Let $(x_{i}, y_{i})_{i=1,\dots,n}$ denote independent samples from a general mixture distribution $\sum_{c\in\mathcal{C}}\rho_{c}P_{c}^{x}$, and consider the hypothesis class of generalized linear models $\hat{y} = F(\Theta^{\top}x)$. In this work, we investigate the asymptotic joint statistics of the family of generalized linear estimators $(\Theta_{1}, \dots, \Theta_{M})$ obtained either from (a) minimizing an empirical risk $\hat{R}_{n}(\Theta;X,y)$ or (b) sampling from the associated Gibbs measure $\exp(-\beta n \hat{R}_{n}(\Theta;X,y))$. Our main contribution is to characterize under which conditions the asymptotic joint statistics of this family depends (on a weak sense) only on the means and covariances of the class conditional features distribution $P_{c}^{x}$. In particular, this allow us to prove the universality of different quantities of interest, such as the training and generalization errors, redeeming a recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis. Finally, we discuss the applications of our results to different machine learning tasks of interest, such as ensembling and uncertainty
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We show that there is no operator that given two state $|\psi\rangle,|\phi\rangle$ compute the transformation: $D|\psi\rangle|\phi\rangle = |\psi\rangle\bigl( \mathbb{I} - 2 |\psi\rangle\langle\psi| \bigr)|\phi\rangle$ The contradiction of the existence follows by showing that using $D$ two players can compute the disjoints of their sets in single round and $O\left( \sqrt{n} \right)$ communication complexity, which shown by Braverman to be impossible \cite{Braverman}.
Penalized logistic regression is extremely useful for binary classification with large number of covariates (higher than the sample size), having several real life applications, including genomic disease classification. However, the existing methods based on the likelihood loss function are sensitive to data contamination and other noise and, hence, robust methods are needed for stable and more accurate inference. In this paper, we propose a family of robust estimators for sparse logistic models utilizing the popular density power divergence based loss function and the general adaptively weighted LASSO penalties. We study the local robustness of the proposed estimators through its influence function and also derive its oracle properties and asymptotic distribution. With extensive empirical illustrations, we demonstrate the significantly improved performance of our proposed estimators over the existing ones with particular gain in robustness. Our proposal is finally applied to analyse four different real datasets for cancer classification, obtaining robust and accurate models, that simultaneously performs gene selection and patient classification.
With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.
We characterise the learning of a mixture of two clouds of data points with generic centroids via empirical risk minimisation in the high dimensional regime, under the assumptions of generic convex loss and convex regularisation. Each cloud of data points is obtained by sampling from a possibly uncountable superposition of Gaussian distributions, whose variance has a generic probability density $\varrho$. Our analysis covers therefore a large family of data distributions, including the case of power-law-tailed distributions with no covariance. We study the generalisation performance of the obtained estimator, we analyse the role of regularisation, and the dependence of the separability transition on the distribution scale parameters.
Explainable Artificial Intelligence (XAI) provides tools to help understanding how the machine learning models work and reach a specific outcome. It helps to increase the interpretability of models and makes the models more trustworthy and transparent. In this context, many XAI methods were proposed being SHAP and LIME the most popular. However, the proposed methods assume that used predictors in the machine learning models are independent which in general is not necessarily true. Such assumption casts shadows on the robustness of the XAI outcomes such as the list of informative predictors. Here, we propose a simple, yet useful proxy that modifies the outcome of any XAI feature ranking method allowing to account for the dependency among the predictors. The proposed approach has the advantage of being model-agnostic as well as simple to calculate the impact of each predictor in the model in presence of collinearity.
