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Recently, bi-level optimization (BLO) has taken center stage in some veryexciting developments in the area of signal processing (SP) and machinelearning (ML). Roughly speaking, BLO is a classical optimization problem thatinvolves two levels of hierarchy (i.e., upper and lower levels), whereinobtaining the solution to the upper-level problem requires solving thelower-level one. BLO has become popular largely because it is powerful inmodeling problems in SP and ML, among others, that involve optimizing nestedobjective functions. Prominent applications of BLO range from resourceallocation for wireless systems to adversarial machine learning. In this work,we focus on a class of tractable BLO problems that often appear in SP and MLapplications. We provide an overview of some basic concepts of this class ofBLO problems, such as their optimality conditions, standard algorithms(including their optimization principles and practical implementations), aswell as how they can be leveraged to obtain state-of-the-art results for anumber of key SP and ML applications. Further, we discuss some recent advancesin BLO theory, its implications for applications, and point out somelimitations of the state-of-the-art that require significant future researchefforts. Overall, we hope that this article can serve to accelerate theadoption of BLO as a generic tool to model, analyze, and innovate on a widearray of emerging SP and ML applications.

An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning