Bayesian neural networks (BNN) promise to combine the predictive performance of neural networks with principled uncertainty modeling important for safety-critical systems and decision making. However, posterior uncertainty estimates depend on the choice of prior, and finding informative priors in weight-space has proven difficult. This has motivated variational inference (VI) methods that pose priors directly on the function generated by the BNN rather than on weights. In this paper, we address a fundamental issue with such function-space VI approaches pointed out by Burt et al. (2020), who showed that the objective function (ELBO) is negative infinite for most priors of interest. Our solution builds on generalized VI (Knoblauch et al., 2019) with the regularized KL divergence (Quang, 2019) and is, to the best of our knowledge, the first well-defined variational objective for function-space inference in BNNs with Gaussian process (GP) priors. Experiments show that our method incorporates the properties specified by the GP prior on synthetic and small real-world data sets, and provides competitive uncertainty estimates for regression, classification and out-of-distribution detection compared to BNN baselines with both function and weight-space priors.
High-dimensional Bayesian optimization (BO) tasks such as molecular design often require 10,000 function evaluations before obtaining meaningful results. While methods like sparse variational Gaussian processes (SVGPs) reduce computational requirements in these settings, the underlying approximations result in suboptimal data acquisitions that slow the progress of optimization. In this paper we modify SVGPs to better align with the goals of BO: targeting informed data acquisition rather than global posterior fidelity. Using the framework of utility-calibrated variational inference, we unify GP approximation and data acquisition into a joint optimization problem, thereby ensuring optimal decisions under a limited computational budget. Our approach can be used with any decision-theoretic acquisition function and is compatible with trust region methods like TuRBO. We derive efficient joint objectives for the expected improvement and knowledge gradient acquisition functions in both the standard and batch BO settings. Our approach outperforms standard SVGPs on high-dimensional benchmark tasks in control and molecular design.
We introduce YOASOVI, an algorithm for performing fast, self-correcting stochastic optimization for Variational Inference (VI) on large Bayesian heirarchical models. To accomplish this, we take advantage of available information on the objective function used for stochastic VI at each iteration and replace regular Monte Carlo sampling with acceptance sampling. Rather than spend computational resources drawing and evaluating over a large sample for the gradient, we draw only one sample and accept it with probability proportional to the expected improvement in the objective. The following paper develops two versions of the algorithm: the first one based on a naive intuition, and another building up the algorithm as a Metropolis-type scheme. Empirical results based on simulations and benchmark datasets for multivariate Gaussian mixture models show that YOASOVI consistently converges faster (in clock time) and within better optimal neighborhoods than both regularized Monte Carlo and Quasi-Monte Carlo VI algorithms.
We consider deep neural networks in a Bayesian framework with a prior distribution sampling the network weights at random. Following a recent idea of Agapiou and Castillo (2023), who show that heavy-tailed prior distributions achieve automatic adaptation to smoothness, we introduce a simple Bayesian deep learning prior based on heavy-tailed weights and ReLU activation. We show that the corresponding posterior distribution achieves near-optimal minimax contraction rates, simultaneously adaptive to both intrinsic dimension and smoothness of the underlying function, in a variety of contexts including nonparametric regression, geometric data and Besov spaces. While most works so far need a form of model selection built-in within the prior distribution, a key aspect of our approach is that it does not require to sample hyperparameters to learn the architecture of the network. We also provide variational Bayes counterparts of the results, that show that mean-field variational approximations still benefit from near-optimal theoretical support.
Reconstructing jets, which provide vital insights into the properties and histories of subatomic particles produced in high-energy collisions, is a main problem in data analyses in collider physics. This intricate task deals with estimating the latent structure of a jet (binary tree) and involves parameters such as particle energy, momentum, and types. While Bayesian methods offer a natural approach for handling uncertainty and leveraging prior knowledge, they face significant challenges due to the super-exponential growth of potential jet topologies as the number of observed particles increases. To address this, we introduce a Combinatorial Sequential Monte Carlo approach for inferring jet latent structures. As a second contribution, we leverage the resulting estimator to develop a variational inference algorithm for parameter learning. Building on this, we introduce a variational family using a pseudo-marginal framework for a fully Bayesian treatment of all variables, unifying the generative model with the inference process. We illustrate our method's effectiveness through experiments using data generated with a collider physics generative model, highlighting superior speed and accuracy across a range of tasks.
Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the \emph{stochastic} setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first $\mathcal{O}(\frac{1}{T})$ non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (\aka black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of $\mathcal{O}(\frac{1}{T})$ is unlikely without some significant new understanding of optimizing the ELBO using natural gradients.
Spatiotemporal processes are a fundamental tool for modeling dynamics across various domains, from heat propagation in materials to oceanic and atmospheric flows. However, currently available neural network-based modeling approaches fall short when faced with data collected randomly over time and space, as is often the case with sensor networks in real-world applications like crowdsourced earthquake detection or pollution monitoring. In response, we developed a new spatiotemporal method that effectively handles such randomly sampled data. Our model integrates techniques from amortized variational inference, neural differential equations, neural point processes, and implicit neural representations to predict both the dynamics of the system and the probabilistic locations and timings of future observations. It outperforms existing methods on challenging spatiotemporal datasets by offering substantial improvements in predictive accuracy and computational efficiency, making it a useful tool for modeling and understanding complex dynamical systems observed under realistic, unconstrained conditions.
The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified. Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the R\"ossler system with different noise intensities and sources. Then the performance of the overall method - named VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics and fluid dynamics.
Few-shot fine-tuning of Diffusion Models (DMs) is a key advancement, significantly reducing training costs and enabling personalized AI applications. However, we explore the training dynamics of DMs and observe an unanticipated phenomenon: during the training process, image fidelity initially improves, then unexpectedly deteriorates with the emergence of noisy patterns, only to recover later with severe overfitting. We term the stage with generated noisy patterns as corruption stage. To understand this corruption stage, we begin by theoretically modeling the one-shot fine-tuning scenario, and then extend this modeling to more general cases. Through this modeling, we identify the primary cause of this corruption stage: a narrowed learning distribution inherent in the nature of few-shot fine-tuning. To tackle this, we apply Bayesian Neural Networks (BNNs) on DMs with variational inference to implicitly broaden the learned distribution, and present that the learning target of the BNNs can be naturally regarded as an expectation of the diffusion loss and a further regularization with the pretrained DMs. This approach is highly compatible with current few-shot fine-tuning methods in DMs and does not introduce any extra inference costs. Experimental results demonstrate that our method significantly mitigates corruption, and improves the fidelity, quality and diversity of the generated images in both object-driven and subject-driven generation tasks.
Semi-implicit variational inference (SIVI) extends traditional variational families with semi-implicit distributions defined in a hierarchical manner. Due to the intractable densities of semi-implicit distributions, classical SIVI often resorts to surrogates of evidence lower bound (ELBO) that would introduce biases for training. A recent advancement in SIVI, named SIVI-SM, utilizes an alternative score matching objective made tractable via a minimax formulation, albeit requiring an additional lower-level optimization. In this paper, we propose kernel SIVI (KSIVI), a variant of SIVI-SM that eliminates the need for lower-level optimization through kernel tricks. Specifically, we show that when optimizing over a reproducing kernel Hilbert space (RKHS), the lower-level problem has an explicit solution. This way, the upper-level objective becomes the kernel Stein discrepancy (KSD), which is readily computable for stochastic gradient descent due to the hierarchical structure of semi-implicit variational distributions. An upper bound for the variance of the Monte Carlo gradient estimators of the KSD objective is derived, which allows us to establish novel convergence guarantees of KSIVI. We demonstrate the effectiveness and efficiency of KSIVI on both synthetic distributions and a variety of real data Bayesian inference tasks.
Neural Processes (NPs) are deep probabilistic models that represent stochastic processes by conditioning their prior distributions on a set of context points. Despite their obvious advantages in uncertainty estimation for complex distributions, NPs enforce parameterization coupling between the conditional prior model and the posterior model, thereby risking introducing a misspecified prior distribution. We hereby revisit the NP objectives and propose R\'enyi Neural Processes (RNP) to ameliorate the impacts of prior misspecification by optimizing an alternative posterior that achieves better marginal likelihood. More specifically, by replacing the standard KL divergence with the R\'enyi divergence between the model posterior and the true posterior, we scale the density ratio $\frac{p}{q}$ by the power of (1-$\alpha$) in the divergence gradients with respect to the posterior. This hyper parameter $\alpha$ allows us to dampen the effects of the misspecified prior for the posterior update, which has been shown to effectively avoid oversmoothed predictions and improve the expressiveness of the posterior model. Our extensive experiments show consistent log-likelihood improvements over state-of-the-art NP family models which adopt both the variational inference or maximum likelihood estimation objectives. We validate the effectiveness of our approach across multiple benchmarks including regression and image inpainting tasks, and show significant performance improvements of RNPs in real-world regression problems where the underlying prior model is misspecifed.
