Variational Inference

Also known as VI

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Latest papers 140

14 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Jun 1, 2026

VEDAL: Variational Error-Driven Asynchronous Learning for 3D Gaussian Splatting Pruning

3D Gaussian Splatting (3DGS) achieves remarkable novel view synthesis quality with real-time rendering, yet suffers from excessive memory consumption due to millions of Gaussian primitives. Existing pruning methods rely on heuristic importance scores or synchronous batch updates, leading to suboptimal compression and training instability. We propose VEDAL, a principled framework that formulates Gaussian pruning as variational free energy minimization. Our approach introduces (1) a prediction-error gating mechanism that asynchronously activates pruning based on per-Gaussian reconstruction uncertainty, and (2) a variational uncertainty head that models pruning decisions as latent variables with learnable priors. The free energy objective naturally balances reconstruction fidelity against model complexity through an information-theoretic lens. Extensive experiments on Mip-NeRF 360, Tanks&Temples, and Deep Blending demonstrate that VEDAL achieves 5.2x compression with only 0.31 dB PSNR drop, outperforming PUP 3D-GS by +0.05 dB at a higher compression ratio and LightGaussian by +0.35 dB at comparable quality, while maintaining real-time rendering at 185 FPS.

Aoduo Li, Jiancheng Li, Huan Ye +5

Guangdong University of Technology · Huizhou Boluo Power Supply Bureau, Guangdong Power Grid Co., Ltd. · Shenzhen Polytechnic University +1

3DGS Compression 3D Gaussian Splatting 3D Gaussian Pruning Variational Inference Neural Network Pruning

Jun 1, 2026

Flow-Transformed Implicit Processes for Function-Space Variational Inference

Implicit-process priors define distributions over functions through flexible generative mechanisms, making them attractive for Bayesian function-space modelling. However, performing posterior inference with such priors is challenging because their induced function-space distributions are typically not available in closed form. One practical strategy is to approximate the prior using a finite collection of sampled functions, and then represent posterior functions as learned combinations of these samples. Existing approaches commonly place a Gaussian variational distribution over the combination weights. While tractable, this choice limits the shapes of posterior uncertainty that can be represented, especially when the true posterior is asymmetric, heavy-tailed, or multimodal. We propose Flow-Transformed Implicit Processes (FTIP), a variational inference method that makes this finite-dimensional function-space approximation more expressive. Instead of using a Gaussian distribution over the combination weights, FTIP uses a normalizing flow to define a richer variational distribution. This induces a flexible posterior distribution over functions while preserving tractable optimization. We train the model using a Black-Box α objective, allowing us to compare mass-covering and mode-seeking variational behaviour. Experiments show that FTIP captures asymmetric and multimodal posterior structure in function space that Gaussian coefficient approximations tend to smooth or collapse.

Luis A. Ortega, Andrés R. Masegosa, Thomas D. Nielsen

Aalborg University

Bayesian Inference Variational Inference

May 28, 2026

Wasserstein Contraction of Coordinate Ascent Variational Inference

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with Pólya--Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.

Rocco Caprio, Adrien Corenflos, Sam Power

University of Warwick and Bocconi University. · University of Warwick. · University of Bristol.

Bayesian Inference Variational Inference Wasserstein Distance

May 28, 2026

Composing Non-Conjugate Factor Graphs with Closed-Form Variational Inference

Stacking probabilistic building blocks into deeper architectures typically breaks closed-form inference. We show that closed-form inference can be preserved. We identify five factor-graph primitives: a bilinear factor, an exponential link, a Gamma prior, a Gaussian likelihood, and an equality node, and prove that any model composed from them admits closed-form variational message passing. The construction works because each primitive preserves a small set of message families: under mean-field factorization, messages on Gaussian variables remain Gaussian and messages on precision variables remain Gamma, while the only non-conjugate interface, the exponential link, remains tractable through the Gaussian moment-generating function and the sufficient statistics of the Gamma family. We demonstrate composition at increasing depth, from static ensembles through input-dependent gating to split-branch routing, and show that stacking routing layers encodes arbitrary decision trees, establishing universal function approximation with closed-form inference. Applied to ensemble time-series forecasting, the framework yields a Bayesian mixture of experts in which gating functions are inferred rather than learned, providing calibrated uncertainty over expert selection across five benchmark datasets.

