This primer is an attempt to provide a detailed summary of the different facets of lifelong learning. We start with Chapter 2 which provides a high-level overview of lifelong learning systems. In this chapter, we discuss prominent scenarios in lifelong learning (Section 2.4), provide 8 Introduction a high-level organization of different lifelong learning approaches (Section 2.5), enumerate the desiderata for an ideal lifelong learning system (Section 2.6), discuss how lifelong learning is related to other learning paradigms (Section 2.7), describe common metrics used to evaluate lifelong learning systems (Section 2.8). This chapter is more useful for readers who are new to lifelong learning and want to get introduced to the field without focusing on specific approaches or benchmarks. The remaining chapters focus on specific aspects (either learning algorithms or benchmarks) and are more useful for readers who are looking for specific approaches or benchmarks. Chapter 3 focuses on regularization-based approaches that do not assume access to any data from previous tasks. Chapter 4 discusses memory-based approaches that typically use a replay buffer or an episodic memory to save subset of data across different tasks. Chapter 5 focuses on different architecture families (and their instantiations) that have been proposed for training lifelong learning systems. Following these different classes of learning algorithms, we discuss the commonly used evaluation benchmarks and metrics for lifelong learning (Chapter 6) and wrap up with a discussion of future challenges and important research directions in Chapter 7.

We develop a multiscale approach to estimate high-dimensional probability distributions from a dataset of physical fields or configurations observed in experiments or simulations. In this way we can estimate energy functions (or Hamiltonians) and efficiently generate new samples of many-body systems in various domains, from statistical physics to cosmology. Our method -- the Wavelet Conditional Renormalization Group (WC-RG) -- proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields. These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis. WC-RG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales. Near phase transitions, it avoids the "critical slowing down" of direct estimation and sampling algorithms. This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and $\varphi^4$ field theories. We show that multiscale WC-RG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales. This is demonstrated for weak-gravitational-lensing fields reflecting dark matter distributions in cosmology, which include long-range interactions with long-tail probability distributions. WC-RG has a large number of potential applications in non-equilibrium systems, where the underlying distribution is not known {\it a priori}. Finally, we discuss the connection between WC-RG and deep network architectures.

Despite the progress observed with model-agnostic explainable AI (XAI), it is the case that model-agnostic XAI can produce incorrect explanations. One alternative are the so-called formal approaches to XAI, that include PI-explanations. Unfortunately, PI-explanations also exhibit important drawbacks, the most visible of which is arguably their size. The computation of relevant features serves to trade off probabilistic precision for the number of features in an explanation. However, even for very simple classifiers, the complexity of computing sets of relevant features is prohibitive. This paper investigates the computation of relevant sets for Naive Bayes Classifiers (NBCs), and shows that, in practice, these are easy to compute. Furthermore, the experiments confirm that succinct sets of relevant features can be obtained with NBCs.

The real-world testing of decisions made using causal machine learning models is an essential prerequisite for their successful application. We focus on evaluating and improving contextual treatment assignment decisions: these are personalised treatments applied to e.g. customers, each with their own contextual information, with the aim of maximising a reward. In this paper we introduce a model-agnostic framework for gathering data to evaluate and improve contextual decision making through Bayesian Experimental Design. Specifically, our method is used for the data-efficient evaluation of the regret of past treatment assignments. Unlike approaches such as A/B testing, our method avoids assigning treatments that are known to be highly sub-optimal, whilst engaging in some exploration to gather pertinent information. We achieve this by introducing an information-based design objective, which we optimise end-to-end. Our method applies to discrete and continuous treatments. Comparing our information-theoretic approach to baselines in several simulation studies demonstrates the superior performance of our proposed approach.