Temporally causal representation learning aims to identify the latent causal process from time series observations, but most methods require the assumption that the latent causal processes do not have instantaneous relations. Although some recent methods achieve identifiability in the instantaneous causality case, they require either interventions on the latent variables or grouping of the observations, which are in general difficult to obtain in real-world scenarios. To fill this gap, we propose an \textbf{ID}entification framework for instantane\textbf{O}us \textbf{L}atent dynamics (\textbf{IDOL}) by imposing a sparse influence constraint that the latent causal processes have sparse time-delayed and instantaneous relations. Specifically, we establish identifiability results of the latent causal process based on sufficient variability and the sparse influence constraint by employing contextual information of time series data. Based on these theories, we incorporate a temporally variational inference architecture to estimate the latent variables and a gradient-based sparsity regularization to identify the latent causal process. Experimental results on simulation datasets illustrate that our method can identify the latent causal process. Furthermore, evaluations on multiple human motion forecasting benchmarks with instantaneous dependencies indicate the effectiveness of our method in real-world settings.
Directed acyclic graph (DAG) learning is a central task in structure discovery and causal inference. Although the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. We address the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel, provably valid distributions that have support directly on the space of sparse DAGs. These distributions, which we use to define our prior and variational posterior, are induced by a projection operation that maps an arbitrary continuous distribution onto the space of sparse weighted acyclic adjacency matrices. While this projection is combinatorial, it can be solved efficiently using recent continuous reformulations of acyclicity constraints. We empirically demonstrate that our method, ProDAG, can outperform state-of-the-art alternatives in both accuracy and uncertainty quantification.
Re-ranking is a process of rearranging ranking list to more effectively meet user demands by accounting for the interrelationships between items. Existing methods predominantly enhance the precision of search results, often at the expense of diversity, leading to outcomes that may not fulfill the varied needs of users. Conversely, methods designed to promote diversity might compromise the precision of the results, failing to satisfy the users' requirements for accuracy. To alleviate the above problems, this paper proposes a Preference-oriented Diversity Model Based on Mutual-information (PODM-MI), which consider both accuracy and diversity in the re-ranking process. Specifically, PODM-MI adopts Multidimensional Gaussian distributions based on variational inference to capture users' diversity preferences with uncertainty. Then we maximize the mutual information between the diversity preferences of the users and the candidate items using the maximum variational inference lower bound to enhance their correlations. Subsequently, we derive a utility matrix based on the correlations, enabling the adaptive ranking of items in line with user preferences and establishing a balance between the aforementioned objectives. Experimental results on real-world online e-commerce systems demonstrate the significant improvements of PODM-MI, and we have successfully deployed PODM-MI on an e-commerce search platform.
Analyzing high-dimensional count data is a challenge and statistical model-based approaches provide an adequate and efficient framework that preserves explainability. The (multivariate) Poisson-Log-Normal (PLN) model is one such model: it assumes count data are driven by an underlying structured latent Gaussian variable, so that the dependencies between counts solely stems from the latent dependencies. However PLN doesn't account for zero-inflation, a feature frequently observed in real-world datasets. Here we introduce the Zero-Inflated PLN (ZIPLN) model, adding a multivariate zero-inflated component to the model, as an additional Bernoulli latent variable. The Zero-Inflation can be fixed, site-specific, feature-specific or depends on covariates. We estimate model parameters using variational inference that scales up to datasets with a few thousands variables and compare two approximations: (i) independent Gaussian and Bernoulli variational distributions or (ii) Gaussian variational distribution conditioned on the Bernoulli one. The method is assessed on synthetic data and the efficiency of ZIPLN is established even when zero-inflation concerns up to $90\%$ of the observed counts. We then apply both ZIPLN and PLN to a cow microbiome dataset, containing $90.6\%$ of zeroes. Accounting for zero-inflation significantly increases log-likelihood and reduces dispersion in the latent space, thus leading to improved group discrimination.