Mykola Lukashchuk, Kyrylo Yemets, Wouter M. Kouw +4

Eindhoven University of Technology, the Netherlands · Lviv Polytechnic National University, Lviv, Ukraine · Lazy Dynamics, Utrecht, the Netherlands +1

Factor Graph Optimization Mixture-of-Experts Inference Probabilistic Graphical Models Variational Inference

May 27, 2026

Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models

We study inference in stochastic block models (SBMs) through the lens of optimal transport (OT). We first establish that maximum likelihood variational inference (MLVI) can be interpreted as a semi-relaxed Gromov-Wasserstein (srGW) projection with entropic regularization. While this formulation yields accurate clustering, the entropic regularization prevents transport plans to be sparse, hindering intrinsic model selection. Consequently, we investigate unregularized srGW estimators, and prove that they consistently recover both the SBM connectivity matrix and latent cluster assignments in the asymptotic regime. However, this asymptotic property does not translate into reliable model selection in finite samples, and calls for additional mechanisms to promote sparsity in the inferred cluster proportions. We empirically show that such a regularized formulation yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem, thereby avoiding costly grid search or heuristic model selection procedures.

Simon Queric, Cédric Vincent-Cuaz, Charles Bouveyron +1

Universit´e Cˆote d’Azur, Inria, CNRS LJAD, Maasai Nice, France · EPFL Lausanne, Switzerland · Universit´e Cˆote d’Azur, Inria, CNRS CEPAM, Maasai Nice, France

Model Selection Latent Variable Models Stochastic Block Model Graph Clustering Variational Inference Optimal Transport

May 26, 2026

Soft Specialists: $α$ -Rényi Ensembles for Uncertainty-Aware LLM Post-Training

Existing training approaches for large language models learn a single set of parameters, based on large volumes of data, which is typically heterogeneous, conflicting and often outright contradictory. As a result, the model is forced to compress conflicting goals, and inherent uncertainties into a single, averaged pattern of behaviour. We propose an

α

-Rényi variational framework for learning distributions over post-training parameters, offering an uncertainty-aware alternative to deep ensemble approaches. The resulting variational objective interpolates between classical variational Bayes and predictively oriented posterior learning, balancing between globally plausible individual models against systems of complementary specialists. We identify local stability criteria, demonstrating how model misspecification can make non-degenerate posterior spread locally favourable, manifesting contradictory or conflicting data as epistemic uncertainty. We apply our framework to LLM post-training, learning an ensemble of LoRA adapters attached to a shared, frozen base model, providing a scalable training procedure for both supervised fine-tuning and preference optimisation. Our approach enables training examples to be softly routed across ensemble members, promoting model specialisation and providing actionable uncertainty estimates across different tasks.

Paula Cordero-Encinar, Georgy Tyukin, Andrew B. Duncan

Department of Mathematics, Imperial College London · Bessemer AI

LLM Uncertainty Estimation Language Model Ensembles LLM Post-Training Variational Inference

May 26, 2026

Variational Inference for Evidential Deep Learning

While Deep Neural Networks (DNNs) achieve remarkable performance, their tendency to produce overconfident predictions. Evidential Deep Learning (EDL) mitigates this by formulating predictions as a Dirichlet distribution over class probabilities to explicitly quantify epistemic uncertainty. However, we found that the conventional EDL suffers from two fundamental limitations: a Kullback-Leibler (KL) penalty that only suppresses the evidence of negative classes, producing excessively high evidence therefore decreasing the model's ability to quantify uncertainty, and an absence in theoretical guarantee of setting Dirichlet parameter

α=e+1

. In this paper, we propose a mathematically principled framework, Variational Inference Evidential Deep Learning (VI-EDL). By reformulating evidential learning through the lens of variational inference, we derive an Evidence Lower Bound (ELBO), which prevents the evidence from growing excessively. Theoretically, we rigorously establish a generalization bound and reveal how the predicted uncertainty, feature and network complexity affect this bound, and why setting

\boldsymbolα = \mathbf{e} + \mathbf{1}

can minimize it. Extensive experiments on standard visual and medical datasets demonstrate that VI-EDL achieves state-of-the-art performance, showing excellent performance in out-of-distribution detection, noise detection and autonomous driving scenario. The code is available in https://github.com/seutjw/VI-EDL.

