The evaluation of noisy binary classifiers on unlabeled data is treated as a streaming task: given a data sketch of the decisions by an ensemble, estimate the true prevalence of the labels as well as each classifier's accuracy on them. Two fully algebraic evaluators are constructed to do this. Both are based on the assumption that the classifiers make independent errors. The first is based on majority voting. The second, the main contribution of the paper, is guaranteed to be correct. But how do we know the classifiers are independent on any given test? This principal/agent monitoring paradox is ameliorated by exploiting the failures of the independent evaluator to return sensible estimates. A search for nearly error independent trios is empirically carried out on the \texttt{adult}, \texttt{mushroom}, and \texttt{two-norm} datasets by using the algebraic failure modes to reject evaluation ensembles as too correlated. The searches are refined by constructing a surface in evaluation space that contains the true value point. The algebra of arbitrarily correlated classifiers permits the selection of a polynomial subset free of any correlation variables. Candidate evaluation ensembles are rejected if their data sketches produce independent estimates too far from the constructed surface. The results produced by the surviving ensembles can sometimes be as good as 1\%. But handling even small amounts of correlation remains a challenge. A Taylor expansion of the estimates produced when independence is assumed but the classifiers are, in fact, slightly correlated helps clarify how the independent evaluator has algebraic `blind spots'.

Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, feasible densities are Gaussian and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. As these updates require high-dimensional integration, this work first proposes an efficient quasirandom quadrature sequence for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratics and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate quadratics. The corresponding variational updates require 4 loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.

One of the core problems of modern Bayesian inference is to compute the posterior distribution, a joint probability measure over unknown quantities, such as model parameters and unobserved latent variables, obtained by combining data information with prior knowledge in a principled manner. Modern statistics often rely on complex models for which the posterior distribution is analytically intractable and requires approximate computation. As a common alternative strategy to conventional Markov chain Monte Carlo (MCMC) sampling approach for approximating the posterior, variational inference (VI, [10]), or variational Bayes [27], finds the closest member in a user specified class of analytically tractable distributions, referred to as the variational (distribution) family, to approximate the target posterior. Although MCMC is asymptotically exact, VI is usually orders of magnitude faster [12, 62] since it turns the sampling or integration into an optimization problem. VI has successfully demonstrated its power in a wide variety of applications, including clustering [11, 23], semi-supervised learning [38], neural-network training [5, 52], and probabilistic modeling [13, 36]. Among various approximating schemes, the mean-field (MF) approximation, which originates from statistical mechanics and uses the approximating family consisting of all fully factorized density functions over (blocks of) the unknown quantities, is the most widely used and representative instance of VI that is conceptually simple yet practically powerful. On the downside, VI still requires certain conditional conjugacy structure to facilitate efficient computation (c.f.

Image-based visual servoing (IBVS) is a widely-used approach in robotics that employs visual information to guide robots towards desired positions. However, occlusions in this approach can lead to visual servoing failure and degrade the control performance due to the obstructed vision feature points that are essential for providing visual feedback. In this paper, we propose a Control Barrier Function (CBF) based controller that enables occlusion-free IBVS tasks by automatically adjusting the robot's configuration to keep the feature points in the field of view and away from obstacles. In particular, to account for measurement noise of the feature points, we develop the Probabilistic Control Barrier Certificates (PrCBC) using control barrier functions that encode the chance-constrained occlusion avoidance constraints under uncertainty into deterministic admissible control space for the robot, from which the resulting configuration of robot ensures that the feature points stay occlusion free from obstacles with a satisfying predefined probability. By integrating such constraints with a Model Predictive Control (MPC) framework, the sequence of optimized control inputs can be derived to achieve the primary IBVS task while enforcing the occlusion avoidance during robot movements. Simulation results are provided to validate the performance of our proposed method.

Graph neural networks (GNNs) have gained an increasing amount of popularity due to their superior capability in learning node embeddings for various graph inference tasks, but training them can raise privacy concerns. To address this, we propose using link local differential privacy over decentralized nodes, enabling collaboration with an untrusted server to train GNNs without revealing the existence of any link. Our approach spends the privacy budget separately on links and degrees of the graph for the server to better denoise the graph topology using Bayesian estimation, alleviating the negative impact of LDP on the accuracy of the trained GNNs. We bound the mean absolute error of the inferred link probabilities against the ground truth graph topology. We then propose two variants of our LDP mechanism complementing each other in different privacy settings, one of which estimates fewer links under lower privacy budgets to avoid false positive link estimates when the uncertainty is high, while the other utilizes more information and performs better given relatively higher privacy budgets. Furthermore, we propose a hybrid variant that combines both strategies and is able to perform better across different privacy budgets. Extensive experiments show that our approach outperforms existing methods in terms of accuracy under varying privacy budgets.

