Despite their incredible performance, it is well reported that deep neural networks tend to be overoptimistic about their prediction confidence. Finding effective and efficient calibration methods for neural networks is therefore an important endeavour towards better uncertainty quantification in deep learning. In this manuscript, we introduce a novel calibration technique named expectation consistency (EC), consisting of a post-training rescaling of the last layer weights by enforcing that the average validation confidence coincides with the average proportion of correct labels. First, we show that the EC method achieves similar calibration performance to temperature scaling (TS) across different neural network architectures and data sets, all while requiring similar validation samples and computational resources. However, we argue that EC provides a principled method grounded on a Bayesian optimality principle known as the Nishimori identity. Next, we provide an asymptotic characterization of both TS and EC in a synthetic setting and show that their performance crucially depends on the target function. In particular, we discuss examples where EC significantly outperforms TS.

Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of the marginal likelihood. The learned harmonic mean estimator learns an importance sampling target distribution that approximates the optimal distribution. While the approximation need not be highly accurate, it is critical that the probability mass of the learned distribution is contained within the posterior in order to avoid the exploding variance problem. In previous work a bespoke optimization problem is introduced when training models in order to ensure this property is satisfied. In the current article we introduce the use of normalizing flows to represent the importance sampling target distribution. A flow-based model is trained on samples from the posterior by maximum likelihood estimation. Then, the probability density of the flow is concentrated by lowering the variance of the base distribution, i.e. by lowering its "temperature", ensuring its probability mass is contained within the posterior. This approach avoids the need for a bespoke optimisation problem and careful fine tuning of parameters, resulting in a more robust method. Moreover, the use of normalizing flows has the potential to scale to high dimensional settings. We present preliminary experiments demonstrating the effectiveness of the use of flows for the learned harmonic mean estimator. The harmonic code implementing the learned harmonic mean, which is publicly available, has been updated to now support normalizing flows.

We introduce SABRE, a novel framework for robust variational Bayesian peer-to-peer federated learning. We analyze the robustness of the known variational Bayesian peer-to-peer federated learning framework (BayP2PFL) against poisoning attacks and subsequently show that BayP2PFL is not robust against those attacks. The new SABRE aggregation methodology is then devised to overcome the limitations of the existing frameworks. SABRE works well in non-IID settings, does not require the majority of the benign nodes over the compromised ones, and even outperforms the baseline algorithm in benign settings. We theoretically prove the robustness of our algorithm against data / model poisoning attacks in a decentralized linear regression setting. Proof-of-Concept evaluations on benchmark data from image classification demonstrate the superiority of SABRE over the existing frameworks under various poisoning attacks.

Change-of-variables (CoV) formulas allow to reduce complicated probability densities to simpler ones by a learned transformation with tractable Jacobian determinant. They are thus powerful tools for maximum-likelihood learning, Bayesian inference, outlier detection, model selection, etc. CoV formulas have been derived for a large variety of model types, but this information is scattered over many separate works. We present a systematic treatment from the unifying perspective of encoder/decoder architectures, which collects 28 CoV formulas in a single place, reveals interesting relationships between seemingly diverse methods, emphasizes important distinctions that are not always clear in the literature, and identifies surprising gaps for future research.

The topology of the large-scale structure of the universe contains valuable information on the underlying cosmological parameters. While persistent homology can extract this topological information, the optimal method for parameter estimation from the tool remains an open question. To address this, we propose a neural network model to map persistence images to cosmological parameters. Through a parameter recovery test, we demonstrate that our model makes accurate and precise estimates, considerably outperforming conventional Bayesian inference approaches.

Neural network pruning is a highly effective technique aimed at reducing the computational and memory demands of large neural networks. In this research paper, we present a novel approach to pruning neural networks utilizing Bayesian inference, which can seamlessly integrate into the training procedure. Our proposed method leverages the posterior probabilities of the neural network prior to and following pruning, enabling the calculation of Bayes factors. The calculated Bayes factors guide the iterative pruning. Through comprehensive evaluations conducted on multiple benchmarks, we demonstrate that our method achieves desired levels of sparsity while maintaining competitive accuracy.