Jiawei Tang, Xinyan Du, Hui Liu +2

School of Computer Science and Engineering Southeast University Nanjing 210096, China · School of Computing Information Sciences Saint Francis University2026 Hong Kong SAR, China · CityDepartmentUniversityof Computerof Hong KongScience Hong Kong SAR, China

Evidential Deep Learning Uncertainty Quantification Variational Inference

May 26, 2026

Amortized Factor Inference Networks for Posterior Inference

Amortized inference promises fast test-time Bayesian inference, but existing methods are inherently tied to fixed models. Extending amortization to unseen models typically requires retraining or costly test-time finetuning. In this paper, we ask: is it possible to build a single inference network capable of generalizing across varying priors, likelihoods, and dimensionality? We introduce Amortized Factor Inference Networks (AFINs), a family of encode-merge-decode inference networks built on dimension-independent modules that map a model specification and its observations to the parameters of a variational posterior. Experimentally, a single trained AFIN achieves posterior accuracy comparable to NUTS and several variational inference methods, while requiring 2 to 4 orders of magnitude less test-time compute. Code is available at https://github.com/joohwanko/AFINs.

Joohwan Ko, Justin Domke

Manning College of Information and Computer Sciences University of Massachusetts Amherst

Amortized Inference Bayesian Inference Variational Inference

May 24, 2026

Unbiased Diffusion Variational Inversion via Principled Posterior Matching

Existing score-based methods for inverse problems often resort to approximate minimization of the KL divergence between the inversion distribution and the Bayesian posterior. Such an approximation leads to severe mode collapse and unreliable uncertainty quantification. In this paper, we propose Principled Posterior Matching (PPM), a framework that returns to the fundamentals of variational inference, rather than using tricky approximations. Instead of relying on heuristic approximations, we rigorously formulate the exact optimization of the KL divergence via the integration of Fisher divergence. We derive a tractable, equivalent gradient form of this integral, enabling precise optimization without the biases introduced by prior approximations. Our analysis clearly reveals that the mode collapse in previous methods stems directly from this approximation gap. Supported by our theoretical solution, PPM unifies two complementary paradigms: (1) In variational inference, PPM adopts mass-covering divergences that significantly improve the inversion diversity and uncertainty quantification; (2) In amortized inference, it enables the training of an efficient reconstruction network for rapid, single-step reconstruction. Furthermore, our formulation naturally extends to a broader family of divergence measures by generalizing the integral of the Fisher divergence. We validate PPM across challenging computational imaging tasks, including inpainting, super-resolution fluorescent microscopy, and radio interferometric black-hole imaging. In all experiments, PPM achieves superior reconstruction fidelity, faithful multimodal posterior recovery, and well-calibrated uncertainty estimates, establishing a robust framework for scientific imaging.

Weimin Bai, Yuxuan Gu, Yifei Wang +2

Peking University

Diffusion Model Inversion Bayesian Inverse Problems Computational Imaging Diffusion-Based Inverse Problems Variational Inference

May 22, 2026

TUBE: Tangent Upper Bound on Evidence for Discrete Diffusion Language Models

Log-likelihood is a standard metric for evaluating generative models. Unfortunately, in contrast to autoregressive models (ARMs), discrete diffusion models generally do not admit exact computation of this quantity. Existing evaluations, therefore, rely on the evidence lower bound (ELBO), leaving unclear how much higher the true value may be. We address this by introducing the Tangent Upper Bound on Evidence (TUBE), a variational upper bound on log-likelihood that admits an unbiased Monte Carlo estimator. Our TUBE extends across latent-variable models, including masked diffusion models (MDMs), any-order ARMs (AO-ARMs), and block variants of both. Applied to block MDMs and block AO-ARMs, TUBE reveals our key empirical finding that these models lie strictly below the exact ARM baseline, showing that ARMs still dominate in likelihood.

