A new particle algorithm improves mean-field variational inference.
problem Efficiently approximating nonparametric posterior distributions in machine learning.
method Introduces PArticle VI (PAVI), a novel particle-based algorithm for nonparametric mean-field approximation.
result Obtains non-asymptotic error bounds for PArticle VI, providing the first end-to-end guarantee for particle-based MFVI.
The mean field algorithm is a widely used approximate inference algorithm for graphical models whose exact inference is intractable. In each iteration of mean field, the approximate marginals for each variable are updated by getting information from the neighbors. This process can be equivalently converted into a feedf…
The mean field variational Bayes method is becoming increasingly popular in statistics and machine learning. Its iterative Coordinate Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent comprehensive review. Despite the popularity of the …
Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
New method improves approximate inference for Bayesian models.
problem Approximate inference for high-dimensional Bayesian models.
method Entropic regularization of mean-field variational inference.
result Improved recovery of true posterior dependency.
We conduct non-asymptotic analysis on the mean-field variational inference for approximating posterior distributions in complex Bayesian models that may involve latent variables. We show that the mean-field approximation to the posterior can be well-approximated relative to the Kullback-Leibler divergence discrepancy m…
Beta process is the standard nonparametric Bayesian prior for latent factor model. In this paper, we derive a structured mean-field variational inference algorithm for a beta process non-negative matrix factorization (NMF) model with Poisson likelihood. Unlike the linear Gaussian model, which is well-studied in the non…
CoRMF uses RNNs to solve Ising models efficiently by ordering critical edges.
problem Solving Ising models efficiently and accurately.
method Criticality-ordered spin sequence and RNNs for mean-field factorization.
result Proves tighter error bounds than naive mean-field.
ALO-CV approximates leave-one-out error in proportional regime.
problem Estimating generalization error in high-dimensional settings.
method Developed new analysis for ALO-CV, showed consistency under strong convexity.
result ALO-CV approximates leave-one-out error up to negligible error.
Deep Bayesian neural nets can use simpler weight approximations without sacrificing performance.
problem The need for complex weight posterior approximations in deep Bayesian neural networks.
method Theoretical and empirical analysis of mean-field variational inference in deep networks.
result Mean-field variational weight posteriors in deep networks can induce similar function-space distributions as complex approximations in shallower networks.
Wide BNNs with odd activations fail to approximate data under mean-field inference.
problem Theoretical limitations of mean-field variational inference in wide, deep Bayesian neural networks.
method Analysis of mean-field variational inference in fully-connected BNNs with odd activation functions and Gaussian likelihood.
result The optimal mean-field variational posterior predictive distribution converges to the prior predictive distribution as network width increases.
The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equa…
New algorithm improves CRF inference and learning.
problem Efficient inference and learning for dense CRFs.
method Regularized Frank-Wolfe algorithm for nonconvex CRF optimization.
result Regularized Frank-Wolfe outperforms mean field and CNN baselines.
Develops a new framework for analyzing MFVI algorithms.
problem Analyzes mean field variational inference (MFVI) formulations.
method Inspired by variational Bayesian formulations, represents MFVI problem in three ways: gradient flow, Fokker-Planck-like equations, and diffusion process.
result Establishes rigorous guarantees for convergence of time-discretized coordinate ascent variational inference algorithms.
Nonnegative Boltzmann machines (NNBMs) are recurrent probabilistic neural network models that can describe multi-modal nonnegative data. NNBMs form rectified Gaussian distributions that appear in biological neural network models, positive matrix factorization, nonnegative matrix factorization, and so on. In this paper,…
Rotates MFVI for better Gaussian approximations.
problem Improving variational approximations for complex distributions.
method Rotated coordinate system, PCA-based rotation, iterative Gaussianization.
result Significantly more accurate approximations with lower computational cost.
Mean-field variational methods are widely used for approximate posterior inference in many probabilistic models. In a typical application, mean-field methods approximately compute the posterior with a coordinate-ascent optimization algorithm. When the model is conditionally conjugate, the coordinate updates are easily …
A Gaussian restricted Boltzmann machine (GRBM) is a Boltzmann machine defined on a bipartite graph and is an extension of usual restricted Boltzmann machines. A GRBM consists of two different layers: a visible layer composed of continuous visible variables and a hidden layer composed of discrete hidden variables. In th…
We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus …
New method infers population dynamics from snapshots using path space optimization.
problem Recover dynamics of a population from its temporal marginals.
method Grid-free algorithm using Schrödinger bridges coupled via noisy gradient descent in mean-field limit.
result Global convergence to min-entropy estimator with end-to-end theoretical guarantees.
New method for variational inference without conjugacy constraints.
problem Efficient variational inference with flexible prior and approximation families.
method Wasserstein gradient flow for mean-field approximation.
result Improved convergence and efficiency of variational inference.
Bayesian neural networks ignore data in infinite units limit.
problem Pathological behavior of posterior in over-parameterized networks.
method Mean-field variational inference in infinite hidden units limit.
result Posterior mean converges to zero, ignoring data.
New algorithm speeds up large-scale statistical inference.
problem Efficiently solving large-scale mean-field variational inference problems.
method Developed a novel primal-dual algorithm (PD-VI) and a block-preconditioned extension (P2D-VI) for mean-field variational inference. result PD-VI and P2D-VI achieve faster convergence and better solution quality compared to existing methods. Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from a theoretical point of view. The mathematical description of learning problems i…
An autonomous variational inference algorithm for arbitrary graphical models requires the ability to optimize variational approximations over the space of model parameters as well as over the choice of tractable families used for the variational approximation. In this paper, we present a novel combination of graph part…
MFVI mode collapse explained; RoVI proposed to mitigate.
problem Mode collapse in MFVI for mixture distributions.
method Introducing ε-separateness, deriving bounds, proposing RoVI.
result MFVI optimizers collapse to a single component when components are ε-separated.
