Feedback control of sparse optimization algorithms

Sparse optimization is the science of learning parsimonious models through the solution of suitable minimization problems. Sparsity is a desirable condition for several purposes. For example, it reduces the numerical complexity and memory footprint of a model for edge deployment and it enhances the physical interpretability. For this motivation, sparsity is studied in several contexts, ranging from system identification to neural networks.Proximal, gradient-based iterative algorithms are a popular approach to sparse optimization problems, because they are straightforward to implement. However, they require the setting of hyperparameters that affect the performance. Typically, this is a sensitive task that entails a tradeoff between estimation accuracy and convergence speed. To overcome this drawback, some recent works propose a time-varying tuning of the hyperparameters according to specific feedback control laws; see, e.g., [1] and references therein. More precisely, iterative sparse optimization algorithms are discrete-time dynamical systems, in which the hyperparameters can serve as control inputs to regulate the behavior and enhance the performance.The goal of this tutorial talk is to propose a complete overview of control-theoretic approaches to sparse optimization algorithms. Specifically, we compare the different results obtained in the recent literature and we discuss possible future directions in this research area.