Variational Inference

Masked discrete diffusion models (MDMs) are a promising new approach to generative modelling, offering the ability for parallel token generation and therefore greater efficiency than autoregressive counterparts. However, achieving an optimal balance between parallel generation and sample quality remains an open problem. Current approaches primarily address this issue through fixed, heuristic parallel sampling methods. There exist some recent learning based approaches to this problem, but its formulation from the perspective of variational inference remains underexplored. In this work, we propose a variational inference framework for learning parallel generation orders for MDMs. As part of our method, we propose a parameterisation for the approximate posterior of generation orders which facilitates parallelism and efficient sampling during training. Using this method, we conduct preliminary experiments on the GSM8K dataset, where our method performs competitively against heuristic sampling strategies in the regime of highly parallel generation. For example, our method achieves 33.1% accuracy with an average of only only 4 generation steps, compared to 23.7-29.0% accuracy achieved by standard competitor methods in the same number of steps. We believe further experiments and analysis of the method will yield valuable insights into the problem of parallel generation with MDMs.

Bayesian deep learning (BDL) has emerged as a principled approach to produce reliable uncertainty estimates by integrating deep neural networks with Bayesian inference, and the selection of informative prior distributions remains a significant challenge. Various function-space variational inference (FSVI) regularisation methods have been presented, assigning meaningful priors over model predictions. However, these methods typically rely on a Gaussian prior, which fails to capture the heavy-tailed statistical characteristics inherent in neural network outputs. By contrast, this work proposes a novel function-space empirical Bayes regularisation framework -- termed ST-FS-EB -- which employs heavy-tailed Student's $t$ priors in both parameter and function spaces. Also, we approximate the posterior distribution through variational inference (VI), inducing an evidence lower bound (ELBO) objective based on Monte Carlo (MC) dropout. Furthermore, the proposed method is evaluated against various VI-based BDL baselines, and the results demonstrate its robust performance in in-distribution prediction, out-of-distribution (OOD) detection and handling distribution shifts.

We develop a Coordinate Ascent Variational Inference (CAVI) algorithm for Bayesian Mixed Data Sampling (MIDAS) regression with linear weight parameterizations. The model separates impact coeffcients from weighting function parameters through a normalization constraint, creating a bilinear structure that renders generic Hamiltonian Monte Carlo samplers unreliable while preserving conditional conjugacy exploitable by CAVI. Each variational update admits a closed-form solution: Gaussian for regression coefficients and weight parameters, Inverse-Gamma for the error variance. The algorithm propagates uncertainty across blocks through second moments, distinguishing it from naive plug-in approximations. In a Monte Carlo study spanning 21 data-generating configurations with up to 50 predictors, CAVI produces posterior means nearly identical to a block Gibbs sampler benchmark while achieving speedups of 107x to 1,772x (Table 9). Generic automatic differentiation VI (ADVI), by contrast, produces bias 714 times larger while being orders of magnitude slower, confirming the value of model-specific derivations. Weight function parameters maintain excellent calibration (coverage above 92%) across all configurations. Impact coefficient credible intervals exhibit the underdispersion characteristic of mean-field approximations, with coverage declining from 89% to 55% as the number of predictors grows a documented trade-off between speed and interval calibration that structured variational methods can address. An empirical application to realized volatility forecasting on S&P 500 daily returns cofirms that CAVI and Gibbs sampling yield virtually identical point forecasts, with CAVI completing each monthly estimation in under 10 milliseconds.

Optimally trading-off exploration and exploitation is the holy grail of reinforcement learning as it promises maximal data-efficiency for solving any task. Bayes-optimal agents achieve this, but obtaining the belief-state and performing planning are both typically intractable. Although deep learning methods can greatly help in scaling this computation, existing methods are still costly to train. To accelerate this, this paper proposes a variational framework for learning and planning in Bayes-adaptive Markov decision processes that coalesces variational belief learning, sequential Monte-Carlo planning, and meta-reinforcement learning. In a single-GPU setup, our new method VariBASeD exhibits favorable scaling to larger planning budgets, improving sample- and runtime-efficiency over prior methods.

