A new finite form of de Finetti's representation theorem is established using elementary information-theoretic tools. The distribution of the first $k$ random variables in an exchangeable vector of $n\geq k$ random variables is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided. This bound is tighter than those obtained via earlier information-theoretic proofs, and its utility extends to random variables taking values in general spaces. The core argument employed has its origins in the quantum information-theoretic literature.
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This paper presents some elements of a new approach to construction of Br\`{e}gman relative entropies over nonreflexive Banach spaces, based on nonlinear mappings into reflexive Banach spaces. We apply it to derive a new family of Br\`{e}gman relative entropies over preduals of any W$^*$-algebras and of semifinite JBW-algebras, induced using the Mazur maps into corresponding noncommutative and nonassociative $L_p$ spaces. We prove generalised pythagorean theorem and norm-to-norm continuity of the corresponding entropic projections, as well as H\"{o}lder continuity of the nonassociative Mazur map on positive parts of unit balls. We also discuss the possibility of extension of these results to base normed spaces in spectral duality, pointing to an open problem of construction of $L_p$ spaces over the corresponding order unit spaces.
In the literature of high-dimensional central limit theorems, there is a gap between results for general limiting correlation matrix $\Sigma$ and the strongly non-degenerate case. For the general case where $\Sigma$ may be degenerate, under certain light-tail conditions, when approximating a normalized sum of $n$ independent random vectors by the Gaussian distribution $N(0,\Sigma)$ in multivariate Kolmogorov distance, the best-known error rate has been $O(n^{-1/4})$, subject to logarithmic factors of the dimension. For the strongly non-degenerate case, that is, when the minimum eigenvalue of $\Sigma$ is bounded away from 0, the error rate can be improved to $O(n^{-1/2})$ up to a $\log n$ factor. In this paper, we show that the $O(n^{-1/2})$ rate up to a $\log n$ factor can still be achieved in the degenerate case, provided that the minimum eigenvalue of the limiting correlation matrix of any three components is bounded away from 0. We prove our main results using Stein's method in conjunction with previously unexplored inequalities for the integral of the first three derivatives of the standard Gaussian density over convex polytopes. These inequalities were previously known only for hyperrectangles. Our proof demonstrates the connection between the three-components condition and the third moment Berry--Esseen bound.
We introduce an information-theoretic quantity with similar properties to mutual information that can be estimated from data without making explicit assumptions on the underlying distribution. This quantity is based on a recently proposed matrix-based entropy that uses the eigenvalues of a normalized Gram matrix to compute an estimate of the eigenvalues of an uncentered covariance operator in a reproducing kernel Hilbert space. We show that a difference of matrix-based entropies (DiME) is well suited for problems involving the maximization of mutual information between random variables. While many methods for such tasks can lead to trivial solutions, DiME naturally penalizes such outcomes. We compare DiME to several baseline estimators of mutual information on a toy Gaussian dataset. We provide examples of use cases for DiME, such as latent factor disentanglement and a multiview representation learning problem where DiME is used to learn a shared representation among views with high mutual information.
For regression tasks, standard Gaussian processes (GPs) provide natural uncertainty quantification, while deep neural networks (DNNs) excel at representation learning. We propose to synergistically combine these two approaches in a hybrid method consisting of an ensemble of GPs built on the output of hidden layers of a DNN. GP scalability is achieved via Vecchia approximations that exploit nearest-neighbor conditional independence. The resulting deep Vecchia ensemble not only imbues the DNN with uncertainty quantification but can also provide more accurate and robust predictions. We demonstrate the utility of our model on several datasets and carry out experiments to understand the inner workings of the proposed method.
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. We provide a functional asymptotic expansion for the empirical angular measure in the bivariate case based on the theory of weak convergence in the space of bounded functions. From the expansion, not only can the known asymptotic distribution of the empirical angular measure be recovered, it also enables to find expansions and weak limits for other statistics based on the associated empirical process or its quantile version.
It is unclear how changing the learning rule of a deep neural network alters its learning dynamics and representations. To gain insight into the relationship between learned features, function approximation, and the learning rule, we analyze infinite-width deep networks trained with gradient descent (GD) and biologically-plausible alternatives including feedback alignment (FA), direct feedback alignment (DFA), and error modulated Hebbian learning (Hebb), as well as gated linear networks (GLN). We show that, for each of these learning rules, the evolution of the output function at infinite width is governed by a time varying effective neural tangent kernel (eNTK). In the lazy training limit, this eNTK is static and does not evolve, while in the rich mean-field regime this kernel's evolution can be determined self-consistently with dynamical mean field theory (DMFT). This DMFT enables comparisons of the feature and prediction dynamics induced by each of these learning rules. In the lazy limit, we find that DFA and Hebb can only learn using the last layer features, while full FA can utilize earlier layers with a scale determined by the initial correlation between feedforward and feedback weight matrices. In the rich regime, DFA and FA utilize a temporally evolving and depth-dependent NTK. Counterintuitively, we find that FA networks trained in the rich regime exhibit more feature learning if initialized with smaller correlation between the forward and backward pass weights. GLNs admit a very simple formula for their lazy limit kernel and preserve conditional Gaussianity of their preactivations under gating functions. Error modulated Hebb rules show very small task-relevant alignment of their kernels and perform most task relevant learning in the last layer.
Double-descent curves in neural networks describe the phenomenon that the generalisation error initially descends with increasing parameters, then grows after reaching an optimal number of parameters which is less than the number of data points, but then descends again in the overparameterized regime. In this paper, we use techniques from random matrix theory to characterize the spectral distribution of the empirical feature covariance matrix as a width-dependent perturbation of the spectrum of the neural network Gaussian process (NNGP) kernel, thus establishing a novel connection between the NNGP literature and the random matrix theory literature in the context of neural networks. Our analytical expression allows us to study the generalisation behavior of the corresponding kernel and GP regression, and provides a new interpretation of the double-descent phenomenon, namely as governed by the discrepancy between the width-dependent empirical kernel and the width-independent NNGP kernel.
In this paper, we present the Bayesian inference procedures for the parameters of the multivariate random effects model derived under the assumption of an elliptically contoured distribution when the Berger and Bernardo reference and the Jeffreys priors are assigned to the model parameters. We develop a new numerical algorithm for drawing samples from the posterior distribution, which is based on the hybrid Gibbs sampler. The new approach is compared to the two Metropolis-Hastings algorithms, which were previously derived in the literature, via an extensive simulation study. The results are implemented in practice by considering ten studies about the effectiveness of hypertension treatment for reducing blood pressure where the treatment effects on both the systolic blood pressure and diastolic blood pressure are investigated.
It is typically understood that the training of modern neural networks is a process of fitting the probability distribution of desired output. However, recent paradoxical observations in a number of language generation tasks let one wonder if this canonical probability-based explanation can really account for the empirical success of deep learning. To resolve this issue, we propose an alternative utility-based explanation to the standard supervised learning procedure in deep learning. The basic idea is to interpret the learned neural network not as a probability model but as an ordinal utility function that encodes the preference revealed in training data. In this perspective, training of the neural network corresponds to a utility learning process. Specifically, we show that for all neural networks with softmax outputs, the SGD learning dynamic of maximum likelihood estimation (MLE) can be seen as an iteration process that optimizes the neural network toward an optimal utility function. This utility-based interpretation can explain several otherwise-paradoxical observations about the neural networks thus trained. Moreover, our utility-based theory also entails an equation that can transform the learned utility values back to a new kind of probability estimation with which probability-compatible decision rules enjoy dramatic (double-digits) performance improvements. These evidences collectively reveal a phenomenon of utility-probability duality in terms of what modern neural networks are (truly) modeling: We thought they are one thing (probabilities), until the unexplainable showed up; changing mindset and treating them as another thing (utility values) largely reconcile the theory, despite remaining subtleties regarding its original (probabilistic) identity.
Recently, inference privacy has attracted increasing attention. The inference privacy concern arises most notably in the widely deployed edge-cloud video analytics systems, where the cloud needs the videos captured from the edge. The video data can contain sensitive information and subject to attack when they are transmitted to the cloud for inference. Many privacy protection schemes have been proposed. Yet, the performance of a scheme needs to be determined by experiments or inferred by analyzing the specific case. In this paper, we propose a new metric, \textit{privacy protectability}, to characterize to what degree a video stream can be protected given a certain video analytics task. Such a metric has strong operational meaning. For example, low protectability means that it may be necessary to set up an overall secure environment. We can also evaluate a privacy protection scheme, e.g., assume it obfuscates the video data, what level of protection this scheme has achieved after obfuscation. Our definition of privacy protectability is rooted in information theory and we develop efficient algorithms to estimate the metric. We use experiments on real data to validate that our metric is consistent with empirical measurements on how well a video stream can be protected for a video analytics task.
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