Understanding the generalisation properties of Artificial Deep Neural Networks (ANN) has recently motivated a number of statistical questions. These models perform well in practice despite perfectly fitting (interpolating) the data, a property that seems at odds with classical statistical theory [49]. This has motivated the investigation of the generalisation performance of methods that achieve zero training error (interpolators) [32, 9, 11, 10, 8] and, in the context of linear least squares, the unique least norm solution to which gradient descent converges [22, 5, 37, 8, 21, 38, 20, 39]. Overparameterized linear models, where the number of variables exceed the number of points, are arguably the simplest and most natural setting where interpolation can be studied. Moreover, in certain regimes ANN can be approximated by suitable linear models [24, 17, 18, 2, 13]. The learning curve (test error versus model capacity) for interpolators has been shown to exhibit a characteristic "Double Descent" [1, 7] shape, where the test error decreases after peaking at the "interpolating" threshold, that is, the model capacity required to interpolate the data. The regime beyond this threshold naturally captures the settings of ANN [49], and thus, has motivated its investigation [36, 44, 39].

Recent articles indicate that deep neural networks are efficient models for various learning problems. However they are often highly sensitive to various changes that cannot be detected by an independent observer. As our understanding of deep neural networks with traditional generalization bounds still remains incomplete, there are several measures which capture the behaviour of the model in case of small changes at a specific state. In this paper we consider adversarial stability in the tangent space and suggest tangent sensitivity in order to characterize stability. We focus on a particular kind of stability with respect to changes in parameters that are induced by individual examples without known labels. We derive several easily computable bounds and empirical measures for feed-forward fully connected ReLU (Rectified Linear Unit) networks and connect tangent sensitivity to the distribution of the activation regions in the input space realized by the network. Our experiments suggest that even simple bounds and measures are associated with the empirical generalization gap.

Normalising flows (NFs) for discrete data are challenging because parameterising bijective transformations of discrete variables requires predicting discrete/integer parameters. Having a neural network architecture predict discrete parameters takes a non-differentiable activation function (eg, the step function) which precludes gradient-based learning. To circumvent this non-differentiability, previous work has employed biased proxy gradients, such as the straight-through estimator. We present an unbiased alternative where rather than deterministically parameterising one transformation, we predict a distribution over latent transformations. With stochastic transformations, the marginal likelihood of the data is differentiable and gradient-based learning is possible via score function estimation. To test the viability of discrete-data NFs we investigate performance on binary MNIST. We observe great challenges with both deterministic proxy gradients and unbiased score function estimation. Whereas the former often fails to learn even a shallow transformation, the variance of the latter could not be sufficiently controlled to admit deeper NFs.

Existing methods for reducing disparate performance of a classifier across different demographic groups assume that one has access to a large data set, thereby focusing on the algorithmic aspect of optimizing overall performance subject to additional constraints. However, poor data collection and imbalanced data sets can severely affect the quality of these methods. In this work, we consider a setting where data collection and optimization are performed simultaneously. In such a scenario, a natural strategy to mitigate the performance difference of the classifier is to provide additional training data drawn from the demographic groups that are worse off. In this paper, we propose to consistently follow this strategy throughout the whole training process and to guide the resulting classifier towards equal performance on the different groups by adaptively sampling each data point from the group that is currently disadvantaged. We provide a rigorous theoretical analysis of our approach in a simplified one-dimensional setting and an extensive experimental evaluation on numerous real-world data sets, including a case study on the data collected during the Flint water crisis.