Gaussian graphical models can capture complex dependency structures among variables. For such models, Bayesian inference is attractive as it provides principled ways to incorporate prior information and to quantify uncertainty through the posterior distribution. However, posterior computation under the conjugate G-Wishart prior distribution on the precision matrix is expensive for general non-decomposable graphs. We therefore propose a new Markov chain Monte Carlo (MCMC) method named the G-Wishart weighted proposal algorithm (WWA). WWA's distinctive features include delayed acceptance MCMC, Gibbs updates for the precision matrix and an informed proposal distribution on the graph space that enables embarrassingly parallel computations. Compared to existing approaches, WWA reduces the frequency of the relatively expensive sampling from the G-Wishart distribution. This results in faster MCMC convergence, improved MCMC mixing and reduced computing time. Numerical studies on simulated and real data show that WWA provides a more efficient tool for posterior inference than competing state-of-the-art MCMC algorithms.
Learning models that are robust to distribution shifts is a key concern in the context of their real-life applicability. Invariant Risk Minimization (IRM) is a popular framework that aims to learn robust models from multiple environments. The success of IRM requires an important assumption: the underlying causal mechanisms/features remain invariant across environments. When not satisfied, we show that IRM can over-constrain the predictor and to remedy this, we propose a relaxation via $\textit{partial invariance}$. In this work, we theoretically highlight the sub-optimality of IRM and then demonstrate how learning from a partition of training domains can help improve invariant models. Several experiments, conducted both in linear settings as well as with deep neural networks on tasks over both language and image data, allow us to verify our conclusions.
Repairing inconsistent knowledge bases is a task that has been assessed, with great advances over several decades, from within the knowledge representation and reasoning and the database theory communities. As information becomes more complex and interconnected, new types of repositories, representation languages and semantics are developed in order to be able to query and reason about it. Graph databases provide an effective way to represent relationships among data, and allow processing and querying these connections efficiently. In this work, we focus on the problem of computing preferred (subset and superset) repairs for graph databases with data values, using a notion of consistency based on a set of Reg-GXPath expressions as integrity constraints. Specifically, we study the problem of computing preferred repairs based on two different preference criteria, one based on weights and the other based on multisets, showing that in most cases it is possible to retain the same computational complexity as in the case where no preference criterion is available for exploitation.
We present \textit{universal} estimators for the statistical mean, variance, and scale (in particular, the interquartile range) under pure differential privacy. These estimators are universal in the sense that they work on an arbitrary, unknown continuous distribution $\mathcal{P}$ over $\mathbb{R}$, while yielding strong utility guarantees except for ill-behaved $\mathcal{P}$. For certain distribution families like Gaussians or heavy-tailed distributions, we show that our universal estimators match or improve existing estimators, which are often specifically designed for the given family and under \textit{a priori} boundedness assumptions on the mean and variance of $\mathcal{P}$. This is the first time these boundedness assumptions are removed under pure differential privacy. The main technical tools in our development are instance-optimal empirical estimators for the mean and quantiles over the unbounded integer domain, which can be of independent interest.
In many important graph data processing applications the acquired information includes both node features and observations of the graph topology. Graph neural networks (GNNs) are designed to exploit both sources of evidence but they do not optimally trade-off their utility and integrate them in a manner that is also universal. Here, universality refers to independence on homophily or heterophily graph assumptions. We address these issues by introducing a new Generalized PageRank (GPR) GNN architecture that adaptively learns the GPR weights so as to jointly optimize node feature and topological information extraction, regardless of the extent to which the node labels are homophilic or heterophilic. Learned GPR weights automatically adjust to the node label pattern, irrelevant on the type of initialization, and thereby guarantee excellent learning performance for label patterns that are usually hard to handle. Furthermore, they allow one to avoid feature over-smoothing, a process which renders feature information nondiscriminative, without requiring the network to be shallow. Our accompanying theoretical analysis of the GPR-GNN method is facilitated by novel synthetic benchmark datasets generated by the so-called contextual stochastic block model. We also compare the performance of our GNN architecture with that of several state-of-the-art GNNs on the problem of node-classification, using well-known benchmark homophilic and heterophilic datasets. The results demonstrate that GPR-GNN offers significant performance improvement compared to existing techniques on both synthetic and benchmark data.
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