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Run-by-run variability in parallel programs caused by floating-point non-associativity (FPNA) has been known to significantly affect reproducibility in iterative algorithms, due to accumulating errors. Non-reproducibility negatively affects efficiency and effectiveness of correctness testing for stochastic programs. Recently, the sensitivity of deep learning (DL) training and inference pipelines to FPNA have been found to be extreme, and can prevent certification for commercial applications, accurate assessment of robustness and sensitivity, and bug detection. New approaches in scientific computing applications have coupled DL models with high-performance computing (HPC) simulations, leading to an aggravation of debugging and testing challenges. Here we perform an investigation of the statistical properties of FPNA within modern parallel programming models, analyze performance and productivity impacts of replacing atomic operations with deterministic alternatives on GPUs, and examine the recently-added deterministic options within the PyTorch framework within the context of GPU deployment, uncovering and quantifying the impacts of input parameters triggering run-by-run variability and reporting on the reliability and completeness of the documentation. Finally, we evaluate the strategy of exploiting automatic determinism provided by deterministic hardware, using the Groq LPU$^{TM}$ accelerator for inference portions of the DL pipeline. We demonstrate the benefits that this strategy can provide within reproducibility and correctness efforts.

Network scientists often use complex dynamic processes to describe network contagions, but tools for fitting contagion models typically assume simple dynamics. Here, we address this gap by developing a nonparametric method to reconstruct a network and dynamics from a series of node states, using a model that breaks the dichotomy between simple pairwise and complex neighborhood-based contagions. We then show that a network is more easily reconstructed when observed through the lens of complex contagions if it is dense or the dynamic saturates, and that simple contagions are better otherwise.

We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets.

Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.

This short study presents an opportunistic approach to a (more) reliable validation method for prediction uncertainty average calibration. Considering that variance-based calibration metrics (ZMS, NLL, RCE...) are quite sensitive to the presence of heavy tails in the uncertainty and error distributions, a shift is proposed to an interval-based metric, the Prediction Interval Coverage Probability (PICP). It is shown on a large ensemble of molecular properties datasets that (1) sets of z-scores are well represented by Student's-$t(\nu)$ distributions, $\nu$ being the number of degrees of freedom; (2) accurate estimation of 95 $\%$ prediction intervals can be obtained by the simple $2\sigma$ rule for $\nu>3$; and (3) the resulting PICPs are more quickly and reliably tested than variance-based calibration metrics. Overall, this method enables to test 20 $\%$ more datasets than ZMS testing. Conditional calibration is also assessed using the PICP approach.

Micro and survey datasets often contain private information about individuals, like their health status, income or political preferences. Previous studies have shown that, even after data anonymization, a malicious intruder could still be able to identify individuals in the dataset by matching their variables to external information. Disclosure risk measures are statistical measures meant to quantify how big such a risk is for a specific dataset. One of the most common measures is the number of sample unique values that are also population-unique. \cite{Man12} have shown how mixed membership models can provide very accurate estimates of this measure. A limitation of that approach is that the number of extreme profiles has to be chosen by the modeller. In this article, we propose a non-parametric version of the model, based on the Hierarchical Dirichlet Process (HDP). The proposed approach does not require any tuning parameter or model selection step and provides accurate estimates of the disclosure risk measure, even with samples as small as 1$\%$ of the population size. Moreover, a data augmentation scheme to address the presence of structural zeros is presented. The proposed methodology is tested on a real dataset from the New York census.

We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.

Do neural network models of vision learn brain-aligned representations because they share architectural constraints and task objectives with biological vision or because they learn universal features of natural image processing? We characterized the universality of hundreds of thousands of representational dimensions from visual neural networks with varied construction. We found that networks with varied architectures and task objectives learn to represent natural images using a shared set of latent dimensions, despite appearing highly distinct at a surface level. Next, by comparing these networks with human brain representations measured with fMRI, we found that the most brain-aligned representations in neural networks are those that are universal and independent of a network's specific characteristics. Remarkably, each network can be reduced to fewer than ten of its most universal dimensions with little impact on its representational similarity to the human brain. These results suggest that the underlying similarities between artificial and biological vision are primarily governed by a core set of universal image representations that are convergently learned by diverse systems.

This dissertation studies a fundamental open challenge in deep learning theory: why do deep networks generalize well even while being overparameterized, unregularized and fitting the training data to zero error? In the first part of the thesis, we will empirically study how training deep networks via stochastic gradient descent implicitly controls the networks' capacity. Subsequently, to show how this leads to better generalization, we will derive {\em data-dependent} {\em uniform-convergence-based} generalization bounds with improved dependencies on the parameter count. Uniform convergence has in fact been the most widely used tool in deep learning literature, thanks to its simplicity and generality. Given its popularity, in this thesis, we will also take a step back to identify the fundamental limits of uniform convergence as a tool to explain generalization. In particular, we will show that in some example overparameterized settings, {\em any} uniform convergence bound will provide only a vacuous generalization bound. With this realization in mind, in the last part of the thesis, we will change course and introduce an {\em empirical} technique to estimate generalization using unlabeled data. Our technique does not rely on any notion of uniform-convergece-based complexity and is remarkably precise. We will theoretically show why our technique enjoys such precision. We will conclude by discussing how future work could explore novel ways to incorporate distributional assumptions in generalization bounds (such as in the form of unlabeled data) and explore other tools to derive bounds, perhaps by modifying uniform convergence or by developing completely new tools altogether.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.