We investigate the problem of estimating the structure of a weighted network from repeated measurements of a Gaussian Graphical Model (GGM) on the network. In this vein, we consider GGMs whose covariance structures align with the geometry of the weighted network on which they are based. Such GGMs have been of longstanding interest in statistical physics, and are referred to as the Gaussian Free Field (GFF). In recent years, they have attracted considerable interest in the machine learning and theoretical computer science. In this work, we propose a novel estimator for the weighted network (equivalently, its Laplacian) from repeated measurements of a GFF on the network, based on the Fourier analytic properties of the Gaussian distribution. In this pursuit, our approach exploits complex-valued statistics constructed from observed data, that are of interest on their own right. We demonstrate the effectiveness of our estimator with concrete recovery guarantees and bounds on the required sample complexity. In particular, we show that the proposed statistic achieves the parametric rate of estimation for fixed network size. In the setting of networks growing with sample size, our results show that for Erdos-Renyi random graphs $G(d,p)$ above the connectivity threshold, we demonstrate that network recovery takes place with high probability as soon as the sample size $n$ satisfies $n \gg d^4 \log d \cdot p^{-2}$.

This paper introduces a first implementation of a novel likelihood-ratio-based approach for constructing confidence intervals for neural networks. Our method, called DeepLR, offers several qualitative advantages: most notably, the ability to construct asymmetric intervals that expand in regions with a limited amount of data, and the inherent incorporation of factors such as the amount of training time, network architecture, and regularization techniques. While acknowledging that the current implementation of the method is prohibitively expensive for many deep-learning applications, the high cost may already be justified in specific fields like medical predictions or astrophysics, where a reliable uncertainty estimate for a single prediction is essential. This work highlights the significant potential of a likelihood-ratio-based uncertainty estimate and establishes a promising avenue for future research.

This article aims to investigate the impact of noise on parameter fitting for an Ornstein-Uhlenbeck process, focusing on the effects of multiplicative and thermal noise on the accuracy of signal separation. To address these issues, we propose algorithms and methods that can effectively distinguish between thermal and multiplicative noise and improve the precision of parameter estimation for optimal data analysis. Specifically, we explore the impact of both multiplicative and thermal noise on the obfuscation of the actual signal and propose methods to resolve them. Firstly, we present an algorithm that can effectively separate thermal noise with comparable performance to Hamilton Monte Carlo (HMC) but with significantly improved speed. Subsequently, we analyze multiplicative noise and demonstrate that HMC is insufficient for isolating thermal and multiplicative noise. However, we show that, with additional knowledge of the ratio between thermal and multiplicative noise, we can accurately distinguish between the two types of noise when provided with a sufficiently large sampling rate or an amplitude of multiplicative noise smaller than thermal noise. This finding results in a situation that initially seems counterintuitive. When multiplicative noise dominates the noise spectrum, we can successfully estimate the parameters for such systems after adding additional white noise to shift the noise balance.

Disordered many-body systems exhibit a wide range of emergent phenomena across different scales. These complex behaviors can be utilized for various information processing tasks such as error correction, learning, and optimization. Despite the empirical success of utilizing these systems for intelligent tasks, the underlying principles that govern their emergent intelligent behaviors remain largely unknown. In this thesis, we aim to characterize such emergent intelligence in disordered systems through statistical physics. We chart a roadmap for our efforts in this thesis based on two axes: learning mechanisms (long-term memory vs. working memory) and learning dynamics (artificial vs. natural). Throughout our journey, we uncover relationships between learning mechanisms and physical dynamics that could serve as guiding principles for designing intelligent systems. We hope that our investigation into the emergent intelligence of seemingly disparate learning systems can expand our current understanding of intelligence beyond neural systems and uncover a wider range of computational substrates suitable for AI applications.