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.
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.
Bayesian few-shot classification has been a focal point in the field of few-shot learning. This paper seamlessly integrates mirror descent-based variational inference into Gaussian process-based few-shot classification, addressing the challenge of non-conjugate inference. By leveraging non-Euclidean geometry, mirror descent achieves accelerated convergence by providing the steepest descent direction along the corresponding manifold. It also exhibits the parameterization invariance property concerning the variational distribution. Experimental results demonstrate competitive classification accuracy, improved uncertainty quantification, and faster convergence compared to baseline models. Additionally, we investigate the impact of hyperparameters and components. Code is publicly available at https://github.com/keanson/MD-BSFC.
Learning expressive stochastic policies instead of deterministic ones has been proposed to achieve better stability, sample complexity, and robustness. Notably, in Maximum Entropy Reinforcement Learning (MaxEnt RL), the policy is modeled as an expressive Energy-Based Model (EBM) over the Q-values. However, this formulation requires the estimation of the entropy of such EBMs, which is an open problem. To address this, previous MaxEnt RL methods either implicitly estimate the entropy, resulting in high computational complexity and variance (SQL), or follow a variational inference procedure that fits simplified actor distributions (e.g., Gaussian) for tractability (SAC). We propose Stein Soft Actor-Critic (S$^2$AC), a MaxEnt RL algorithm that learns expressive policies without compromising efficiency. Specifically, S$^2$AC uses parameterized Stein Variational Gradient Descent (SVGD) as the underlying policy. We derive a closed-form expression of the entropy of such policies. Our formula is computationally efficient and only depends on first-order derivatives and vector products. Empirical results show that S$^2$AC yields more optimal solutions to the MaxEnt objective than SQL and SAC in the multi-goal environment, and outperforms SAC and SQL on the MuJoCo benchmark. Our code is available at: https://github.com/SafaMessaoud/S2AC-Energy-Based-RL-with-Stein-Soft-Actor-Critic
Predictive uncertainty quantification is crucial for reliable decision-making in various applied domains. Bayesian neural networks offer a powerful framework for this task. However, defining meaningful priors and ensuring computational efficiency remain significant challenges, especially for complex real-world applications. This paper addresses these challenges by proposing a novel neural adaptive empirical Bayes (NA-EB) framework. NA-EB leverages a class of implicit generative priors derived from low-dimensional distributions. This allows for efficient handling of complex data structures and effective capture of underlying relationships in real-world datasets. The proposed NA-EB framework combines variational inference with a gradient ascent algorithm. This enables simultaneous hyperparameter selection and approximation of the posterior distribution, leading to improved computational efficiency. We establish the theoretical foundation of the framework through posterior and classification consistency. We demonstrate the practical applications of our framework through extensive evaluations on a variety of tasks, including the two-spiral problem, regression, 10 UCI datasets, and image classification tasks on both MNIST and CIFAR-10 datasets. The results of our experiments highlight the superiority of our proposed framework over existing methods, such as sparse variational Bayesian and generative models, in terms of prediction accuracy and uncertainty quantification.
The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation $\mathbf{t}$ by striking a trade-off between a compression term $I(\mathbf{x};\mathbf{t})$ and a prediction term $I(y;\mathbf{t})$, where $I(\cdot;\cdot)$ refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at \url{https://github.com/SJYuCNEL/Cauchy-Schwarz-Information-Bottleneck}.
We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks.
The application of artificial intelligence (AI) models in fields such as engineering is limited by the known difficulty of quantifying the reliability of an AI's decision. A well-calibrated AI model must correctly report its accuracy on in-distribution (ID) inputs, while also enabling the detection of out-of-distribution (OOD) inputs. A conventional approach to improve calibration is the application of Bayesian ensembling. However, owing to computational limitations and model misspecification, practical ensembling strategies do not necessarily enhance calibration. This paper proposes an extension of variational inference (VI)-based Bayesian learning that integrates calibration regularization for improved ID performance, confidence minimization for OOD detection, and selective calibration to ensure a synergistic use of calibration regularization and confidence minimization. The scheme is constructed successively by first introducing calibration-regularized Bayesian learning (CBNN), then incorporating out-of-distribution confidence minimization (OCM) to yield CBNN-OCM, and finally integrating also selective calibration to produce selective CBNN-OCM (SCBNN-OCM). Selective calibration rejects inputs for which the calibration performance is expected to be insufficient. Numerical results illustrate the trade-offs between ID accuracy, ID calibration, and OOD calibration attained by both frequentist and Bayesian learning methods. Among the main conclusions, SCBNN-OCM is seen to achieve best ID and OOD performance as compared to existing state-of-the-art approaches at the cost of rejecting a sufficiently large number of inputs.