Differentially private gradient descent (DP-GD) has been extremely effective both theoretically, and in practice, for solving private empirical risk minimization (ERM) problems. In this paper, we focus on understanding the impact of the clipping norm, a critical component of DP-GD, on its convergence. We provide the first formal convergence analysis of clipped DP-GD. More generally, we show that the value which one sets for clipping really matters: done wrong, it can dramatically affect the resulting quality; done properly, it can eliminate the dependence of convergence on the model dimensionality. We do this by showing a dichotomous behavior of the clipping norm. First, we show that if the clipping norm is set smaller than the optimal, even by a constant factor, the excess empirical risk for convex ERMs can increase from $O(1/n)$ to $\Omega(1)$, where $n$ is the number of data samples. Next, we show that, regardless of the value of the clipping norm, clipped DP-GD minimizes a well-defined convex objective over an unconstrained space, as long as the underlying ERM is a generalized linear problem. Furthermore, if the clipping norm is set within at most a constant factor higher than the optimal, then one can obtain an excess empirical risk guarantee that is independent of the dimensionality of the model space. Finally, we extend our result to non-convex generalized linear problems by showing that DP-GD reaches a first-order stationary point as long as the loss is smooth, and the convergence is independent of the dimensionality of the model space.

We study the computation of non-regularized Wasserstein barycenters of probability measures supported on the finite set. The first result gives a stochastic optimization algorithm for the discrete distribution over the probability measures which is comparable with the current best algorithms. The second result extends the previous one to the arbitrary distribution using kernel methods. Moreover, this new algorithm has a total complexity better than the Stochastic Averaging approach via the Sinkhorn algorithm in many cases.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. (2016) provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on {\em nonsmooth} convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case Feldman et al. (2020).

Data augmentation has been widely used to improve generalizability of machine learning models. However, comparatively little work studies data augmentation for graphs. This is largely due to the complex, non-Euclidean structure of graphs, which limits possible manipulation operations. Augmentation operations commonly used in vision and language have no analogs for graphs. Our work studies graph data augmentation for graph neural networks (GNNs) in the context of improving semi-supervised node-classification. We discuss practical and theoretical motivations, considerations and strategies for graph data augmentation. Our work shows that neural edge predictors can effectively encode class-homophilic structure to promote intra-class edges and demote inter-class edges in given graph structure, and our main contribution introduces the GAug graph data augmentation framework, which leverages these insights to improve performance in GNN-based node classification via edge prediction. Extensive experiments on multiple benchmarks show that augmentation via GAug improves performance across GNN architectures and datasets.

In many real-world scenarios, decision makers seek to efficiently optimize multiple competing objectives in a sample-efficient fashion. Multi-objective Bayesian optimization (BO) is a common approach, but many existing acquisition functions do not have known analytic gradients and suffer from high computational overhead. We leverage recent advances in programming models and hardware acceleration for multi-objective BO using Expected Hypervolume Improvement (EHVI)---an algorithm notorious for its high computational complexity. We derive a novel formulation of $q$-Expected Hypervolume Improvement ($q$EHVI), an acquisition function that extends EHVI to the parallel, constrained evaluation setting. $q$EHVI is an exact computation of the joint EHVI of $q$ new candidate points (up to Monte-Carlo (MC) integration error). Whereas previous EHVI formulations rely on gradient-free acquisition optimization or approximated gradients, we compute exact gradients of the MC estimator via auto-differentiation, thereby enabling efficient and effective optimization using first-order and quasi-second-order methods. Lastly, our empirical evaluation demonstrates that $q$EHVI is computationally tractable in many practical scenarios and outperforms state-of-the-art multi-objective BO algorithms at a fraction of their wall time.

This research proposes and evaluates scoring and assessment methods for Virtual Reality (VR) training simulators. VR simulators capture detailed n-dimensional human motion data which is useful for performance analysis. Custom made medical haptic VR training simulators were developed and used to record data from 271 trainees of multiple clinical experience levels. DTW Multivariate Prototyping (DTW-MP) is proposed. VR data was classified as Novice, Intermediate or Expert. Accuracy of algorithms applied for time-series classification were: dynamic time warping 1-nearest neighbor (DTW-1NN) 60%, nearest centroid SoftDTW classification 77.5%, Deep Learning: ResNet 85%, FCN 75%, CNN 72.5% and MCDCNN 28.5%. Expert VR data recordings can be used for guidance of novices. Assessment feedback can help trainees to improve skills and consistency. Motion analysis can identify different techniques used by individuals. Mistakes can be detected dynamically in real-time, raising alarms to prevent injuries.