Causal background knowledge about the existence or the absence of causal edges and paths is frequently encountered in observational studies. The shared directed edges and links of a subclass of Markov equivalent DAGs refined due to background knowledge can be represented by a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and give a minimal representation of a causal MPDAG. Then, we introduce a novel representation called direct causal clause (DCC) to represent all types of causal background knowledge in a unified form. Using DCCs, we study the consistency and equivalency of causal background knowledge and show that any causal background knowledge set can be equivalently decomposed into a causal MPDAG plus a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking the consistency, equivalency, and finding the decomposed MPDAG and residual DCCs. Finally, with causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that causal background knowledge can significantly improve the identifiability of causal effects.

The book covers a broad array of topics not usually included in introductory machine learning texts, including supervised learning, Bayesian decision theory, parametric methods, semiparametric methods, nonparametric methods, multivariate analysis, hidden Markov models, reinforcement learning, kernel machines, graphical models, Bayesian estimation, and statistical testing. The fourth edition offers a new chapter on deep learning that discusses training, regularizing, and structuring deep neural networks such as convolutional and generative adversarial networks; new material in the chapter on reinforcement learning that covers the use of deep networks, the policy gradient methods, and deep reinforcement learning; new material in the chapter on multilayer perceptrons on autoencoders and the word2vec network; and discussion of a popular method of dimensionality reduction, t-SNE. New appendixes offer background material on linear algebra and optimization. End-of-chapter exercises help readers to apply concepts learned. Introduction to Machine Learning can be used in courses for advanced undergraduate and graduate students and as a reference for professionals.

This paper studies the Variational Inference (VI) used for training Bayesian Neural Networks (BNN) in the overparameterized regime, i.e., when the number of neurons tends to infinity. More specifically, we consider overparameterized two-layer BNN and point out a critical issue in the mean-field VI training. This problem arises from the decomposition of the lower bound on the evidence (ELBO) into two terms: one corresponding to the likelihood function of the model and the second to the Kullback-Leibler (KL) divergence between the prior distribution and the variational posterior. In particular, we show both theoretically and empirically that there is a trade-off between these two terms in the overparameterized regime only when the KL is appropriately re-scaled with respect to the ratio between the the number of observations and neurons. We also illustrate our theoretical results with numerical experiments that highlight the critical choice of this ratio.

Bayesian hierarchical models are well-suited to analyzing the often noisy data from electroencephalography experiments in cognitive neuroscience: these models provide an intuitive framework to account for structures and correlations in the data, and they allow a straightforward handling of uncertainty. In a typical neurolinguistic experiment, event-related potentials show only very small effect sizes and frequentist approaches to data analysis fail to establish the significance of some of these effects. Here, we present a Bayesian approach to analyzing event-related potentials using as an example data from an experiment which relates word surprisal and neural response. Our model is able to estimate the effect of word surprisal on most components of the event-related potential and provides a richer description of the data. The Bayesian framework also allows easier comparison between estimates based on surprisal values calculated using different language models.

A declarative action model is a compact representation of the state transitions of dynamic systems that generalizes over world objects. The specification of declarative action models is often a complex hand-crafted task. In this paper we formulate declarative action models via state constraints, and present the learning of such models as a combinatorial search. The comprehensive framework presented here allows us to connect the learning of declarative action models to well-known problem solving tasks. In addition, our framework allows us to characterize the existing work in the literature according to four dimensions: (1) the target action models, in terms of the state transitions they define; (2) the available learning examples; (3) the functions used to guide the learning process, and to evaluate the quality of the learned action models; (4) the learning algorithm. Last, the paper lists relevant successful applications of the learning of declarative actions models and discusses some open challenges with the aim of encouraging future research work.

Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm -- inspired by variational methods used in computational physics -- is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a variational Bayes (VB) inference algorithm based on a non-traditional quantum annealing approach -- referred to as quantum annealing variational Bayes (QAVB) inference -- and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key concepts from quantum mechanics: (i) the ground state of the Hamiltonian of a quantum system -- defined from the given variational Bayes (VB) problem -- corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; (ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and (iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using $\lceil \log K \rceil$ qubits and $\mathcal{O} (K)$ operations per step. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.