Differentiable annealed importance sampling (DAIS), proposed by Geffner & Domke (2021) and Zhang et al. (2021), allows optimizing over the initial distribution of AIS. In this paper, we show that, in the limit of many transitions, DAIS minimizes the symmetrized Kullback-Leibler divergence between the initial and target distribution. Thus, DAIS can be seen as a form of variational inference (VI) as its initial distribution is a parametric fit to an intractable target distribution. We empirically evaluate the usefulness of the initial distribution as a variational distribution on synthetic and real-world data, observing that it often provides more accurate uncertainty estimates than VI (optimizing the reverse KL divergence), importance weighted VI, and Markovian score climbing (optimizing the forward KL divergence).
In environments with delayed observation, state augmentation by including actions within the delay window is adopted to retrieve Markovian property to enable reinforcement learning (RL). However, state-of-the-art (SOTA) RL techniques with Temporal-Difference (TD) learning frameworks often suffer from learning inefficiency, due to the significant expansion of the augmented state space with the delay. To improve learning efficiency without sacrificing performance, this work introduces a novel framework called Variational Delayed Policy Optimization (VDPO), which reformulates delayed RL as a variational inference problem. This problem is further modelled as a two-step iterative optimization problem, where the first step is TD learning in the delay-free environment with a small state space, and the second step is behaviour cloning which can be addressed much more efficiently than TD learning. We not only provide a theoretical analysis of VDPO in terms of sample complexity and performance, but also empirically demonstrate that VDPO can achieve consistent performance with SOTA methods, with a significant enhancement of sample efficiency (approximately 50\% less amount of samples) in the MuJoCo benchmark.
This paper addresses the problem of diversity-aware sign language production, where we want to give an image (or sequence) of a signer and produce another image with the same pose but different attributes (\textit{e.g.} gender, skin color). To this end, we extend the variational inference paradigm to include information about the pose and the conditioning of the attributes. This formulation improves the quality of the synthesised images. The generator framework is presented as a UNet architecture to ensure spatial preservation of the input pose, and we include the visual features from the variational inference to maintain control over appearance and style. We generate each body part with a separate decoder. This architecture allows the generator to deliver better overall results. Experiments on the SMILE II dataset show that the proposed model performs quantitatively better than state-of-the-art baselines regarding diversity, per-pixel image quality, and pose estimation. Quantitatively, it faithfully reproduces non-manual features for signers.
Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances in machine learning and variational inference (VI) have lowered the computational barrier by learning from examples. Two VI paradigms have emerged that represent different tradeoffs: amortized and non-amortized. Amortized VI can produce fast results but due to generalizing to many observed datasets it produces suboptimal inference results. Non-amortized VI is slower at inference but finds better posterior approximations since it is specialized towards a single observed dataset. Current amortized VI techniques run into a sub-optimality wall that can not be improved without more expressive neural networks or extra training data. We present a solution that enables iterative improvement of amortized posteriors that uses the same networks architectures and training data. The benefits of our method requires extra computations but these remain frugal since they are based on physics-hybrid methods and summary statistics. Importantly, these computations remain mostly offline thus our method maintains cheap and reusable online evaluation while bridging the approximation gap these two paradigms. We denote our proposed method ASPIRE - Amortized posteriors with Summaries that are Physics-based and Iteratively REfined. We first validate our method on a stylized problem with a known posterior then demonstrate its practical use on a high-dimensional and nonlinear transcranial medical imaging problem with ultrasound. Compared with the baseline and previous methods from the literature our method stands out as an computationally efficient and high-fidelity method for posterior inference.
Black Box Variational Inference is a promising framework in a succession of recent efforts to make Variational Inference more ``black box". However, in basic version it either fails to converge due to instability or requires some fine-tuning of the update steps prior to execution that hinder it from being completely general purpose. We propose a method for regulating its parameter updates by reframing stochastic gradient ascent as a multivariate estimation problem. We examine the properties of the James-Stein estimator as a replacement for the arithmetic mean of Monte Carlo estimates of the gradient of the evidence lower bound. The proposed method provides relatively weaker variance reduction than Rao-Blackwellization, but offers a tradeoff of being simpler and requiring no fine tuning on the part of the analyst. Performance on benchmark datasets also demonstrate a consistent performance at par or better than the Rao-Blackwellized approach in terms of model fit and time to convergence.