Machine learning algorithms and systems are progressively becoming part of our societies, leading to a growing need of building a vast multitude of accurate, reliable and interpretable models which should possibly exploit similarities among tasks. Automating segments of machine learning itself seems to be a natural step to undertake to deliver increasingly capable systems able to perform well in both the big-data and the few-shot learning regimes. Hyperparameter optimization (HPO) and meta-learning (MTL) constitute two building blocks of this growing effort. We explore these two topics under a unifying perspective, presenting a mathematical framework linked to bilevel programming that captures existing similarities and translates into procedures of practical interest rooted in algorithmic differentiation. We discuss the derivation, applicability and computational complexity of these methods and establish several approximation properties for a class of objective functions of the underlying bilevel programs. In HPO, these algorithms generalize and extend previous work on gradient-based methods. In MTL, the resulting framework subsumes classic and emerging strategies and provides a starting basis from which to build and analyze novel techniques. A series of examples and numerical simulations offer insight and highlight some limitations of these approaches. Experiments on larger-scale problems show the potential gains of the proposed methods in real-world applications. Finally, we develop two extensions of the basic algorithms apt to optimize a class of discrete hyperparameters (graph edges) in an application to relational learning and to tune online learning rate schedules for training neural network models, an old but crucially important issue in machine learning
