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Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

Generalized linear models (GLMs) are routinely used for modeling relationships between a response variable and a set of covariates. The simple form of a GLM comes with easy interpretability, but also leads to concerns about model misspecification impacting inferential conclusions. A popular semi-parametric solution adopted in the frequentist literature is quasi-likelihood, which improves robustness by only requiring correct specification of the first two moments. We develop a robust approach to Bayesian inference in GLMs through quasi-posterior distributions. We show that quasi-posteriors provide a coherent generalized Bayes inference method, while also approximating so-called coarsened posteriors. In so doing, we obtain new insights into the choice of coarsening parameter. Asymptotically, the quasi-posterior converges in total variation to a normal distribution and has important connections with the loss-likelihood bootstrap posterior. We demonstrate that it is also well-calibrated in terms of frequentist coverage. Moreover, the loss-scale parameter has a clear interpretation as a dispersion, and this leads to a consolidated method of moments estimator.

We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem, we propose a Bayesian strategy based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.

Causal representation learning algorithms discover lower-dimensional representations of data that admit a decipherable interpretation of cause and effect; as achieving such interpretable representations is challenging, many causal learning algorithms utilize elements indicating prior information, such as (linear) structural causal models, interventional data, or weak supervision. Unfortunately, in exploratory causal representation learning, such elements and prior information may not be available or warranted. Alternatively, scientific datasets often have multiple modalities or physics-based constraints, and the use of such scientific, multimodal data has been shown to improve disentanglement in fully unsupervised settings. Consequently, we introduce a causal representation learning algorithm (causalPIMA) that can use multimodal data and known physics to discover important features with causal relationships. Our innovative algorithm utilizes a new differentiable parametrization to learn a directed acyclic graph (DAG) together with a latent space of a variational autoencoder in an end-to-end differentiable framework via a single, tractable evidence lower bound loss function. We place a Gaussian mixture prior on the latent space and identify each of the mixtures with an outcome of the DAG nodes; this novel identification enables feature discovery with causal relationships. Tested against a synthetic and a scientific dataset, our results demonstrate the capability of learning an interpretable causal structure while simultaneously discovering key features in a fully unsupervised setting.

The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather prediction, and urban planning. Neural Partial Differential Equations (Neural PDEs), which combine deep learning (DL) techniques with domain expertise (e.g., governing equations) for parameterization, have proven to be effective in capturing valuable correlations within spatiotemporal datasets. However, sparse and noisy measurements coupled with modeling approximation introduce aleatoric and epistemic uncertainties. Therefore, quantifying uncertainties propagated from model inputs to outputs remains a challenge and an essential goal for establishing the trustworthiness of Neural PDEs. This work evaluates various Uncertainty Quantification (UQ) approaches for both Forward and Inverse Problems in scientific applications. Specifically, we investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach, Deep Ensembles (DE). To illustrate their performance, we take two canonical PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate that Neural PDEs can effectively reconstruct flow systems and predict the associated unknown parameters. However, it is noteworthy that the results derived from Bayesian methods, based on our observations, tend to display a higher degree of certainty in their predictions as compared to those obtained using the DE. This elevated certainty in predictions suggests that Bayesian techniques might underestimate the true underlying uncertainty, thereby appearing more confident in their predictions than the DE approach.

Generation of simulated detector response to collision products is crucial to data analysis in particle physics, but computationally very expensive. One subdetector, the calorimeter, dominates the computational time due to the high granularity of its cells and complexity of the interactions. Generative models can provide more rapid sample production, but currently require significant effort to optimize performance for specific detector geometries, often requiring many models to describe the varying cell sizes and arrangements, without the ability to generalize to other geometries. We develop a $\textit{geometry-aware}$ autoregressive model, which learns how the calorimeter response varies with geometry, and is capable of generating simulated responses to unseen geometries without additional training. The geometry-aware model outperforms a baseline unaware model by over $50\%$ in several metrics such as the Wasserstein distance between the generated and the true distributions of key quantities which summarize the simulated response. A single geometry-aware model could replace the hundreds of generative models currently designed for calorimeter simulation by physicists analyzing data collected at the Large Hadron Collider. This proof-of-concept study motivates the design of a foundational model that will be a crucial tool for the study of future detectors, dramatically reducing the large upfront investment usually needed to develop generative calorimeter models.

We combine Kronecker products, and quantitative information flow, to give a novel formal analysis for the fine-grained verification of utility in complex privacy pipelines. The combination explains a surprising anomaly in the behaviour of utility of privacy-preserving pipelines -- that sometimes a reduction in privacy results also in a decrease in utility. We use the standard measure of utility for Bayesian analysis, introduced by Ghosh at al., to produce tractable and rigorous proofs of the fine-grained statistical behaviour leading to the anomaly. More generally, we offer the prospect of formal-analysis tools for utility that complement extant formal analyses of privacy. We demonstrate our results on a number of common privacy-preserving designs.

Percolation theory investigates systems of interconnected units, their resilience to damage and their propensity to propagation. For random networks we can solve the percolation problems analytically using the generating function formalism. Yet, with the introduction of higher order networks, the generating function calculations are becoming difficult to perform and harder to validate. Here, I illustrate the mapping of percolation in higher order networks to percolation in chygraphs. Chygraphs are defined as a set of complexes where complexes are hypergraphs with vertex sets in the set of complexes. In a previous work I reported the generating function formalism to percolation in chygraphs and obtained an analytical equation for the order parameter. Taking advantage of this result, I recapitulate analytical results for percolation problems in higher order networks and report extensions to more complex scenarios using symbolic calculations. The code for symbolic calculations can be found at //github.com/av2atgh/chygraph.

This paper introduces an assumption-lean method that constructs valid and efficient lower predictive bounds (LPBs) for survival times with censored data. We build on recent work by Cand\`es et al. (2021), whose approach first subsets the data to discard any data points with early censoring times, and then uses a reweighting technique (namely, weighted conformal inference (Tibshirani et al., 2019)) to correct for the distribution shift introduced by this subsetting procedure. For our new method, instead of constraining to a fixed threshold for the censoring time when subsetting the data, we allow for a covariate-dependent and data-adaptive subsetting step, which is better able to capture the heterogeneity of the censoring mechanism. As a result, our method can lead to LPBs that are less conservative and give more accurate information. We show that in the Type I right-censoring setting, if either of the censoring mechanism or the conditional quantile of survival time is well estimated, our proposed procedure achieves nearly exact marginal coverage, where in the latter case we additionally have approximate conditional coverage. We evaluate the validity and efficiency of our proposed algorithm in numerical experiments, illustrating its advantage when compared with other competing methods. Finally, our method is applied to a real dataset to generate LPBs for users' active times on a mobile app.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.