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Due to the unsupervised nature of anomaly detection, the key to fueling deep models is finding supervisory signals. Different from current reconstruction-guided generative models and transformation-based contrastive models, we devise novel data-driven supervision for tabular data by introducing a characteristic -- scale -- as data labels. By representing varied sub-vectors of data instances, we define scale as the relationship between the dimensionality of original sub-vectors and that of representations. Scales serve as labels attached to transformed representations, thus offering ample labeled data for neural network training. This paper further proposes a scale learning-based anomaly detection method. Supervised by the learning objective of scale distribution alignment, our approach learns the ranking of representations converted from varied subspaces of each data instance. Through this proxy task, our approach models inherent regularities and patterns within data, which well describes data "normality". Abnormal degrees of testing instances are obtained by measuring whether they fit these learned patterns. Extensive experiments show that our approach leads to significant improvement over state-of-the-art generative/contrastive anomaly detection methods.

In this paper, we focus on the distributed set-membership filtering (SMFing) problem for a multi-agent system with absolute (taken from agents themselves) and relative (taken from neighbors) measurements. In the literature, the relative measurements are difficult to deal with, and the SMFs highly rely on specific set descriptions. As a result, establishing the general distributed SMFing framework having relative measurements is still an open problem. To solve this problem, first, we provide the set description based on uncertain variables determined by the relative measurements between two agents as the foundation. Surprisingly, the accurate description requires only a single calculation step rather than multiple iterations, which can effectively reduce computational complexity. Based on the derived set description, called the uncertain range, we propose two distributed SMFing frameworks: one calculates the joint uncertain range of the agent itself and its neighbors, while the other only computes the marginal uncertain range of each local system. Furthermore, we compare the performance of our proposed two distributed SMFing frameworks and the benchmark -- centralized SMFing framework. A rigorous set analysis reveals that the distributed SMF can be essentially considered as the process of computing the marginal uncertain range to outer bound the projection of the uncertain range obtained by the centralized SMF in the corresponding subspace. Simulation results corroborate the effectiveness of our proposed distributed frameworks and verify our theoretical analysis.

We establish a strong law of large numbers and a central limit theorem in the Bures-Wasserstein space of covariance operators -- or equivalently centred Gaussian measures -- over a general separable Hilbert space. Specifically, we show that under a minimal first-moment condition, empirical barycentre sequences indexed by sample size are almost certainly relatively compact, with accumulation points comprising population barycentres. We give a sufficient regularity condition for the limit to be unique. When the limit is unique, we also establish a central limit theorem under a refined pair of moment and regularity conditions. Finally, we prove strong operator convergence of the empirical optimal transport maps to their population counterparts. Though our results naturally extend finite-dimensional counterparts, including associated regularity conditions, our techniques are distinctly different owing to the functional nature of the problem in the general setting. A key element is the elicitation of a class of compact sets that reflect an \emph{ordered} Heine-Borel property of the Bures-Wasserstein space.

Several problems in statistics involve the combination of high-variance unbiased estimators with low-variance estimators that are only unbiased under strong assumptions. A notable example is the estimation of causal effects while combining small experimental datasets with larger observational datasets. There exist a series of recent proposals on how to perform such a combination, even when the bias of the low-variance estimator is unknown. To build intuition for the differing trade-offs of competing approaches, we argue for examining the finite-sample estimation error of each approach as a function of the unknown bias. This includes understanding the bias threshold -- the largest bias for which a given approach improves over using the unbiased estimator alone. Though this lens, we review several recent proposals, and observe in simulation that different approaches exhibits qualitatively different behavior. We also introduce a simple alternative approach, which compares favorably in simulation to recent alternatives, having a higher bias threshold and generally making a more conservative trade-off between best-case performance (when the bias is zero) and worst-case performance (when the bias is adversarially chosen). More broadly, we prove that for any amount of (unknown) bias, the MSE of this estimator can be bounded in a transparent way that depends on the variance / covariance of the underlying estimators that are being combined.

Data imbalance is ubiquitous when applying machine learning to real-world problems, particularly regression problems. If training data are imbalanced, the learning is dominated by the densely covered regions of the target distribution, consequently, the learned regressor tends to exhibit poor performance in sparsely covered regions. Beyond standard measures like over-sampling or re-weighting, there are two main directions to handle learning from imbalanced data. For regression, recent work relies on the continuity of the distribution; whereas for classification there has been a trend to employ mixture-of-expert models and let some ensemble members specialize in predictions for the sparser regions. Here, we adapt the mixture-of-experts approach to the regression setting. A main question when using this approach is how to fuse the predictions from multiple experts into one output. Drawing inspiration from recent work on probabilistic deep learning, we propose to base the fusion on the aleatoric uncertainties of individual experts, thus obviating the need for a separate aggregation module. In our method, dubbed MOUV, each expert predicts not only an output value but also its uncertainty, which in turn serves as a statistically motivated criterion to rely on the right experts. We compare our method with existing alternatives on multiple public benchmarks and show that MOUV consistently outperforms the prior art, while at the same time producing better calibrated uncertainty estimates. Our code is available at link-upon-publication.

