The application of neural network models to scientific machine learning tasks has proliferated in recent years. In particular, neural network models have proved to be adept at modeling processes with spatial-temporal complexity. Nevertheless, these highly parameterized models have garnered skepticism in their ability to produce outputs with quantified error bounds over the regimes of interest. Hence there is a need to find uncertainty quantification methods that are suitable for neural networks. In this work we present comparisons of the parametric uncertainty quantification of neural networks modeling complex spatial-temporal processes with Hamiltonian Monte Carlo and Stein variational gradient descent and its projected variant. Specifically we apply these methods to graph convolutional neural network models of evolving systems modeled with recurrent neural network and neural ordinary differential equations architectures. We show that Stein variational inference is a viable alternative to Monte Carlo methods with some clear advantages for complex neural network models. For our exemplars, Stein variational interference gave similar uncertainty profiles through time compared to Hamiltonian Monte Carlo, albeit with generally more generous variance.Projected Stein variational gradient descent also produced similar uncertainty profiles to the non-projected counterpart, but large reductions in the active weight space were confounded by the stability of the neural network predictions and the convoluted likelihood landscape.
Bayesian optimization is a widely used technique for optimizing black-box functions, with Expected Improvement (EI) being the most commonly utilized acquisition function in this domain. While EI is often viewed as distinct from other information-theoretic acquisition functions, such as entropy search (ES) and max-value entropy search (MES), our work reveals that EI can be considered a special case of MES when approached through variational inference (VI). In this context, we have developed the Variational Entropy Search (VES) methodology and the VES-Gamma algorithm, which adapts EI by incorporating principles from information-theoretic concepts. The efficacy of VES-Gamma is demonstrated across a variety of test functions and read datasets, highlighting its theoretical and practical utilities in Bayesian optimization scenarios.
Bayesian neural networks (BNNs) offer uncertainty quantification but come with the downside of substantially increased training and inference costs. Sparse BNNs have been investigated for efficient inference, typically by either slowly introducing sparsity throughout the training or by post-training compression of dense BNNs. The dilemma of how to cut down massive training costs remains, particularly given the requirement to learn about the uncertainty. To solve this challenge, we introduce Sparse Subspace Variational Inference (SSVI), the first fully sparse BNN framework that maintains a consistently highly sparse Bayesian model throughout the training and inference phases. Starting from a randomly initialized low-dimensional sparse subspace, our approach alternately optimizes the sparse subspace basis selection and its associated parameters. While basis selection is characterized as a non-differentiable problem, we approximate the optimal solution with a removal-and-addition strategy, guided by novel criteria based on weight distribution statistics. Our extensive experiments show that SSVI sets new benchmarks in crafting sparse BNNs, achieving, for instance, a 10-20x compression in model size with under 3\% performance drop, and up to 20x FLOPs reduction during training compared with dense VI training. Remarkably, SSVI also demonstrates enhanced robustness to hyperparameters, reducing the need for intricate tuning in VI and occasionally even surpassing VI-trained dense BNNs on both accuracy and uncertainty metrics.
BlackJAX is a library implementing sampling and variational inference algorithms commonly used in Bayesian computation. It is designed for ease of use, speed, and modularity by taking a functional approach to the algorithms' implementation. BlackJAX is written in Python, using JAX to compile and run NumpPy-like samplers and variational methods on CPUs, GPUs, and TPUs. The library integrates well with probabilistic programming languages by working directly with the (un-normalized) target log density function. BlackJAX is intended as a collection of low-level, composable implementations of basic statistical 'atoms' that can be combined to perform well-defined Bayesian inference, but also provides high-level routines for ease of use. It is designed for users who need cutting-edge methods, researchers who want to create complex sampling methods, and people who want to learn how these work.
This paper is intended to appear as a chapter for the Handbook of Markov Chain Monte Carlo. The goal of this chapter is to unify various problems at the intersection of Markov chain Monte Carlo (MCMC) and machine learning$\unicode{x2014}$which includes black-box variational inference, adaptive MCMC, normalizing flow construction and transport-assisted MCMC, surrogate-likelihood MCMC, coreset construction for MCMC with big data, Markov chain gradient descent, Markovian score climbing, and more$\unicode{x2014}$within one common framework. By doing so, the theory and methods developed for each may be translated and generalized.
