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-minimax optimal bounds for the variational risk. 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.

翻译：变分推断作为马尔可夫链蒙特卡罗采样的替代方法，在实现复杂贝叶斯模型的可扩展计算方面发挥了变革性作用。然而，现有方法通常依赖于僵化的模型特定形式或随机黑盒优化流程。正切近似是一类结构化的变分方法，其原理基于对底层概率模型几何特性的利用。然而，该方法的应用范围主要局限于逻辑回归及相关建模体系。本文针对具有强超高斯似然特性的广义概率模型类别，提出了一种基于正切变换的新型变分推断框架。该方法利用凸对偶性构建对数似然的正切下界，进而在原本难以处理的设置中诱导模型参数与高斯先验的共轭性。在对数据生成机制施加温和假设的条件下，我们建立了算法收敛性保证，这一理论贡献与通常黑盒变分方法有限的理论保证形成鲜明对比。此外，我们推导了变分风险的近似极小极大最优界。通过模拟数据和真实数据场景的实验，展示了所提方法在可扩展性和一致捕捉复杂底层数据结构能力方面对现有先进变分算法的优越性能。