A Computable Measure of Suboptimality for Entropy-Regularised Variational Objectives

2026-07-13 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The paper introduces Gradient Discrepancy (GD), specifically Kernel Gradient Discrepancy (KGD), as a novel, computable measure of suboptimality for entropy-regularized variational objectives. These objectives are common in emerging post-Bayesian methods where explicit unnormalized target densities are unavailable, posing a computational challenge. KGD extends Kernel Stein Discrepancy (KSD) to this broader context, offering a new interpretation of KSD as a variational gradient measure. The authors propose several new sampling algorithms, including a generalization of Stein Variational Gradient Descent (SVGD), and demonstrate applications in mean-field neural networks and prediction-centric uncertainty quantification. Theoretically, the work establishes sufficient conditions for KGD's desirable properties, such as continuity and convergence control, proving it characterizes \"alpha\"-convergence.

Key takeaway

For Machine Learning Engineers developing or deploying post-Bayesian methods with nonlinear loss functions, you should consider integrating Kernel Gradient Discrepancy (KGD) into your workflow. KGD offers a robust, computable metric for assessing sample quality and tuning algorithms like Mean Field Langevin Dynamics, even when traditional unnormalized densities are inaccessible. This enables principled step size selection and the development of novel, more accurate sampling schemes, potentially improving model performance and uncertainty quantification in complex applications like mean-field neural networks.

Key insights

KGD provides a computable measure of suboptimality for entropy-regularized variational objectives lacking explicit unnormalized densities.

Principles

• KGD generalizes KSD, interpreting it as a variational gradient measure.
• Minimizing KGD offers a principled criterion for algorithm tuning and development.
• KGD characterizes \"alpha\"-convergence, ensuring small values imply closeness to the target.

Method

KGD is computed in closed-form for finite support distributions by leveraging reproducing kernel Hilbert spaces, transforming the variational gradient into a kernel-based integral.

In practice

• Tune step sizes in Mean Field Langevin Dynamics (MFLD).
• Develop extensible sampling algorithms for variational targets.
• Enable parametric variational inference without explicit densities.

Topics

• Kernel Gradient Discrepancy
• Variational Inference
• Post-Bayesian Methods
• Mean Field Neural Networks
• Uncertainty Quantification
• Stein's Method

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.