Arseny Ivanov, Sergei Kholkin, Vladislav Gromadskii +3

AXXX, Russia · Applied AI Institute, Moscow, Russia · MIRAI, Russia

Masked Diffusion Models Autoregressive Language Modeling Generative Modeling Diffusion Language Models Variational Inference Diffusion Models

May 22, 2026

Onsager-Machlup Posterior Transport for Deep Gaussian Processes

Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density

q_φ(\bU)

by an ELBO (DSVI, IPVI, DDVI, DBVI) or sample by MCMC (SGHMC). We instead frame DGP inference as posterior transport: learn a deterministic sampler that maps a tractable reference measure to posterior-relevant inducing variables, regularised by a path prior derived from the Doob-bridged reference diffusion. Our realisation, OM-Path (formally FBVI-bridge-Path), uses Song's probability-flow ODE applied to DBVI's Doob-bridged forward SDE; the reference drift is closed-form from the bridge marginal coefficients (no score matching) and the path regulariser is the Onsager--Machlup action. At the finite-

ε

value used at training, the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior, and Theorem 1 identifies it with the same posterior's small-noise MAP path via the Freidlin--Wentzell LDP. Two strict path-space ELBO variants on the same bridge backbone (FFJORD log-det; OM-regularised CNF) are derived as ablations. Under a matched-seed paired Wilcoxon test against DBVI on seven UCI regression benchmarks, OM-Path delivers statistically significant wins on the two largest datasets (power:

p\!=\!0.014

, NLL

\mathbf{0.012}

matching the DSVI baseline of

0.017

; protein:

p\!=\!0.002

, RMSE

\mathbf{0.716}

vs.\

0.764

, NLL

\mathbf{1.086}

vs.\

1.149

), statistical ties on yacht / qsar, and concedes boston / energy / concrete to DBVI on small-

N

noisy data. The strict-ELBO variants do not clear DBVI on any UCI metric: in this regime, reducing the variance of the path objective dominates exact-density tracking.

Jian Xu, Delu Zeng, John Paisley +1

RIKEN iTHEMS · RIKEN AIP · South China University of Technology +1

Gaussian Processes Variational Inference Gaussian Process Regression

May 20, 2026

Efficient Learning of Deep State Space Models via Importance Smoothing

Latent state space systems are ubiquitous in statistical modelling, arising naturally when time series are observed through noisy measurements. However, training deep state space models (DSSMs) at scale remains difficult. Two largely distinct strategies have emerged for training DSSMs. The first, auto-encoding DSSMs, trains generative models by optimising a variational lower bound. The second backpropagates through the outputs of classical sequential Monte Carlo (SMC) algorithms. Such approaches can train DSSMs for both discriminative and generative tasks, but their inherently sequential forward passes scale poorly on modern hardware. We propose parallel variational Monte Carlo (PVMC), a new training method that bridges these paradigms and robustly trains DSSMs for both discriminative and generative tasks. Across a set of benchmark experiments, PVMC matches or exceeds state-of-the-art performance while training

10\times

faster than the fastest competing SMC-based approach.

John-Joseph Brady, Nikolas Nusken, Yunpeng Li

Centre for Oral, Clinical and Translational Sciences, King’s College London, London, United Kingdom · Department of Mathematics, King’s College London, London, United Kingdom

Variational Inference State Space Models

May 19, 2026

Closed-form predictive coding via hierarchical Gaussian filters

Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.

Aleksandrs Baskakovs, Sylvain Estebe, Kenneth Enevoldsen +3

Center for Humanities Computing Aarhus University Aarhus, Denmark · Interacting Minds Center Aarhus University Aarhus, Denmark

Predictive Coding Variational Inference Local Learning

May 18, 2026

Distributionally Robust Control via Stein Variational Inference for Contact-Rich Manipulation

Reliable robotic manipulation requires control policies that can accurately represent and adapt to uncertainty arising from contact-rich interactions. Modern data-driven methods mitigate uncertainty through large-scale training and computation, and degrade significantly in performance with limited number of training samples. By contrast, classical model-based controllers are computationally efficient and reliable, but their limited ability to represent task-relevant uncertainty can hinder performance in contact-rich interactions. In this work, we propose to expand the capabilities of model-based manipulation control through more flexible uncertainty modeling that retains performance while exactly adapting to uncertainty. Our approach casts the manipulation problem as a distributionally robust control optimization and proposes a novel deterministic formulation based on Stein variational inference that preserves performance while explicitly modeling task-sensitive parameter uncertainty. As a result, the derived controllers are more aware of task sensitivities to uncertainty, yielding high reliability without compromising performance. Experimental results demonstrate up to 3

\times

improved robustness across a range of contact-rich manipulation tasks under broad parametric uncertainty, outperforming existing model-based control methods.

