This final chapter covers topics that build on the material discussed in the book, with the aim of pointing to avenues for further study and research. The selection of topics is clearly a matter of personal choice, but care has been taken to present both well-established topics, such as probabilistic graphical models, and emerging ones, such as causality and quantum machine learning. The topics are distinct, and each section can be read separately. The presentation is brief, and only meant as a launching pad for exploration.

As discussed so far in this book, the standard formulation of machine learning makes the following two basic assumptions: 1. Statistical equivalence of training and testing. The statistical properties of the data observed during training match those to be experienced during testing – i.e., the population distribution underlying the generation of the data is the same during both training and testing. 2. Separation of learning tasks. Training is carried out separately for each separate learning task – i.e., for any new data set and/or loss function, training is viewed as a new problem to be addressed from scratch.

In this chapter, we use the optimization tools presented in Chapter 5 to develop supervised learning algorithms that move beyond the simple settings studied in Chapter 4 for which the training problem could be solved exactly, typically by addressing an LS problem. We will focus specifically on binary and multi-class classification, with a brief discussion at the end of the chapter about the (direct) extension to regression problems. Following Chapter 4, the presentation will mostly concentrate on parametric model classes, but we will also touch upon mixture models and non-parametric methods.

This chapter focuses on three key problems that underlie the formulation of many machine learning methods for inference and learning, namely variational inference (VI), amortized VI, and variational expectation maximization (VEM). We have already encountered these problems in simplified forms in previous chapters, and they will be essential in developing the more advanced techniques to be covered in the rest of the book. Notably, VI and amortized VI underpin optimal Bayesian inference, which was used, e.g., in Chapter 6 to design optimal predictors for generative models; and VEM generalizes the EM algorithm that was introduced in Chapter 7 for training directed generative latent-variable models.

The previous chapters have adopted a limited range of probabilistic models, namely Bernoulli and categorical distributions for discrete rvs and Gaussian distributions for continuous rvs. While these are common modeling choices, they clearly do not represent many important situations of interest for machine learning applications. For instance, discrete data may a priori take arbitrarily large values, making categorical models unsuitable. Continuous data may need to satisfy certain constraints, such as non-negativity, rendering Gaussian models far from ideal.

So far, this book has focused on conventional centralized learning settings in which data are collected at a central server, which carries out training. When data originate at distributed agents, such as personal devices, organizations, or factories run by different companies, this approach has two clear drawbacks: • First, it requires transferring data from the agents to the server, which may incur a prohibitive communication load. • Second, in the process of transferring, storing, and processing the agents’ data, sensitive information may be exposed or exploited.

This appendix collects a series of analytical methods that are needed in various parts of the book. All the tools are oriented toward the diagonalization of Hamiltonian, including some cases that allow a complete analytical diagonalization (e.g., coupled resonators) as well as perturbation theory methods for cases where the fully interacting model is too complex for an exact solution.