Latent Dirichlet Allocation (LDA) is a foundational model for discovering latent thematic structure in discrete data, but its Dirichlet prior cannot represent the rich correlations and hierarchical relationships often present among topics. We introduce the framework of Latent Dirichlet-Tree Allocation (LDTA), a generalization of LDA that replaces the Dirichlet prior with an arbitrary Dirichlet-Tree (DT) distribution. LDTA preserves LDA's generative structure but enables expressive, tree-structured priors over topic proportions. To perform inference, we develop universal mean-field variational inference and Expectation Propagation, providing tractable updates for all DT. We reveal the vectorized nature of the two inference methods through theoretical development, and perform fully vectorized, GPU-accelerated implementations. The resulting framework substantially expands the modeling capacity of LDA while maintaining scalability and computational efficiency.

For approximating a target distribution given only its unnormalized log-density, stochastic gradient-based variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.

We propose a proof of concept for a variational distributional neuron: a compute unit formulated as a VAE brick, explicitly carrying a prior, an amortized posterior and a local ELBO. The unit is no longer a deterministic scalar but a distribution: computing is no longer about propagating values, but about contracting a continuous space of possibilities under constraints. Each neuron parameterizes a posterior, propagates a reparameterized sample and is regularized by the KL term of a local ELBO - hence, the activation is distributional. This "contraction" becomes testable through local constraints and can be monitored via internal measures. The amount of contextual information carried by the unit, as well as the temporal persistence of this information, are locally tuned by distinct constraints. This proposal addresses a structural tension: in sequential generation, causality is predominantly organized in the symbolic space and, even when latents exist, they often remain auxiliary, while the effective dynamics are carried by a largely deterministic decoder. In parallel, probabilistic latent models capture factors of variation and uncertainty, but that uncertainty typically remains borne by global or parametric mechanisms, while units continue to propagate scalars - hence the pivot question: if uncertainty is intrinsic to computation, why does the compute unit not carry it explicitly? We therefore draw two axes: (i) the composition of probabilistic constraints, which must be made stable, interpretable and controllable; and (ii) granularity: if inference is a negotiation of distributions under constraints, should the primitive unit remain deterministic or become distributional? We analyze "collapse" modes and the conditions for a "living neuron", then extend the contribution over time via autoregressive priors over the latent, per unit.

In variational inference (VI), the practitioner approximates a high-dimensional distribution $π$ with a simple surrogate one, often a (product) Gaussian distribution. However, in many cases of practical interest, Gaussian distributions might not capture the correct radial profile of $π$, resulting in poor coverage. In this work, we approach the VI problem from the perspective of optimizing over these radial profiles. Our algorithm radVI is a cheap, effective add-on to many existing VI schemes, such as Gaussian (mean-field) VI and Laplace approximation. We provide theoretical convergence guarantees for our algorithm, owing to recent developments in optimization over the Wasserstein space--the space of probability distributions endowed with the Wasserstein distance--and new regularity properties of radial transport maps in the style of Caffarelli (2000).

Graph generation aims to sample discrete node and edge attributes while satisfying coupled structural constraints. Diffusion models for graphs often adopt largely factorized forward-noising, and many flow-matching methods start from factorized reference noise and coordinate-wise interpolation, so node-edge coupling is not encoded by the generative geometry and must be recovered implicitly by the core network, which can be brittle after discrete decoding. Bayesian Flow Networks (BFNs) evolve distribution parameters and naturally support discrete generation. But classical BFNs typically rely on factorized beliefs and independent channels, which limit geometric evidence fusion. We propose Variational Bayesian Flow Network (VBFN), which performs a variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions. Each Bayesian update reduces to solving a symmetric positive definite linear system, enabling coupled node and edge updates within a single fusion step. We construct sample-agnostic sparse precisions from a representation-induced dependency graph, thereby avoiding label leakage while enforcing node-edge consistency. On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods. The reproducible code is available at https://github.com/Cello2195/VBFN.