The prior drift is crucial in Continual Test-Time Adaptation (CTTA) methods that only use unlabeled test data, as it can cause significant error propagation. In this paper, we introduce VCoTTA, a variational Bayesian approach to measure uncertainties in CTTA. At the source stage, we transform a pre-trained deterministic model into a Bayesian Neural Network (BNN) via a variational warm-up strategy, injecting uncertainties into the model. During the testing time, we employ a mean-teacher update strategy using variational inference for the student model and exponential moving average for the teacher model. Our novel approach updates the student model by combining priors from both the source and teacher models. The evidence lower bound is formulated as the cross-entropy between the student and teacher models, along with the Kullback-Leibler (KL) divergence of the prior mixture. Experimental results on three datasets demonstrate the method's effectiveness in mitigating prior drift within the CTTA framework.
This paper questions the effectiveness of a modern predictive uncertainty quantification approach, called \emph{evidential deep learning} (EDL), in which a single neural network model is trained to learn a meta distribution over the predictive distribution by minimizing a specific objective function. Despite their perceived strong empirical performance on downstream tasks, a line of recent studies by Bengs et al. identify limitations of the existing methods to conclude their learned epistemic uncertainties are unreliable, e.g., in that they are non-vanishing even with infinite data. Building on and sharpening such analysis, we 1) provide a sharper understanding of the asymptotic behavior of a wide class of EDL methods by unifying various objective functions; 2) reveal that the EDL methods can be better interpreted as an out-of-distribution detection algorithm based on energy-based-models; and 3) conduct extensive ablation studies to better assess their empirical effectiveness with real-world datasets. Through all these analyses, we conclude that even when EDL methods are empirically effective on downstream tasks, this occurs despite their poor uncertainty quantification capabilities. Our investigation suggests that incorporating model uncertainty can help EDL methods faithfully quantify uncertainties and further improve performance on representative downstream tasks, albeit at the cost of additional computational complexity.
Standard diffusion models involve an image transform -- adding Gaussian noise -- and an image restoration operator that inverts this degradation. We observe that the generative behavior of diffusion models is not strongly dependent on the choice of image degradation, and in fact an entire family of generative models can be constructed by varying this choice. Even when using completely deterministic degradations (e.g., blur, masking, and more), the training and test-time update rules that underlie diffusion models can be easily generalized to create generative models. The success of these fully deterministic models calls into question the community's understanding of diffusion models, which relies on noise in either gradient Langevin dynamics or variational inference, and paves the way for generalized diffusion models that invert arbitrary processes. Our code is available at https://github.com/arpitbansal297/Cold-Diffusion-Models
Longitudinal data are important in numerous fields, such as healthcare, sociology and seismology, but real-world datasets present notable challenges for practitioners because they can be high-dimensional, contain structured missingness patterns, and measurement time points can be governed by an unknown stochastic process. While various solutions have been suggested, the majority of them have been designed to account for only one of these challenges. In this work, we propose a flexible and efficient latent-variable model that is capable of addressing all these limitations. Our approach utilizes Gaussian processes to capture temporal correlations between samples and their associated missingness masks as well as to model the underlying point process. We construct our model as a variational autoencoder together with deep neural network parameterised encoder and decoder models, and develop a scalable amortised variational inference approach for efficient model training. We demonstrate competitive performance using both simulated and real datasets.
Current clustering priors for deep latent variable models (DLVMs) require defining the number of clusters a-priori and are susceptible to poor initializations. Addressing these deficiencies could greatly benefit deep learning-based scRNA-seq analysis by performing integration and clustering simultaneously. We adapt the VampPrior (Tomczak & Welling, 2018) into a Dirichlet process Gaussian mixture model, resulting in the VampPrior Mixture Model (VMM), a novel prior for DLVMs. We propose an inference procedure that alternates between variational inference and Empirical Bayes to cleanly distinguish variational and prior parameters. Using the VMM in a Variational Autoencoder attains highly competitive clustering performance on benchmark datasets. Augmenting scVI (Lopez et al., 2018), a popular scRNA-seq integration method, with the VMM significantly improves its performance and automatically arranges cells into biologically meaningful clusters.
Estimating the structure of a Bayesian network, in the form of a directed acyclic graph (DAG), from observational data is a statistically and computationally hard problem with essential applications in areas such as causal discovery. Bayesian approaches are a promising direction for solving this task, as they allow for uncertainty quantification and deal with well-known identifiability issues. From a probabilistic inference perspective, the main challenges are (i) representing distributions over graphs that satisfy the DAG constraint and (ii) estimating a posterior over the underlying combinatorial space. We propose an approach that addresses these challenges by formulating a joint distribution on an augmented space of DAGs and permutations. We carry out posterior estimation via variational inference, where we exploit continuous relaxations of discrete distributions. We show that our approach performs competitively when compared with a wide range of Bayesian and non-Bayesian benchmarks on a range of synthetic and real datasets.