We propose an analytical solution for approximating the gradient of the Evidence Lower Bound (ELBO) in variational inference problems where the statistical model is a Bayesian network consisting of observations drawn from a mixture of a Gaussian distribution embedded in unrelated clutter, known as the clutter problem. The method employs the reparameterization trick to move the gradient operator inside the expectation and relies on the assumption that, because the likelihood factorizes over the observed data, the variational distribution is generally more compactly supported than the Gaussian distribution in the likelihood factors. This allows efficient local approximation of the individual likelihood factors, which leads to an analytical solution for the integral defining the gradient expectation. We integrate the proposed gradient approximation as the expectation step in an EM (Expectation Maximization) algorithm for maximizing ELBO and test against classical deterministic approaches in Bayesian inference, such as the Laplace approximation, Expectation Propagation and Mean-Field Variational Inference. The proposed method demonstrates good accuracy and rate of convergence together with linear computational complexity.
Current state-of-the-art optimizers are adaptive gradient-based optimization methods such as Adam. Recently, there has been an increasing interest in formulating gradient-based optimizers in a probabilistic framework for better modeling the uncertainty of the gradients. Here, we propose to combine both approaches, resulting in the Variational Stochastic Gradient Descent (VSGD) optimizer. We model gradient updates as a probabilistic model and utilize stochastic variational inference (SVI) to derive an efficient and effective update rule. Further, we show how our VSGD method relates to other adaptive gradient-based optimizers like Adam. Lastly, we carry out experiments on two image classification datasets and four deep neural network architectures, where we show that VSGD outperforms Adam and SGD.
Forecasting human trajectories in traffic scenes is critical for safety within mixed or fully autonomous systems. Human future trajectories are driven by two major stimuli, social interactions, and stochastic goals. Thus, reliable forecasting needs to capture these two stimuli. Edge-based relation modeling represents social interactions using pairwise correlations from precise individual states. Nevertheless, edge-based relations can be vulnerable under perturbations. To alleviate these issues, we propose a region-based relation learning paradigm that models social interactions via region-wise dynamics of joint states, i.e., the changes in the density of crowds. In particular, region-wise agent joint information is encoded within convolutional feature grids. Social relations are modeled by relating the temporal changes of local joint information from a global perspective. We show that region-based relations are less susceptible to perturbations. In order to account for the stochastic individual goals, we exploit a conditional variational autoencoder to realize multi-goal estimation and diverse future prediction. Specifically, we perform variational inference via the latent distribution, which is conditioned on the correlation between input states and associated target goals. Sampling from the latent distribution enables the framework to reliably capture the stochastic behavior in test data. We integrate multi-goal estimation and region-based relation learning to model the two stimuli, social interactions, and stochastic goals, in a prediction framework. We evaluate our framework on the ETH-UCY dataset and Stanford Drone Dataset (SDD). We show that the diverse prediction better fits the ground truth when incorporating the relation module. Our framework outperforms the state-of-the-art models on SDD by $27.61\%$/$18.20\%$ of ADE/FDE metrics.
Out-of-distribution (OOD) detection plays a crucial role in ensuring the safety and reliability of deep neural networks in various applications. While there has been a growing focus on OOD detection in visual data, the field of textual OOD detection has received less attention. Only a few attempts have been made to directly apply general OOD detection methods to natural language processing (NLP) tasks, without adequately considering the characteristics of textual data. In this paper, we delve into textual OOD detection with Transformers. We first identify a key problem prevalent in existing OOD detection methods: the biased representation learned through the maximization of the conditional likelihood $p(y\mid x)$ can potentially result in subpar performance. We then propose a novel variational inference framework for OOD detection (VI-OOD), which maximizes the likelihood of the joint distribution $p(x, y)$ instead of $p(y\mid x)$. VI-OOD is tailored for textual OOD detection by efficiently exploiting the representations of pre-trained Transformers. Through comprehensive experiments on various text classification tasks, VI-OOD demonstrates its effectiveness and wide applicability. Our code has been released at \url{https://github.com/liam0949/LLM-OOD}.
Variational Inference (VI) is a commonly used technique for approximate Bayesian inference and uncertainty estimation in deep learning models, yet it comes at a computational cost, as it doubles the number of trainable parameters to represent uncertainty. This rapidly becomes challenging in high-dimensional settings and motivates the use of alternative techniques for inference, such as Monte Carlo Dropout (MCD) or Spectral-normalized Neural Gaussian Process (SNGP). However, such methods have seen little adoption in survival analysis, and VI remains the prevalent approach for training probabilistic neural networks. In this paper, we investigate how to train deep probabilistic survival models in large datasets without introducing additional overhead in model complexity. To achieve this, we adopt three probabilistic approaches, namely VI, MCD, and SNGP, and evaluate them in terms of their prediction performance, calibration performance, and model complexity. In the context of probabilistic survival analysis, we investigate whether non-VI techniques can offer comparable or possibly improved prediction performance and uncertainty calibration compared to VI. In the MIMIC-IV dataset, we find that MCD aligns with VI in terms of the concordance index (0.748 vs. 0.743) and mean absolute error (254.9 vs. 254.7) using hinge loss, while providing C-calibrated uncertainty estimates. Moreover, our SNGP implementation provides D-calibrated survival functions in all datasets compared to VI (4/4 vs. 2/4, respectively). Our work encourages the use of techniques alternative to VI for survival analysis in high-dimensional datasets, where computational efficiency and overhead are of concern.