We present a class of algorithms for state estimation in nonlinear, non-Gaussian state-space models. Our approach is based on a variational Lagrangian formulation that casts Bayesian inference as a sequence of entropic trust-region updates subject to dynamic consistency constraints. This framework gives rise to a family of forward-backward algorithms whose structure is determined by the chosen factorization of the variational posterior. By focusing on Gauss--Markov approximations, we derive recursive schemes with favorable computational complexity. For general nonlinear, non-Gaussian models, we close the recursions using generalized statistical linear regression and Fourier--Hermite moment matching.

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. It has been empirically observed that MFVI optimizers often suffer from mode collapse. For example, when the target measure $π$ is a mixture $π= w P_0 + (1 - w) P_1$, the MFVI optimizer tends to place most of its mass near a single component of the mixture. This work provides the first theoretical explanation of mode collapse in MFVI. Starting from two-component mixtures, we introduce the notion of \emph{$\varepsilon$-separateness} to quantify the separation of the mixture components. We derive explicit bounds on the mass that any MFVI optimizer assigns to each component, when $P_0$ and $P_1$ are $\varepsilon$-separated for sufficiently small $\varepsilon$. We further extend this analysis to mixture target distributions with multiple components. Our results reveal that the occurrence of mode collapse crucially depends on the relative position of the components. To address this issue, we propose rotational variational inference (RoVI), which augments MFVI with a rotation matrix. The numerical studies support our theoretical findings and illustrate improved mode coverage with RoVI in Bayesian mixture posteriors, including a real-data example.

In many applications, one must propagate parameter uncertainty from an earlier (upstream) analysis, available as samples, to subsequent (downstream) analyses without feedback. This problem is called cutting feedback or cut-Bayes, and the cut-posterior, the optimal posterior preserving information-flow constraints, is well characterized. However, sampling from it (e.g., via nested MCMC) is computationally intensive, while existing variational inference methods for cut-Bayes require access to upstream data and model, often unavailable. We propose a modular and provably accurate cut-Bayes approach requiring no access to upstream data or model. We leverage the characterization of the cut-posterior as the minimizer of the expected downstream conditional Kullback-Leibler divergence over the upstream posterior, replacing the expectation with the sample average over upstream draws. Our method, NeVI-Cut (neural variational inference for cut-Bayes), employs conditional normalizing flows as the variational family for downstream parameters. We provide fixed-data convergence rates of NeVI-Cut in terms of the richness of neural architecture and complexity of the cut-posterior. We establish, to our knowledge, first results on uniform Kullback-Leibler approximation rates of conditional distributions by common flow classes, yielding widely applicable fixed-data error rates for variational flows. A stochastic algorithm implements NeVI-Cut efficiently, and we demonstrate its speed and accuracy on multiple applications.

Variational inference, as an alternative to Markov chain Monte Carlo sampling, has played a transformative role in enabling scalable computation for complex Bayesian models. Nevertheless, existing approaches often depend on either rigid model-specific formulations or stochastic black-box optimization routines. Tangent approximation is a principled class of structured variational methods that exploits the geometry of the underlying probability model. However, its utility has largely been confined to logistic regression and related modeling regimes. In this article, we propose a novel variational framework based on tangent transformation for a broad class of probability models characterized by strongly super-Gaussian likelihoods. Our method leverages convex duality to construct tangent minorants of the log-likelihood, thereby inducing conjugacy with Gaussian priors over model parameters in an otherwise intractable setup. Under mild assumptions on the data-generating mechanism, we establish algorithmic convergence guarantees, a contribution that stands in contrast to the limited theoretical assurances typically available for black-box variational methods. Additionally, we derive near-parametric variational risk bounds. Superior performance of our proposed methodology is illustrated on simulated and real-data scenarios that challenge state-of-the-art variational algorithms in terms of scalability and their ability to consistently capture complex underlying data structure.