We propose a systematic training-free method to transform the probability flow of a "linear" stochastic process characterized by the equation X_{t}=a_{t}X_{0}+\sigma_{t}X_{1} into a straight constant-speed (SC) flow, reminiscent of Rectified Flow. This transformation facilitates fast sampling along the original probability flow via the Euler method without training a new model of the SC flow. The flexibility of our approach allows us to extend our transformation to inter-convert two posterior flows of two distinct linear stochastic processes. Moreover, we can easily integrate high-order numerical solvers into the transformed SC flow, further enhancing the sampling accuracy and efficiency. Rigorous theoretical analysis and extensive experimental results substantiate the advantages of our framework. Our code is available at this [https://github.com/clarken92/VFM||link].
The partially observable generalized linear model (POGLM) is a powerful tool for understanding neural connectivity under the assumption of existing hidden neurons. With spike trains only recorded from visible neurons, existing works use variational inference to learn POGLM meanwhile presenting the difficulty of learning this latent variable model. There are two main issues: (1) the sampled Poisson hidden spike count hinders the use of the pathwise gradient estimator in VI; and (2) the existing design of the variational model is neither expressive nor time-efficient, which further affects the performance. For (1), we propose a new differentiable POGLM, which enables the pathwise gradient estimator, better than the score function gradient estimator used in existing works. For (2), we propose the forward-backward message-passing sampling scheme for the variational model. Comprehensive experiments show that our differentiable POGLMs with our forward-backward message passing produce a better performance on one synthetic and two real-world datasets. Furthermore, our new method yields more interpretable parameters, underscoring its significance in neuroscience.
A crucial task in predictive maintenance is estimating the remaining useful life of physical systems. In the last decade, deep learning has improved considerably upon traditional model-based and statistical approaches in terms of predictive performance. However, in order to optimally plan maintenance operations, it is also important to quantify the uncertainty inherent to the predictions. This issue can be addressed by turning standard frequentist neural networks into Bayesian neural networks, which are naturally capable of providing confidence intervals around the estimates. Several methods exist for training those models. Researchers have focused mostly on parametric variational inference and sampling-based techniques, which notoriously suffer from limited approximation power and large computational burden, respectively. In this work, we use Stein variational gradient descent, a recently proposed algorithm for approximating intractable distributions that overcomes the drawbacks of the aforementioned techniques. In particular, we show through experimental studies on simulated run-to-failure turbofan engine degradation data that Bayesian deep learning models trained via Stein variational gradient descent consistently outperform with respect to convergence speed and predictive performance both the same models trained via parametric variational inference and their frequentist counterparts trained via backpropagation. Furthermore, we propose a method to enhance performance based on the uncertainty information provided by the Bayesian models. We release the source code at https://github.com/lucadellalib/bdl-rul-svgd.
While the great capability of Transformers significantly boosts prediction accuracy, it could also yield overconfident predictions and require calibrated uncertainty estimation, which can be commonly tackled by Gaussian processes (GPs). Existing works apply GPs with symmetric kernels under variational inference to the attention kernel; however, omitting the fact that attention kernels are in essence asymmetric. Moreover, the complexity of deriving the GP posteriors remains high for large-scale data. In this work, we propose Kernel-Eigen Pair Sparse Variational Gaussian Processes (KEP-SVGP) for building uncertainty-aware self-attention where the asymmetry of attention kernels is tackled by Kernel SVD (KSVD) and a reduced complexity is acquired. Through KEP-SVGP, i) the SVGP pair induced by the two sets of singular vectors from KSVD w.r.t. the attention kernel fully characterizes the asymmetry; ii) using only a small set of adjoint eigenfunctions from KSVD, the derivation of SVGP posteriors can be based on the inversion of a diagonal matrix containing singular values, contributing to a reduction in time complexity; iii) an evidence lower bound is derived so that variational parameters and network weights can be optimized with it. Experiments verify our excellent performances and efficiency on in-distribution, distribution-shift and out-of-distribution benchmarks.
