See, e.g., Resnick and Varian (1997); Schafer et al. (2007); Koren et al. (2009); Davidson et al. (2010); McAuley and Leskovec (2013); Das et al. (2017). Consider, more specifically, representing the ratings of d 1 users on d 2 products/items by a d 1 d 2 matrix. For all practical purposes, both d 1 and d 2 can be very large yet only a rather small number of the entries can be observed. The idea is that if the interaction between users and products can be approximately captured by a handful of latent user-specific and product-specific characteristics, then it is possible to infer the whole user-item rating matrix from these sparsely observed entries, and hence recommend products to users who may be genuinely interested in them. Since the pioneering works of Cand` es and Tao (2009); Candes and Plan (2010); Candes and Recht (2012), a lot of impressive progress has been made to make these techniques more accurate and scalable, and to better understand the statistical and computational underpinnings of the problem.

Graph contrastive learning has shown great promise when labeled data is scarce, but large unlabeled datasets are available. However, it often does not take uncertainty estimation into account. We show that a variational Bayesian neural network approach can be used to improve not only the uncertainty estimates but also the downstream performance on semi-supervised node-classification tasks. Moreover, we propose a new measure of uncertainty for contrastive learning, that is based on the disagreement in likelihood due to different positive samples.

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

Why does a phenomenon occur? Addressing this question is central to most scientific inquiries based on empirical observations, and often heavily relies on simulations of scientific models. As models become more intricate, deciphering the causes behind these phenomena in high-dimensional spaces of interconnected variables becomes increasingly challenging. Causal machine learning may assist scientists in the discovery of relevant and interpretable patterns of causation in simulations. We introduce Targeted Causal Reduction (TCR), a method for turning complex models into a concise set of causal factors that explain a specific target phenomenon. We derive an information theoretic objective to learn TCR from interventional data or simulations and propose algorithms to optimize this objective efficiently. TCR's ability to generate interpretable high-level explanations from complex models is demonstrated on toy and mechanical systems, illustrating its potential to assist scientists in the study of complex phenomena in a broad range of disciplines.

Prior works have shown that semi-supervised learning algorithms can leverage unlabeled data to improve over the labeled sample complexity of supervised learning (SL) algorithms. However, existing theoretical analyses focus on regimes where the unlabeled data is sufficient to learn a good decision boundary using unsupervised learning (UL) alone. This begs the question: Can SSL algorithms simultaneously improve upon both UL and SL? To this end, we derive a tight lower bound for 2-Gaussian mixture models that explicitly depends on the labeled and the unlabeled dataset size as well as the signal-to-noise ratio of the mixture distribution. Surprisingly, our result implies that no SSL algorithm can improve upon the minimax-optimal statistical error rates of SL or UL algorithms for these distributions. Nevertheless, we show empirically on real-world data that SSL algorithms can still outperform UL and SL methods. Therefore, our work suggests that, while proving performance gains for SSL algorithms is possible, it requires careful tracking of constants.

Mixed linear regression (MLR) is a powerful model for characterizing nonlinear relationships by utilizing a mixture of linear regression sub-models. The identification of MLR is a fundamental problem, where most of the existing results focus on offline algorithms, rely on independent and identically distributed (i.i.d) data assumptions, and provide local convergence results only. This paper investigates the online identification and data clustering problems for two basic classes of MLRs, by introducing two corresponding new online identification algorithms based on the expectation-maximization (EM) principle. It is shown that both algorithms will converge globally without resorting to the traditional i.i.d data assumptions. The main challenge in our investigation lies in the fact that the gradient of the maximum likelihood function does not have a unique zero, and a key step in our analysis is to establish the stability of the corresponding differential equation in order to apply the celebrated Ljung's ODE method. It is also shown that the within-cluster error and the probability that the new data is categorized into the correct cluster are asymptotically the same as those in the case of known parameters. Finally, numerical simulations are provided to verify the effectiveness of our online algorithms.