Hrishikesh Sathyanarayan, Victor Vantilborgh, Harish Ravichandar +2

Yale University, New Haven, CT, USA · Ghent University, Belgium · Georgia Institute of Technology, Atlanta, GA, USA +1

Contact-Rich Robotic Manipulation Robust Control Distributionally Robust Optimization Variational Inference

May 15, 2026

Variational Autoregressive Networks with probability priors

Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.

Piotr Białas, Piotr Korcyl, Tomasz Stebel +1

Institute of Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348, Kraków, Poland · Institute of Theoretical Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348, Kraków, Poland · Doctoral School of Exact and Natural Sciences, Jagiellonian University, ul. Łojasiewicza 11, 30-348, Kraków, Poland

Autoregressive Generation Physics-Informed ML Variational Inference

May 14, 2026

$f$ -Trajectory Balance: A Loss Family for Tuning GFlowNets, Generative Models, and LLMs with Off- and On-Policy Data

In GFlowNets and variational inference, it has been shown that the mean square error between target and model log probabilities is an effective, low variance, surrogate loss for training generative models. This loss has the property that when evaluated on-policy its gradients correspond to those of the KL divergence, while off-policy it remains a valid loss with the same global minimizer. In this work, we demonstrate that this construction can be extended to the whole family of

f

-divergences, leading to a family of losses whose on-policy gradients are that of the corresponding

f

-divergence, but retain the same global minimizer off-policy. Specifically, we show that the on-policy gradients lead to a one to one correspondence between translation invariant loss functions on the target and model log probabilities, and

f

-divergences. This equivalence allows us to design new surrogate loss functions for tuning a wide class of generative models that inherit the properties of the corresponding

f

-divergence, such as being more mode covering, whilst being applicable to off-policy data. We apply our losses on a range of tasks, including classic synthetic examples, SynFlowNets for molecule discovery, and asynchronous large language model (LLM) tuning, demonstrating that our models retain their predicted properties on- and off-policy in a wide class of generative models.

Jake Fawkes, Jason Hartford

Department of Statistics, University College London, UK · Valence Labs, London, UK · Recursion, London, UK.

Generative Flow Networks Variational Inference

May 11, 2026

Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.

Yaman Kindap, Manfred Opper, Benjamin Dupuis +2

1Imperial College London, UK · 2Technical University of Berlin, Germany · 3INRIA, CNRS, Département d’Informatique de l’Ecole Normale Supérieure / PSL, France

Stochastic Differential Equations Bayesian Inference Variational Inference

May 10, 2026

Efficient LLM Reasoning via Variational Posterior Guidance with Efficiency Awareness

Although large language models rely on chain-of-thought for complex reasoning, the overthinking phenomenon severely degrades inference efficiency. Existing reinforcement learning methods compress reasoning chains by designing elaborate reward functions, which renders high-quality samples extremely sparse in the exploration space and creates a sampling bottleneck for the prior policy. Inspired by cognitive science, we theoretically prove that a posterior distribution guided by reference answers achieves higher expected utility than the prior distribution, thus capable of breaking through the sampling bottleneck of high-quality samples. However, the posterior distribution is unavailable during inference. To this end, we formalize efficient reasoning as a variational inference problem and introduce an efficiency-aware evidence lower bound as the theoretical foundation. Based on this, we propose the VPG-EA framework. It adopts a parameter-shared dual-stream architecture to instantiate both the posterior distribution and the prior policy; after filtering out pseudo-efficient paths via cross-view evaluation, it unidirectionally transfers the posterior's efficient patterns to the prior policy through variational distillation. Experiments on DeepSeek-R1-Distill-Qwen-1.5B and 7B scales demonstrate that VPG-EA improves the comprehensive efficiency metric epsilon cubed by 8.73% and 12.37% over the strongest baselines on each model size, respectively.

Zizhao Chen, Yuying Li, Siting Lin +1

Guangdong University of Foreign Studies, Guangzhou, China.