Activity detection is an important task in the next generation grant-free multiple access. While there are a number of existing algorithms designed for this purpose, they mostly require precise information about the network, such as large-scale fading coefficients, small-scale fading channel statistics, noise variance at the access points, and user activity probability. Acquiring these information would take a significant overhead and their estimated values might not be accurate. This problem is even more severe in cell-free networks as there are many of these parameters to be acquired. Therefore, this paper sets out to investigate the activity detection problem without the above-mentioned information. In order to handle so many unknown parameters, this paper employs the Bayesian approach, where the unknown variables are endowed with prior distributions which effectively act as regularizations. Together with the likelihood function, a maximum a posteriori (MAP) estimator and a variational inference algorithm are derived. Extensive simulations demonstrate that the proposed methods, even without the knowledge of these system parameters, perform better than existing state-of-the-art methods, such as covariance-based and approximate message passing methods.
Given a hyperspectral image, the problem of hyperspectral unmixing (HU) is to identify the endmembers (or materials) and the abundance (or endmembers' contributions on pixels) that underlie the image. HU can be seen as a matrix factorization problem with a simplex structure in the abundance matrix factor. In practice, hyperspectral images may exhibit endmember variability (EV) effects -- the endmember matrix factor varies from one pixel to another. In this paper we consider a multilayer simplex-structured matrix factorization model to account for the EV effects. Our multilayer model is based on the postulate that if we arrange the varied endmembers as an expanded endmember matrix, that matrix exhibits a low-rank structure. A variational inference-based maximum-likelihood estimation method is employed to tackle the multilayer factorization problem. Simulation results are provided to demonstrate the performance of our multilayer factorization method.
Deep learning has revolutionized the last decade, being at the forefront of extraordinary advances in a wide range of tasks including computer vision, natural language processing, and reinforcement learning, to name but a few. However, it is well-known that deep models trained via maximum likelihood estimation tend to be overconfident and give poorly-calibrated predictions. Bayesian deep learning attempts to address this by placing priors on the model parameters, which are then combined with a likelihood to perform posterior inference. Unfortunately, for deep models, the true posterior is intractable, forcing the user to resort to approximations. In this thesis, we explore the use of variational inference (VI) as an approximation, as it is unique in simultaneously approximating the posterior and providing a lower bound to the marginal likelihood. If tight enough, this lower bound can be used to optimize hyperparameters and to facilitate model selection. However, this capacity has rarely been used to its full extent for Bayesian neural networks, likely because the approximate posteriors typically used in practice can lack the flexibility to effectively bound the marginal likelihood. We therefore explore three aspects of Bayesian learning for deep models: 1) we ask whether it is necessary to perform inference over as many parameters as possible, or whether it is reasonable to treat many of them as optimizable hyperparameters; 2) we propose a variational posterior that provides a unified view of inference in Bayesian neural networks and deep Gaussian processes; 3) we demonstrate how VI can be improved in certain deep Gaussian process models by analytically removing symmetries from the posterior, and performing inference on Gram matrices instead of features. We hope that our contributions will provide a stepping stone to fully realize the promises of VI in the future.
Variational families with full-rank covariance approximations are known not to work well in black-box variational inference (BBVI), both empirically and theoretically. In fact, recent computational complexity results for BBVI have established that full-rank variational families scale poorly with the dimensionality of the problem compared to e.g. mean-field families. This is particularly critical to hierarchical Bayesian models with local variables; their dimensionality increases with the size of the datasets. Consequently, one gets an iteration complexity with an explicit $\mathcal{O}(N^2)$ dependence on the dataset size $N$. In this paper, we explore a theoretical middle ground between mean-field variational families and full-rank families: structured variational families. We rigorously prove that certain scale matrix structures can achieve a better iteration complexity of $\mathcal{O}\left(N\right)$, implying better scaling with respect to $N$. We empirically verify our theoretical results on large-scale hierarchical models.
An intelligent driving system should dynamically formulate appropriate driving strategies based on the current environment and vehicle status while ensuring system security and reliability. However, methods based on reinforcement learning and imitation learning often suffer from high sample complexity, poor generalization, and low safety. To address these challenges, this paper introduces an efficient and generalized end-to-end autonomous driving system (EGADS) for complex and varied scenarios. The RL agent in our EGADS combines variational inference with normalizing flows, which are independent of distribution assumptions. This combination allows the agent to capture historical information relevant to driving in latent space effectively, thereby significantly reducing sample complexity. Additionally, we enhance safety by formulating robust safety constraints and improve generalization and performance by integrating RL with expert demonstrations. Experimental results demonstrate that, compared to existing methods, EGADS significantly reduces sample complexity, greatly improves safety performance, and exhibits strong generalization capabilities in complex urban scenarios. Particularly, we contributed an expert dataset collected through human expert steering wheel control, specifically using the G29 steering wheel.