Gaussian process latent variable models (GPLVMs) are a versatile family of unsupervised learning models commonly used for dimensionality reduction. However, common challenges in modeling data with GPLVMs include inadequate kernel flexibility and improper selection of the projection noise, leading to a type of model collapse characterized by vague latent representations that do not reflect the underlying data structure. This paper addresses these issues by, first, theoretically examining the impact of projection variance on model collapse through the lens of a linear GPLVM. Second, we tackle model collapse due to inadequate kernel flexibility by integrating the spectral mixture (SM) kernel and a differentiable random Fourier feature (RFF) kernel approximation, which ensures computational scalability and efficiency through off-the-shelf automatic differentiation tools for learning the kernel hyperparameters, projection variance, and latent representations within the variational inference framework. The proposed GPLVM, named advisedRFLVM, is evaluated across diverse datasets and consistently outperforms various salient competing models, including state-of-the-art variational autoencoders (VAEs) and other GPLVM variants, in terms of informative latent representations and missing data imputation.
The data bottleneck has emerged as a fundamental challenge in learning based image restoration methods. Researchers have attempted to generate synthesized training data using paired or unpaired samples to address this challenge. This study proposes SeNM-VAE, a semi-supervised noise modeling method that leverages both paired and unpaired datasets to generate realistic degraded data. Our approach is based on modeling the conditional distribution of degraded and clean images with a specially designed graphical model. Under the variational inference framework, we develop an objective function for handling both paired and unpaired data. We employ our method to generate paired training samples for real-world image denoising and super-resolution tasks. Our approach excels in the quality of synthetic degraded images compared to other unpaired and paired noise modeling methods. Furthermore, our approach demonstrates remarkable performance in downstream image restoration tasks, even with limited paired data. With more paired data, our method achieves the best performance on the SIDD dataset.
Given an intractable distribution $p$, the problem of variational inference (VI) is to find the best approximation from some more tractable family $Q$. Commonly, one chooses $Q$ to be a family of factorized distributions (i.e., the mean-field assumption), even though~$p$ itself does not factorize. We show that this mismatch leads to an impossibility theorem: if $p$ does not factorize, then any factorized approximation $q\in Q$ can correctly estimate at most one of the following three measures of uncertainty: (i) the marginal variances, (ii) the marginal precisions, or (iii) the generalized variance (which can be related to the entropy). In practice, the best variational approximation in $Q$ is found by minimizing some divergence $D(q,p)$ between distributions, and so we ask: how does the choice of divergence determine which measure of uncertainty, if any, is correctly estimated by VI? We consider the classic Kullback-Leibler divergences, the more general $\alpha$-divergences, and a score-based divergence which compares $\nabla \log p$ and $\nabla \log q$. We provide a thorough theoretical analysis in the setting where $p$ is a Gaussian and $q$ is a (factorized) Gaussian. We show that all the considered divergences can be \textit{ordered} based on the estimates of uncertainty they yield as objective functions for~VI. Finally, we empirically evaluate the validity of this ordering when the target distribution $p$ is not Gaussian.
For training an encoder network to perform amortized variational inference, the Kullback-Leibler (KL) divergence from the exact posterior to its approximation, known as the inclusive or forward KL, is an increasingly popular choice of variational objective due to the mass-covering property of its minimizer. However, minimizing this objective is challenging. A popular existing approach, Reweighted Wake-Sleep (RWS), suffers from heavily biased gradients and a circular pathology that results in highly concentrated variational distributions. As an alternative, we propose SMC-Wake, a procedure for fitting an amortized variational approximation that uses likelihood-tempered sequential Monte Carlo samplers to estimate the gradient of the inclusive KL divergence. We propose three gradient estimators, all of which are asymptotically unbiased in the number of iterations and two of which are strongly consistent. Our method interleaves stochastic gradient updates, SMC samplers, and iterative improvement to an estimate of the normalizing constant to reduce bias from self-normalization. In experiments with both simulated and real datasets, SMC-Wake fits variational distributions that approximate the posterior more accurately than existing methods.
Stereo matching is a core task for many computer vision and robotics applications. Despite their dominance in traditional stereo methods, the hand-crafted Markov Random Field (MRF) models lack sufficient modeling accuracy compared to end-to-end deep models. While deep learning representations have greatly improved the unary terms of the MRF models, the overall accuracy is still severely limited by the hand-crafted pairwise terms and message passing. To address these issues, we propose a neural MRF model, where both potential functions and message passing are designed using data-driven neural networks. Our fully data-driven model is built on the foundation of variational inference theory, to prevent convergence issues and retain stereo MRF's graph inductive bias. To make the inference tractable and scale well to high-resolution images, we also propose a Disparity Proposal Network (DPN) to adaptively prune the search space of disparity. The proposed approach ranks $1^{st}$ on both KITTI 2012 and 2015 leaderboards among all published methods while running faster than 100 ms. This approach significantly outperforms prior global methods, e.g., lowering D1 metric by more than 50% on KITTI 2015. In addition, our method exhibits strong cross-domain generalization and can recover sharp edges. The codes at https://github.com/aeolusguan/NMRF