LLM Inference Efficiency Efficient Language Model Reasoning Variational Inference LLM Reasoning

May 10, 2026

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the

d

-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in

L^2

-norm, we obtain explicit polynomial convergence rates in

L^2

-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

Lénaïc Chizat, Maria Colombo, Roberto Colombo +1

Dynamical Systems Optimization Convergence Analysis Mean-Field Theory Variational Inference Stein Variational Gradient Descent

May 8, 2026

Direct Bethe Free Energy Minimization for Bayesian Neural Networks

Bayesian neural networks are typically trained against the evidence lower bound (ELBO), whose Jensen gap closes only when the variational posterior is exact. We instead train by local consistency: gradient descent on the Bethe free energy, driving the belief at every factor toward agreement with its neighbours rather than placing a loss on the output. The resulting objective scores each observation by its own predictive density: a strictly proper rule whose optimum is the true conditional, for any likelihood with a tractable predictive convolution. Instantiated with a Gaussian last layer over a deterministic backbone, exact inference appears as one known corner: the neural-linear marginal likelihood. That corner is evidence-optimal; the shared-cavity, free-routed interior is predictive-optimal, improving NLL and calibration over it. This instance, SCROLL (Shared-Cavity fRee-rOuting Last-Layer), is a single-pass Bayesian neural network: batchable, any-likelihood, and implicitly empirical-Bayes-prior precision, observation noise, covariance, and backbone fit in one gradient pass. Prior work enters the interior only through the ELBO and its Jensen gap, even in this conjugate setting. At a single training run and forward pass per architecture-where the validation-tuned conventional references cross-validate

λ

and ensembles pay

5

-

50\times

at inference-a fixed SCROLL variant is best-or-tied on NLL and calibration on 7/8 UCI regression benchmarks, and best on 4/5 across three large tabular datasets (up to 515k examples) and two frozen text/vision embeddings.

Pavel Prochazka

Cisco Inc.

Bayesian Neural Networks Variational Inference

May 8, 2026

SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching

Black-Box Variational Inference (BBVI) typically relies on Stochastic Gradient Descent (SGD) to optimize the Evidence Lower Bound (ELBO). However, the stochastic gradients in BBVI inherently exhibit unbounded variance, violating standard assumptions and instead satisfying the weaker Blum-Gladyshev (BG) condition, where variance grows quadratically with distance from the optimum. In this paper, we bridge the gap between stochastic optimization theory and the practical instances of BBVI. Focusing on the broad elliptic location-scale family of parameterized distributions, we offer two main contributions. First, we prove the existence of an ELBO solution, a foundational property usually assumed a priori in the literature. Second, we establish comprehensive convergence guarantees spanning finite-time and asymptotic regimes for Minibatch Projected SGD (PSGD) equipped with dynamic batching and preconditioning under the BG condition. Our theoretical framework demonstrates that dynamic batching combined with preconditioning systematically enables rigorous guarantees even in complex settings. We illustrate our theoretical findings with numerical results, highlighting the efficacy of our approach for modern inference tasks.

Hippolyte Labarrière, Cesare Molinari, Silvia Villa +1

MaLGa, DIBRIS Università degli Studi di Genova Genoa, Italy · MaLGa, DIMA Università degli Studi di Genova Genoa, Italy · MaLGa, DIBRIS, IIT Università degli Studi di Genova Genoa, Italy

Stochastic Optimization Convergence Variational Inference Stochastic Gradient Descent

May 8, 2026

PISTO: Proximal Inference for Stochastic Trajectory Optimization

Stochastic trajectory optimization methods like STOMP enable planning with non-differentiable costs, offering substantial flexibility over gradient-based approaches. We show that STOMP implicitly minimizes the KL divergence from a Boltzmann trajectory distribution, revealing an elegant Variational Inference (VI) structure underlying its updates. Building on this insight, we propose the Proximal Inference for Stochastic Trajectory Optimization (PISTO) algorithm that stabilizes the updates by augmenting the objective with a KL regularization between successive Gaussian proposals. This proximal formulation admits a trust-region interpretation and yields closed-form mean updates computable as expectations under a surrogate distribution. We estimate these expectations via importance-weighted Monte Carlo sampling, producing a simple, derivative-free algorithm that inherits STOMP's ability to handle non-differentiable and discontinuous costs without modification. On robot arm motion planning benchmarks, PISTO achieves an 89% success rate -- outperforming CHOMP (63%) and STOMP (68%) -- while producing shorter, smoother paths at twice the speed of competing stochastic methods. We further validate PISTO on contact-rich MuJoCo locomotion and manipulation tasks, where it consistently outperforms both CEM and MPPI baselines in reward.

Hongzhe Yu, Zinuo Chang, Yongxin Chen

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA · School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA

Trajectory Optimization Trajectory Generation Variational Inference Robot Motion Planning

May 8, 2026

Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.