Existing statistical methods for compositional data analysis are inadequate for many modern applications for two reasons. First, modern compositional datasets, for example in microbiome research, display traits such as high-dimensionality and sparsity that are poorly modelled with traditional approaches. Second, assessing -- in an unbiased way -- how summary statistics of a composition (e.g., racial diversity) affect a response variable is not straightforward. In this work, we propose a framework based on hypothetical data perturbations that addresses both issues. Unlike existing methods for compositional data, we do not transform the data and instead use perturbations to define interpretable statistical functionals on the compositions themselves, which we call average perturbation effects. These average perturbation effects, which can be employed in many applications, naturally account for confounding that biases frequently used marginal dependence analyses. We show how average perturbation effects can be estimated efficiently by deriving a perturbation-dependent reparametrization and applying semiparametric estimation techniques. We analyze the proposed estimators empirically on simulated data and demonstrate advantages over existing techniques on US census and microbiome data. For all proposed estimators, we provide confidence intervals with uniform asymptotic coverage guarantees.

Active Learning has received significant attention in the field of machine learning for its potential in selecting the most informative samples for labeling, thereby reducing data annotation costs. However, we show that the reported lifts in recent literature generalize poorly to other domains leading to an inconclusive landscape in Active Learning research. Furthermore, we highlight overlooked problems for reproducing AL experiments that can lead to unfair comparisons and increased variance in the results. This paper addresses these issues by providing an Active Learning framework for a fair comparison of algorithms across different tasks and domains, as well as a fast and performant oracle algorithm for evaluation. To the best of our knowledge, we propose the first AL benchmark that tests algorithms in 3 major domains: Tabular, Image, and Text. We report empirical results for 6 widely used algorithms on 7 real-world and 2 synthetic datasets and aggregate them into a domain-specific ranking of AL algorithms.

Science and Engineering applications are typically associated with expensive optimization problem to identify optimal design solutions and states of the system of interest. Bayesian optimization and active learning compute surrogate models through efficient adaptive sampling schemes to assist and accelerate this search task toward a given optimization goal. Both those methodologies are driven by specific infill/learning criteria which quantify the utility with respect to the set goal of evaluating the objective function for unknown combinations of optimization variables. While the two fields have seen an exponential growth in popularity in the past decades, their dualism and synergy have received relatively little attention to date. This paper discusses and formalizes the synergy between Bayesian optimization and active learning as symbiotic adaptive sampling methodologies driven by common principles. In particular, we demonstrate this unified perspective through the formalization of the analogy between the Bayesian infill criteria and active learning criteria as driving principles of both the goal-driven procedures. To support our original perspective, we propose a general classification of adaptive sampling techniques to highlight similarities and differences between the vast families of adaptive sampling, active learning, and Bayesian optimization. Accordingly, the synergy is demonstrated mapping the Bayesian infill criteria with the active learning criteria, and is formalized for searches informed by both a single information source and multiple levels of fidelity. In addition, we provide guidelines to apply those learning criteria investigating the performance of different Bayesian schemes for a variety of benchmark problems to highlight benefits and limitations over mathematical properties that characterize real-world applications.

In supervised learning -- for instance in image classification -- modern massive datasets are commonly labeled by a crowd of workers. The obtained labels in this crowdsourcing setting are then aggregated for training, generally leveraging a per-worker trust score. Yet, such workers oriented approaches discard the tasks' ambiguity. Ambiguous tasks might fool expert workers, which is often harmful for the learning step. In standard supervised learning settings - with one label per task - the Area Under the Margin (AUM) was tailored to identify mislabeled data. We adapt the AUM to identify ambiguous tasks in crowdsourced learning scenarios, introducing the Weighted Areas Under the Margin (WAUM). The WAUM is an average of AUMs weighted according to task-dependent scores. We show that the WAUM can help discarding ambiguous tasks from the training set, leading to better generalization performance. We report improvements over existing strategies for learning with a crowd, both on simulated settings, and on real datasets such as CIFAR-10H (a crowdsourced dataset with a high number of answered labels), LabelMe and Music (two datasets with few answered votes).