This work introduces a fixed-point optimization for variational inference.
problem Improving quantified uncertainty in predictions by optimizing a simplified distribution over parameters.
method Projective integral updates for high-dimensional variational inference.
result Efficient quasirandom quadrature sequence for mean-field distributions, leading to quasi-Newton variational Bayes (QNVB).
The paper introduces structured variational families to improve scalability in black-box variational inference.
problem Scalability issues in black-box variational inference, especially for large datasets and hierarchical models.
method Developed structured variational families that achieve better iteration complexity of O(N) compared to full-rank families.
result Structured variational families can achieve better scaling with respect to dataset size N, improving iteration complexity from O(N^2) to O(N).
This paper analyzes MFVBI for GMM using statistical mechanics.
problem Approximate fast computation of Gaussian Mixture Model.
method Statistical mechanics and MFVBI applied to GMM.
result Rigorous analysis and mathematical foundation for MFVBI applied to GMM.
Inference of hidden classes in stochastic block model is a classical problem with important applications. Most commonly used methods for this problem involve naïve mean field approaches or heuristic spectral methods. Recently, belief propagation was proposed for this problem. In this contribution we perform a comparati…
Ensemble Kalman Filter improves GPSSM inference for online learning.
problem Non-mean-field variational inference issues in GPSSM.
method Combining EnKF with NMF variational inference.
result Improved online learning performance and data-fitting accuracy.
Many modern unsupervised or semi-supervised machine learning algorithms rely on Bayesian probabilistic models. These models are usually intractable and thus require approximate inference. Variational inference (VI) lets us approximate a high-dimensional Bayesian posterior with a simpler variational distribution by solv…
Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.
problem Deviation from true posterior mean and underestimation of posterior uncertainty in variational inference.
method Minimization of TAP free energy in a high-dimensional asymptotic framework, showing geometric and statistical properties.
result Local minimizer of TAP free energy provides consistent estimate of posterior marginals and correctly calibrated posterior inference.
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
The paper models asset pricing in a partially observed market using mean field game theory and exponential quadratic Gaussian framework.
problem Asset pricing in a market with partial observation and heterogeneous agents.
method Mean field game theory, exponential quadratic Gaussian framework, Kalman-Bucy filtering theory.
result Characterization of equilibrium risk premium through mean field BSDE and construction of unobservable risk premium process.
Compact parameterization improves Bayesian neural network performance.
problem Improving performance of Bayesian neural networks using variational methods.
method Restricting variational distribution to a k-tied Normal distribution with low-rank factorization.
result Compact parameterization improves signal-to-noise ratio and convergence speed.
Bayesian model selection via mean-field variational approximation improves efficiency and accuracy.
problem Bayesian model selection under model mis-specification and latent variables.
method Mean-field variational approximation with non-asymptotic properties and geometric convergence.
result ELBO tends to select models closer to the true model than BIC as sample size increases.
Variational inference with a factorized Gaussian posterior estimate is a widely used approach for learning parameters and hidden variables. Empirically, a regularizing effect can be observed that is poorly understood. In this work, we show how mean field inference improves generalization by limiting mutual information …
CAVI converges for log-concave measures via optimal transport.
problem Finding the closest product measure to a log-concave measure via CAVI.
method Adapting coordinate descent techniques from Euclidean space to optimal transport for log-concave densities.
result Proves convergence of CAVI for log-concave densities and provides rates of convergence under additional conditions.
New method for high-dimensional linear regression using empirical Bayes.
problem Estimating prior in high-dimensional linear regression.
method Variational empirical Bayes approach with NPMLE and mean field approximation.
result Established asymptotic consistency and computational efficiency of the method.
A scalable method for accurate inference of low-dimensional parameters in high-dimensional linear regression.
problem Statistical inference for low-dimensional parameters in high-dimensional linear regression models.
method Mean-field variational Bayes approach, focusing on nuisance parameters and conditional distributions.
result Competitive numerical performance and theoretical guarantees for estimation and uncertainty quantification.
BBVI converges nearly dimensionally independent for log-concave targets.
problem Efficiently optimizing variational parameters in high-dimensional spaces.
method Proved convergence rate of BBVI with reparametrization gradient for log-concave targets.
result BBVI converges with nearly independent dimension dependence for log-concave targets.
A new neural network approach for diffusion on networks.
problem Inference and estimation of diffusion on network structures.
method Neural mean-field dynamics derived from Mori-Zwanzig formalism, approximated by learnable time convolution operators.
result Significantly outperforms existing approaches in accuracy and efficiency.
Unified theory for deep and recurrent networks using Gaussian processes.
problem Understanding capabilities and limitations of different network architectures.
method Unified derivation of mean-field theory from statistical physics of disordered systems.
result Gaussian processes yield identical Gaussian kernels for both architectures at a single time point or layer.
In intractable, undirected graphical models, an intuitive way of creating structured mean field approximations is to select an acyclic tractable subgraph. We show that the hardness of computing the objective function and gradient of the mean field objective qualitatively depends on a simple graph property. If the tract…
Paper introduces f-divergence variational inference for broader application.
problem Variational inference limited to specific divergences.
method Generalizes variational inference to all f-divergences using f-divergence minimization.
result Unified framework for variational inference with arbitrary f-divergences.
Mean field inference in probabilistic models is generally a highly nonconvex problem. Existing optimization methods, e.g., coordinate ascent algorithms, can only generate local optima. In this work we propose provable mean filed methods for probabilistic log-submodular models and its posterior agreement (PA) with stron…