been trained on. We provide PAC-Bayes generalization bounds on unseen data for common classes of fixed-point operators: contractive, linearly convergent, and averaged. Applying this framework to well-known applications in control, statistics, and signal processing, we observe a significant reduction in the number of iterations and solution time required to solve these problems, through learned warm starts.

翻译：我们提出了一种机器学习框架，用于冷启动定点优化算法。该架构包含一个将问题参数映射为冷启动点的神经网络，随后执行预定义次数的定点迭代。我们设计了两种损失函数，分别用于最小化定点残差或最小化与真实解之间的距离。通过这种方式，神经网络以端到端方式预测冷启动点，旨在最小化下游损失。该架构的一个重要特性是其灵活性：它能预测任意步数定点算法的冷启动点，而不受训练步数的限制。针对常见的定点算子类别（收缩算子、线性收敛算子和平均算子），我们提供了关于未见数据的PAC-Bayes泛化界。将该框架应用于控制、统计和信号处理领域的经典问题中，我们观察到通过学习的冷启动点，解决这些问题所需的迭代次数和求解时间显著减少。