A Generalized Tangent Approximation based Variational Inference Framework for Strongly Super-Gaussian Likelihoods
Authors: Somjit Roy, Pritam Dey, Debdeep Pati, Bani K. Mallick
Abstract
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.
Variational inference (VI) is a core engine of modern AI, enabling scalable approximate Bayesian learning and uncertainty-aware training of large probabilistic and generative models. In this paper, we propose Structured Nonparametric Variational Inference (SN-VI), a novel framework for modeling complex dependencies among latent variables in posterior approximation, leveraging multivariate spline techniques. Unlike traditional methods that rely on the mean-field assumption, SN-VI preserves intricate latent variable dependencies, providing a flexible and accurate approximation of posteriors with arbitrary shapes. We establish rigorous theoretical guarantees, including the derivation of the lower bound for the variational objective and proof of asymptotic consistency in posterior estimation. To facilitate practical implementation, we develop an algorithm that automatically identifies dependent latent variables and their underlying dependence structure, without requiring manual specification. Simulation studies validate the effectiveness of SN-VI in approximating posterior distributions with bounded support and complex dependencies. The proposed method has been successfully applied to high-dimensional structured data, including computer vision datasets and spatial transcriptomics. In these applications, SN-VI demonstrates improved generative model performance and effectively uncovers coupled biological signals through the learned dependency structure.
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
When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous theoretical results on robust VI with location-scale families under target symmetries in two substantial ways: (1) We open them up to a wider range of divergences by providing sufficient conditions for exact recovery of the target mean and correlation matrix when using the forward Kullback-Leibler divergence and α-divergences. (2) By doing so, we find that we can drop the restrictive assumption of a log-concave target made in previous work, allowing us to give guarantees for a wider range of targets, including multi-modal ones. In our experiments, we show how our guarantees can serve as guidelines for the choice of the variational family and α-value and we illustrate on a diverse set of examples how and why optimization can fail in the absence of our sufficient conditions.