A Computable Measure of Suboptimality for Entropy-Regularised Variational Objectives

2025-09-12

AI Summary

Evaluating optimality in entropy-regularized variational inference is challenging when the unnormalized target density is unavailable. Method: We propose the Kernelized Gradient Discrepancy (KGD), a computable suboptimality measure that integrates variational gradient magnitude with kernel methods; KGD recovers the Kernel Stein Discrepancy (KSD) in standard Bayesian settings and naturally extends to scenarios where the unnormalized density is inaccessible. Contributions/Results: Theoretically, we establish KGD’s consistency, convergence, and statistical robustness. Algorithmically, we introduce KGD-driven variants of Stein variational gradient descent and other optimization schemes. Empirically, we validate KGD’s effectiveness on mean-field neural network training and predictive uncertainty quantification. This work provides a unified, differentiable, and scalable framework for both evaluation and optimization in entropy-regularized variational inference and sampling algorithms.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

Several emerging post-Bayesian methods target a probability distribution for which an entropy-regularised variational objective is minimised. This increased flexibility introduces a computational challenge, as one loses access to an explicit unnormalised density for the target. To mitigate this difficulty, we introduce a novel measure of suboptimality called 'gradient discrepancy', and in particular a 'kernel gradient discrepancy' (KGD) that can be explicitly computed. In the standard Bayesian context, KGD coincides with the kernel Stein discrepancy (KSD), and we obtain a novel charasterisation of KSD as measuring the size of a variational gradient. Outside this familiar setting, KGD enables novel sampling algorithms to be developed and compared, even when unnormalised densities cannot be obtained. To illustrate this point several novel algorithms are proposed, including a natural generalisation of Stein variational gradient descent, with applications to mean-field neural networks and prediction-centric uncertainty quantification presented. On the theoretical side, our principal contribution is to establish sufficient conditions for desirable properties of KGD, such as continuity and convergence control.

Problem

• Measuring suboptimality for entropy-regularised variational objectives
• Computing gradient discrepancies without unnormalised densities
• Developing novel sampling algorithms for flexible distributions

Innovation

• Kernel gradient discrepancy for variational objectives
• Generalized Stein variational gradient descent algorithm
• Computable measure without unnormalized densities