Dataset Distillation is the task of synthesizing small datasets from large ones while still retaining comparable predictive accuracy to the original uncompressed dataset. Despite significant empirical progress in recent years, there is little understanding of the theoretical limitations/guarantees of dataset distillation, specifically, what excess risk is achieved by distillation compared to the original dataset, and how large are distilled datasets? In this work, we take a theoretical view on kernel ridge regression (KRR) based methods of dataset distillation such as Kernel Inducing Points. By transforming ridge regression in random Fourier features (RFF) space, we provide the first proof of the existence of small (size) distilled datasets and their corresponding excess risk for shift-invariant kernels. We prove that a small set of instances exists in the original input space such that its solution in the RFF space coincides with the solution of the original data. We further show that a KRR solution can be generated using this distilled set of instances which gives an approximation towards the KRR solution optimized on the full input data. The size of this set is linear in the dimension of the RFF space of the input set or alternatively near linear in the number of effective degrees of freedom, which is a function of the kernel, number of datapoints, and the regularization parameter $\lambda$. The error bound of this distilled set is also a function of $\lambda$. We verify our bounds analytically and empirically.

We consider a social choice setting in which agents and alternatives are represented by points in a metric space, and the cost of an agent for an alternative is the distance between the corresponding points in the space. The goal is to choose a single alternative to (approximately) minimize the social cost (cost of all agents) or the maximum cost of any agent, when only limited information about the preferences of the agents is given. Previous work has shown that the best possible distortion one can hope to achieve is $3$ when access to the ordinal preferences of the agents is given, even when the distances between alternatives in the metric space are known. We improve upon this bound of $3$ by designing deterministic mechanisms that exploit a bit of cardinal information. We show that it is possible to achieve distortion $1+\sqrt{2}$ by using the ordinal preferences of the agents, the distances between alternatives, and a threshold approval set per agent that contains all alternatives for whom her cost is within an appropriately chosen factor of her cost for her most-preferred alternative. We show that this bound is the best possible for any deterministic mechanism in general metric spaces, and also provide improved bounds for the fundamental case of a line metric.

The dominating NLP paradigm of training a strong neural predictor to perform one task on a specific dataset has led to state-of-the-art performance in a variety of applications (eg. sentiment classification, span-prediction based question answering or machine translation). However, it builds upon the assumption that the data distribution is stationary, ie. that the data is sampled from a fixed distribution both at training and test time. This way of training is inconsistent with how we as humans are able to learn from and operate within a constantly changing stream of information. Moreover, it is ill-adapted to real-world use cases where the data distribution is expected to shift over the course of a model's lifetime. The first goal of this thesis is to characterize the different forms this shift can take in the context of natural language processing, and propose benchmarks and evaluation metrics to measure its effect on current deep learning architectures. We then proceed to take steps to mitigate the effect of distributional shift on NLP models. To this end, we develop methods based on parametric reformulations of the distributionally robust optimization framework. Empirically, we demonstrate that these approaches yield more robust models as demonstrated on a selection of realistic problems. In the third and final part of this thesis, we explore ways of efficiently adapting existing models to new domains or tasks. Our contribution to this topic takes inspiration from information geometry to derive a new gradient update rule which alleviate catastrophic forgetting issues during adaptation.

Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.

Ensembles over neural network weights trained from different random initialization, known as deep ensembles, achieve state-of-the-art accuracy and calibration. The recently introduced batch ensembles provide a drop-in replacement that is more parameter efficient. In this paper, we design ensembles not only over weights, but over hyperparameters to improve the state of the art in both settings. For best performance independent of budget, we propose hyper-deep ensembles, a simple procedure that involves a random search over different hyperparameters, themselves stratified across multiple random initializations. Its strong performance highlights the benefit of combining models with both weight and hyperparameter diversity. We further propose a parameter efficient version, hyper-batch ensembles, which builds on the layer structure of batch ensembles and self-tuning networks. The computational and memory costs of our method are notably lower than typical ensembles. On image classification tasks, with MLP, LeNet, and Wide ResNet 28-10 architectures, our methodology improves upon both deep and batch ensembles.