Ryoichiro Agata, Tomohisa Okazaki

Japan Agency for Marine-Earth Science and Technology, Japan · Disaster Prevention Research Institute, Kyoto University, Japan · RIKEN Center for Advanced Intelligence Project, Japan

Bayesian Inverse Problems Physics-Informed ML Variational Inference PDE Inverse Problems

May 7, 2026

Variational Smoothing and Inference for SDEs from Sparse Data with Dynamic Neural Flows

Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.

Yu Wang, Arnab Ganguly

Department of Mathematics Louisiana State University

Stochastic Differential Equations Bayesian Inference Variational Inference

May 5, 2026

Amortized Variational Inference for Joint Posterior and Predictive Distributions in Bayesian Uncertainty Quantification

Bayesian predictive inference propagates parameter uncertainty to quantities of interest through the posterior-predictive distribution. In practice, this is typically performed using a two-stage procedure: first approximating the posterior distribution of model parameters, and then propagating posterior samples through the predictive model via Monte Carlo simulation. This sequential workflow can be computationally demanding, particularly for high-fidelity models such as those governed by partial differential equations. We propose a variational Bayesian framework that directly targets the posterior-predictive distribution and jointly learns variational approximations of both the posterior and the corresponding predictive distribution. The formulation introduces a variational upper bound on the Kullback--Leibler divergence together with moment-based regularization terms. The variational distributions are trained in an amortized manner, shifting computational effort to an offline stage and enabling efficient online inference. Numerical experiments ranging from analytical benchmarks to a finite-element solid mechanics problem demonstrate that the proposed method achieves more accurate predictive distributions than conventional two-stage variational inference, while substantially reducing the cost of online predictive inference.

Nan Feng, Xun Huan

Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, United States

Uncertainty Quantification Amortized Inference Bayesian Inference Variational Inference

May 1, 2026

Deep Variational Inference Symbolic Regression

Symbolic regression discovers explicit, interpretable equations without assuming a functional form in advance. A Bayesian approach strengthens this through probability distributions over candidate expressions, thus quantifying uncertainty in the presence of noisy and limited data. Deep Symbolic Regression (DSR) uses a neural network to generate symbolic expressions, but it is designed to identify a single best-fitting expression rather than infer a posterior distribution over models. We introduce Deep Variational Inference Symbolic Regression (DVISR), a variational Bayesian extension of DSR. DVISR replaces the original reward with the integrand of the evidence lower bound. It also extends the network architecture to output distributions over constants within expressions, enabling posterior inference over both expression trees and their associated constants. We show that DVISR can recover the true posterior in simple settings, both with and without constant tokens, and we examine how its performance changes as the size of the expression space increases. These results position DVISR as a step toward scalable Bayesian symbolic regression with uncertainty over full symbolic models.

James Butterworth, Gevik Grigorian, Alejandro DiazDelaO

Clinical Operational Research Unit University College London London, United Kingdom · Department of Computer Science University College London London, United Kingdom

Uncertainty Quantification Variational Inference Symbolic Regression

May 1, 2026

Robust volatility updates for Hierarchical Gaussian Filtering

Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.

Christoph Mathys, Nicolas Legrand, Peter Thestrup Waade +2

Interacting Minds Centre, Aarhus University, 8000 Aarhus C, Denmark · Centre for Humanities Computing, Aarhus University, 8000 Aarhus C, Denmark · Translational Neuromodeling Unit, University of Zurich & ETH Zurich, 8032 Zurich, Switzerland +1

Bayesian Filtering Hierarchical Bayesian Modeling Variational Inference

Apr 30, 2026

A Collective Variational Principle Unifying Bayesian Inference, Game Theory, and Thermodynamics

Collective intelligence emerges across biological, physical, and artificial systems without central coordination, yet a unifying principle governing such behaviour remains elusive. The Free Energy Principle explains how individual agents adapt through variational inference, while game theory formalises strategic interactions. Here we introduce the Game-Theoretic Free Energy Principle, a unified framework showing that multi-agent systems performing local free-energy minimisation implicitly implement a stochastic game. We prove that, under bounded rationality and local information constraints, stationary points of collective free energy correspond to approximate Nash equilibria of an induced game. Conversely, a broad class of cooperative games admits a variational representation in which equilibria arise as Gibbs distributions over coalitions, establishing a bridge between Bayesian inference and strategic interaction. To characterise higher-order effects, we introduce a free-energy formulation of the Harsanyi dividend, isolating irreducible multi-agent synergy. This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence. We validate this prediction across neural, biological, and artificial multi-agent systems. These results identify a common variational principle underlying inference, thermodynamics, and game-theoretic equilibrium.

