Safeguarded Learned Convex Optimization

Many applications require repeatedly solving a certain type of optimization problem, each time with new (but similar) data. Data-driven algorithms can "learn to optimize" (L2O) with much fewer iterations and with similar cost per iteration as general-purpose optimization algorithms. L2O algorithms are often derived from general-purpose algorithms, but with the inclusion of (possibly many) tunable parameters. Exceptional performance has been demonstrated when the parameters are optimized for a particular distribution of data. Unfortunately, it is impossible to ensure all L2O algorithms always converge to a solution. However, we present a framework that uses L2O updates together with a safeguard to guarantee convergence for convex problems with proximal and/or gradient oracles. The safeguard is simple and computationally cheap to implement, and it should be activated only when the current L2O updates would perform poorly or appear to diverge. This approach yields the numerical benefits of employing machine learning methods to create rapid L2O algorithms while still guaranteeing convergence. Our numerical examples demonstrate the efficacy of this approach for existing and new L2O schemes

Learning to Warm-Start Fixed-Point Optimization Algorithms

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