Dynamic real-time optimization (DRTO) is a challenging task due to the fact that optimal operating conditions must be computed in real time. The main bottleneck in the industrial application of DRTO is the presence of uncertainty. Many stochastic systems present the following obstacles: 1) plant-model mismatch, 2) process disturbances, 3) risks in violation of process constraints. To accommodate these difficulties, we present a constrained reinforcement learning (RL) based approach. RL naturally handles the process uncertainty by computing an optimal feedback policy. However, no state constraints can be introduced intuitively. To address this problem, we present a chance-constrained RL methodology. We use chance constraints to guarantee the probabilistic satisfaction of process constraints, which is accomplished by introducing backoffs, such that the optimal policy and backoffs are computed simultaneously. Backoffs are adjusted using the empirical cumulative distribution function to guarantee the satisfaction of a joint chance constraint. The advantage and performance of this strategy are illustrated through a stochastic dynamic bioprocess optimization problem, to produce sustainable high-value bioproducts.

Combining symbolic and neural approaches has gained considerable attention in the AI community, as it is often argued that the strengths and weaknesses of these approaches are complementary. One such trend in the literature are weakly supervised learning techniques that employ operators from fuzzy logics. In particular, they use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks (or grounding them), this background knowledge can be added to regular loss functions, hence making reasoning a part of learning. In this paper, we investigate how implications from the fuzzy logic literature behave in a differentiable setting. In such a setting, we analyze the differences between the formal properties of these fuzzy implications. It turns out that various fuzzy implications, including some of the most well-known, are highly unsuitable for use in a differentiable learning setting. A further finding shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and show that sigmoidal implications outperform other choices of fuzzy implications.

This has led to the development of a plethora of domain-dependent and context-specific methods for dealing with the interpretation of machine learning (ML) models and the formation of explanations for humans. Unfortunately, this trend is far from being over, with an abundance of knowledge in the field which is scattered and needs organisation. The goal of this article is to systematically review research works in the field of XAI and to try to define some boundaries in the field. From several hundreds of research articles focused on the concept of explainability, about 350 have been considered for review by using the following search methodology. In a first phase, Google Scholar was queried to find papers related to "explainable artificial intelligence", "explainable machine learning" and "interpretable machine learning". Subsequently, the bibliographic section of these articles was thoroughly examined to retrieve further relevant scientific studies. The first noticeable thing, as shown in figure 2 (a), is the distribution of the publication dates of selected research articles: sporadic in the 70s and 80s, receiving preliminary attention in the 90s, showing raising interest in 2000 and becoming a recognised body of knowledge after 2010. The first research concerned the development of an explanation-based system and its integration in a computer program designed to help doctors make diagnoses [3]. Some of the more recent papers focus on work devoted to the clustering of methods for explainability, motivating the need for organising the XAI literature [4, 5, 6].

We introduce a stochastic variational inference procedure for training scalable Gaussian process (GP) models whose per-iteration complexity is independent of both the number of training points, $n$, and the number basis functions used in the kernel approximation, $m$. Our central contributions include an unbiased stochastic estimator of the evidence lower bound (ELBO) for a Gaussian likelihood, as well as a stochastic estimator that lower bounds the ELBO for several other likelihoods such as Laplace and logistic. Independence of the stochastic optimization update complexity on $n$ and $m$ enables inference on huge datasets using large capacity GP models. We demonstrate accurate inference on large classification and regression datasets using GPs and relevance vector machines with up to $m = 10^7$ basis functions.

We explore if it is possible to learn a directed acyclic graph (DAG) from data without imposing explicitly the acyclicity constraint. In particular, for Gaussian distributions, we frame structural learning as a sparse matrix factorization problem and we empirically show that solving an $\ell_1$-penalized optimization yields to good recovery of the true graph and, in general, to almost-DAG graphs. Moreover, this approach is computationally efficient and is not affected by the explosion of combinatorial complexity as in classical structural learning algorithms.