Djamel Bouchaffra, Faycal Ykhlef, Mustapha Lebbah +1

1DAVID Lab, University of Paris-Saclay, UVSQ, 78035 Versailles, France. · 2TSAM, ASM, Centre for Development of Advanced Technologies, Algiers, Algeria. · 3LIPN, UMR CNRS 7030, Sorbonne Paris Nord University, Villetaneuse, France.

Nash Equilibrium Cooperative Game Theory Game Theory Bayesian Inference Free Energy Principle Multi-Agent Systems+2

Apr 29, 2026

Exploring the Potential of Probabilistic Transformer for Time Series Modeling: A Report on the ST-PT Framework

The Probabilistic Transformer (PT) establishes that the Transformer's self-attention plus its feed-forward block is mathematically equivalent to Mean-Field Variational Inference (MFVI) on a Conditional Random Field (CRF). Under this equivalence the Transformer ceases to be a black-box neural network and becomes a programmable factor graph: graph topology, factor potentials, and the message-passing schedule are all explicit and inspectable primitives that can be engineered. PT was originally developed for natural language and in this report we investigate its potential for time series. We first lift PT into the Spatial-Temporal Probabilistic Transformer (ST-PT) to repair PT's missing channel axis and weak per-step semantics, and adopt ST-PT as a shared cornerstone backbone. We then identify three distinct properties that PT/ST-PT offers as a factor-graph model and derive three Research Questions, one per property, that probe how each property can be exploited in time series: RQ1. The graph topology and potentials are direct programmable primitives. Can this be used to inject symbolic time-series priors into ST-PT through structural graph modifications, especially under data scarcity and noise? RQ2. The CRF's factor matrices are the operator's potentials. Can an external condition program these factor matrices on a per-sample basis, so that conditional generation becomes structural rather than feature-level modulation of a fixed one? RQ3. Each MFVI iteration is a Bayesian posterior update on the factor graph. Can this turn the latent transition of latent-space AutoRegressive (AR) forecasting from an opaque MLP into a principled posterior update, and can a CRF teacher distill its latents into the AR student to counter cumulative error? We give one empirical study per question. Together, these three studies position ST-PT as a programmable framework for time-series modeling.

Zhangzhi Xiong, Haoyi Wu, You Wu +3

School of Information Science and Technology, ShanghaiTech University

Transformer Conditional Random Fields Time Series Forecasting Time Series Generation Variational Inference

Apr 28, 2026

Fitting Large Nonlinear Mixed Effects Models Using Variational Expectation Maximization

Nonlinear Mixed Effects (NLME) models are widely used in pharmacometrics and related fields to analyze hierarchical and longitudinal data. However, as the number of parameters and random effects increases, traditional methods for maximizing the marginal likelihood become computationally expensive. This paper explores the Variational Expectation Maximization (VEM) algorithm, a scalable alternative for fitting NLME models. Originally introduced in the context of probabilistic graphical models and later popularized through variational autoencoders, VEM has not been extensively applied to NLME modeling. By leveraging flexible variational families and reverse-mode automatic differentiation, VEM can efficiently maximize the marginal likelihood, scaling to NLME models with over 15,000 population parameters. This work provides a detailed description of VEM, compares it to other NLME fitting algorithms, and highlights its scalability through computational experiments. Using the Pumas statistical software, we fit two test models: 1) a standard warfarin model, and 2) an unnecessarily over-parameterized DeepNLME Friberg model with 15,410 population parameters and 16 random effects. The warfarin model was fitted to completion to demonstrate the correctness of VEM, while the DeepNLME Friberg model instead demonstrates VEM's scalability on a toy but large model. VEM improves the log likelihood steadily over hundreds of iterations at a practical per-iteration cost, while FOCE fails to complete even one iteration within a day. The model is deliberately over-parameterized for its small dataset and over-fits it, so what this experiment establishes is that VEM optimizes the objective of a model of this size at a practical cost. Applying VEM to large models that are genuinely useful is left to future work.

Mohamed Tarek, Pedro Afonso

PumasAI, USA

Expectation-Maximization Variational Inference Mixed-Effects Models