Employing a forward diffusion chain to gradually map the data to a noise distribution, diffusion-based generative models learn how to generate the data by inferring a reverse diffusion chain. However, this approach is slow and costly because it needs many forward and reverse steps. We propose a faster and cheaper approach that adds noise not until the data become pure random noise, but until they reach a hidden noisy-data distribution that we can confidently learn. Then, we use fewer reverse steps to generate data by starting from this hidden distribution that is made similar to the noisy data. We reveal that the proposed model can be cast as an adversarial auto-encoder empowered by both the diffusion process and a learnable implicit prior. Experimental results show even with a significantly smaller number of reverse diffusion steps, the proposed truncated diffusion probabilistic models can provide consistent improvements over the non-truncated ones in terms of performance in both unconditional and text-guided image generations. Generating photo-realistic images with probabilistic models is a challenging and important task in machine learning and computer vision, with many potential applications in data augmentation, image editing, style transfer, etc. This new modeling class, which includes both score-based and diffusion-based generative models, uses noise injection to gradually corrupt the data distribution into a simple noise distribution that can be easily sampled from, and then uses a denoising network to reverse the noise injection to generate photo-realistic images. From the perspective of score matching (Hyvärinen & Dayan, 2005; Vincent, 2011) and Langevin dynamics (Neal, 2011; Welling & Teh, 2011), the denoising network is trained by matching the score function, which is the gradient of the log-density of the data, of the corrupted data distribution and that of the generator distribution at different noise levels (Song & Ermon, 2019). This training objective can also be formulated under diffusion-based generative models (Sohl-Dickstein et al., 2015; Ho et al., 2020). These two types of models have been further unified by Song et al. (2021b) under the framework of discretized stochastic differential equations.

For many decades, experimental solid mechanics has played a crucial role in characterizing and understanding the mechanical properties of natural and novel materials. Recent advances in machine learning (ML) provide new opportunities for the field, including experimental design, data analysis, uncertainty quantification, and inverse problems. As the number of papers published in recent years in this emerging field is exploding, it is timely to conduct a comprehensive and up-to-date review of recent ML applications in experimental solid mechanics. Here, we first provide an overview of common ML algorithms and terminologies that are pertinent to this review, with emphasis placed on physics-informed and physics-based ML methods. Then, we provide thorough coverage of recent ML applications in traditional and emerging areas of experimental mechanics, including fracture mechanics, biomechanics, nano- and micro-mechanics, architected materials, and 2D material. Finally, we highlight some current challenges of applying ML to multi-modality and multi-fidelity experimental datasets and propose several future research directions. This review aims to provide valuable insights into the use of ML methods as well as a variety of examples for researchers in solid mechanics to integrate into their experiments.

Causal machine learning (ML) algorithms recover graphical structures that tell us something about cause-and-effect relationships. The causal representation praovided by these algorithms enables transparency and explainability, which is necessary for decision making in critical real-world problems. Yet, causal ML has had limited impact in practice compared to associational ML. This paper investigates the challenges of causal ML with application to COVID-19 UK pandemic data. We collate data from various public sources and investigate what the various structure learning algorithms learn from these data. We explore the impact of different data formats on algorithms spanning different classes of learning, and assess the results produced by each algorithm, and groups of algorithms, in terms of graphical structure, model dimensionality, sensitivity analysis, confounding variables, predictive and interventional inference. We use these results to highlight open problems in causal structure learning and directions for future research. To facilitate future work, we make all graphs, models, data sets, and source code publicly available online.

This paper introduces Bayesian frameworks for tackling various aspects of multi-criteria decision-making (MCDM) problems, leveraging a probabilistic interpretation of MCDM methods and challenges. By harnessing the flexibility of Bayesian models, the proposed frameworks offer statistically elegant solutions to key challenges in MCDM, such as group decision-making problems and criteria correlation. Additionally, these models can accommodate diverse forms of uncertainty in decision makers' (DMs) preferences, including normal and triangular distributions, as well as interval preferences. To address large-scale group MCDM scenarios, a probabilistic mixture model is developed, enabling the identification of homogeneous subgroups of DMs. Furthermore, a probabilistic ranking scheme is devised to assess the relative importance of criteria and alternatives based on DM(s) preferences. Through experimentation on various numerical examples, the proposed frameworks are validated, demonstrating their effectiveness and highlighting their distinguishing features in comparison to alternative methods.

Bayesian inference for neural networks, or Bayesian deep learning, has the potential to provide well-calibrated predictions with quantified uncertainty and robustness. However, the main hurdle for Bayesian deep learning is its computational complexity due to the high dimensionality of the parameter space. In this work, we propose a novel scheme that addresses this limitation by constructing a low-dimensional subspace of the neural network parameters-referred to as an active subspace-by identifying the parameter directions that have the most significant influence on the output of the neural network. We demonstrate that the significantly reduced active subspace enables effective and scalable Bayesian inference via either Monte Carlo (MC) sampling methods, otherwise computationally intractable, or variational inference. Empirically, our approach provides reliable predictions with robust uncertainty estimates for various regression tasks.