Gaussian Mixture Models (GMMs) range among the most frequently used models in machine learning. However, training large, general GMMs becomes computationally prohibitive for data sets that have many data points $N$ of high-dimensionality $D$. For GMMs with arbitrary covariances, we here derive a highly efficient variational approximation, which is then integrated with mixtures of factor analyzers (MFAs). For GMMs with $C$ components, our proposed algorithm substantially reduces runtime complexity from $\mathcal{O}(NCD^2)$ per iteration to a complexity scaling linearly with $D$ and sublinearly with $NC$. In numerical experiments, we first validate that the complexity reduction results in a sublinear scaling for the entire GMM optimization process. Second, we show on large-scale benchmarks that the sublinear algorithm results in speed-ups of an order-of-magnitude compared to the state-of-the-art. Third, as a proof of concept, we finally train GMMs with over 10 billion parameters on about 100 million images, observing training times of less than nine hours on a single state-of-the-art CPU. Finally, and fourth, we demonstrate the effectiveness of large-scale GMMs on the task of zero-shot image denoising, where sublinear training results in state-of-the-art denoising times while competitive denoising performance is maintained.

Pareto fronts are useful to find good task-mixing strategies for multitask finetuning, but they are also costly to compute. To reduce costs, recent works have used existing model merging methods to help train cheap surrogate models to estimate the Pareto fronts. However, no work has yet considered designing new model-merging methods to directly, and provably, improve the quality of Pareto fronts. Here, we fill this gap by proposing a new Bayesian approach called Variational Model Merging. In this approach, existing model-merging methods are obtained as special cases of "posterior-merging" when Gaussian posteriors are used and new model-merging strategies can be derived by using non-Gaussian posteriors. Our main theoretical result is to show that more flexible posteriors necessarily yield better estimates of Pareto fronts. For instance, a Pareto front estimate obtained by merging full-Gaussian posteriors is expected to be better than that obtained by using isotropic Gaussian posteriors. We validate the theory through extensive empirical results on vision and language transformers where better Gaussian families consistently yields better or comparable Pareto fronts. Our work is a rare instance where Bayesian ideas are used to improve Pareto analysis.

Otto's Wasserstein gradient flow of the inclusive (forward) Kullback--Leibler (KL) divergence offers a principled framework for analyzing statistical inference algorithms, yet algorithms targeting the exclusive (reverse) KL divergence are rarely studied with such tools. We establish a unified gradient-flow and PDF framework for inclusive KL inference. We show that maximum mean discrepancy minimization can be viewed as inclusive KL inference with an approximate gradient estimator, and we develop the Fisher--Rao and Wasserstein--Fisher--Rao gradient flows that directly target the inclusive KL divergence. Restricting these flows to the manifold of Gaussian distributions yields explicit gradient-flow ODEs, providing a foundation for Gaussian variational inference. Building on this viewpoint, we further introduce a local-estimator Wasserstein gradient flow whose velocity is obtained by local nonparametric regression, free of density-ratio evaluation or kernel gradients, improving the algorithmic performance over the MMD-based particle method.

Differential privacy (DP) provides robust privacy guarantees for statistical inference, but this can lead to unreliable results and biases in downstream applications. While several noise-aware approaches have been proposed which integrate DP perturbation into the inference, they are limited to specific types of simple probabilistic models. In this work, we propose a novel method for noise-aware approximate Bayesian inference based on stochastic gradient variational inference which can also be applied to high-dimensional and non-conjugate models. We also propose a more accurate evaluation method for noise-aware posteriors. Empirically, our inference method has similar performance to existing methods in the domain where they are applicable. Outside this domain, we obtain accurate coverages on high-dimensional Bayesian linear regression and well-calibrated predictive probabilities on Bayesian logistic regression with the UCI Adult dataset.

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Matérn' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

This paper builds bridges between two families of probabilistic algorithms: (hierarchical) variational inference (VI), which is typically used to model distributions over continuous spaces, and generative flow networks (GFlowNets), which have been used for distributions over discrete structures such as graphs. We demonstrate that, in certain cases, VI algorithms are equivalent to special cases of GFlowNets in the sense of equality of expected gradients of their learning objectives. We then point out the differences between the two families and show how these differences emerge experimentally. Notably, GFlowNets, which borrow ideas from reinforcement learning, are more amenable than VI to off-policy training without the cost of high gradient variance induced by importance sampling. We argue that this property of GFlowNets can provide advantages for capturing diversity in multimodal target distributions.