Gaussian Mixture models (GMMs) are a powerful tool for clustering, classification and density estimation when clustering structures are embedded in the data. The presence of missing values can largely impact the GMMs estimation process, thus handling missing data turns out to be a crucial point in clustering, classification and density estimation. Several techniques have been developed to impute the missing values before model estimation. Among these, multiple imputation is a simple and useful general approach to handle missing data. In this paper we propose two different methods to fit Gaussian mixtures in the presence of missing data. Both methods use a variant of the Monte Carlo Expectation-Maximisation (MCEM) algorithm for data augmentation. Thus, multiple imputations are performed during the E-step, followed by the standard M-step for a given eigen-decomposed component-covariance matrix. We show that the proposed methods outperform the multiple imputation approach, both in terms of clusters identification and density estimation.

Recent work in cognitive science has uncovered a diversity of explanatory values, or dimensions along which we judge explanations as better or worse. We propose a Bayesian account of how these values fit together to guide explanation. The resulting taxonomy provides a set of predictors for which explanations people prefer and shows how core values from psychology, statistics, and the philosophy of science emerge from a common mathematical framework. In addition to operationalizing the explanatory virtues associated with, for example, scientific argument-making, this framework also enables us to reinterpret the explanatory vices that drive conspiracy theories, delusions, and extremist ideologies. Intuitively, philosophically, and as seen in laboratory experiments, explanations are judged as better or worse on the basis of many different criteria. These explanatory values appear in early childhood [1, 2, 3, 4, 5] and their influence extends to some of the most sophisticated social knowledge formation processes we know [6]. We lack, however, an understanding of the origin of these values or an account of how they fit together to guide belief formation. The multiplicity of values also appears to conflict with Bayesian models of cognition, which speak solely in terms of degrees of beliefs and suggest we judge explanations as better or worse on the basis of a single quantity, the posterior likelihood (see Glossary). In this opinion, we show how to resolve these conflicts by arguing that previously-identified explanatory values capture different components of a full Bayesian calculation and, when considered together and weighed appropriately, implement Bayesian cognition. This framework shows how key explanatory values identified by laboratory experiments and philosophers of science--co-explanation, descriptiveness, precision, unification, power, and simplicity--emerge naturally from the mathematical structure of probabilistic inference, thereby reconciling them with Bayesian models of cognition [7, 8]. Second, it shows how these values combine to produce preferences for one explanation over another.

We propose to explain the behavior of black-box prediction methods (e.g., deep neural networks trained on image pixel data) using causal graphical models. Specifically, we explore learning the structure of a causal graph where the nodes represent prediction outcomes along with a set of macro-level "interpretable" features, while allowing for arbitrary unmeasured confounding among these variables. The resulting graph may indicate which of the interpretable features, if any, are possible causes of the prediction outcome and which may be merely associated with prediction outcomes due to confounding. The approach is motivated by a counterfactual theory of causal explanation wherein good explanations point to factors which are "difference-makers" in an interventionist sense. The resulting analysis may be useful in algorithm auditing and evaluation, by identifying features which make a causal difference to the algorithm's output.

Most existing black-box optimization methods assume that all variables in the system being optimized have equal cost and can change freely at each iteration. However, in many real world systems, inputs are passed through a sequence of different operations or modules, making variables in earlier stages of processing more costly to update. Such structure imposes a cost on switching variables in early parts of a data processing pipeline. In this work, we propose a new algorithm for switch cost-aware optimization called Lazy Modular Bayesian Optimization (LaMBO). This method efficiently identifies the global optimum while minimizing cost through a passive change of variables in early modules. The method is theoretical grounded and achieves vanishing regret when augmented with switching cost. We apply LaMBO to multiple synthetic functions and a three-stage image segmentation pipeline used in a neuroscience application, where we obtain promising improvements over prevailing cost-aware Bayesian optimization algorithms. Our results demonstrate that LaMBO is an effective strategy for black-box optimization that is capable of minimizing switching costs in modular systems.

In this article, we consider the problem of high-dimensional conditional independence testing, which is a key building block in statistics and machine learning. We propose a double generative adversarial networks (GANs)-based inference procedure. We first introduce a double GANs framework to learn two generators, and integrate the two generators to construct a doubly-robust test statistic. We next consider multiple generalized covariance measures, and take their maximum as our test statistic. Finally, we obtain the empirical distribution of our test statistic through multiplier bootstrap. We show that our test controls type-I error, while the power approaches one asymptotically. More importantly, these theoretical guarantees are obtained under much weaker and practically more feasible conditions compared to existing tests. We demonstrate the efficacy of our test through both synthetic and real datasets.