Mean-Field Approximation to Gaussian-Softmax Integral with Application to Uncertainty Estimation

TL;DR

This paper introduces a novel single-model approach for uncertainty estimation in neural networks by analytically approximating the Gaussian-softmax integral using mean-field theory, achieving competitive results without ensemble training.

Contribution

The paper proposes a mean-field approximation method to analytically compute Gaussian-softmax integrals for uncertainty estimation, eliminating the need for ensemble models.

Findings

Performs competitively with state-of-the-art uncertainty methods

Outperforms many methods in out-of-distribution detection

Enables closed-form approximate inference with only first and second moments

Abstract

Many methods have been proposed to quantify the predictive uncertainty associated with the outputs of deep neural networks. Among them, ensemble methods often lead to state-of-the-art results, though they require modifications to the training procedures and are computationally costly for both training and inference. In this paper, we propose a new single-model based approach. The main idea is inspired by the observation that we can "simulate" an ensemble of models by drawing from a Gaussian distribution, with a form similar to those from the asymptotic normality theory, infinitesimal Jackknife, Laplacian approximation to Bayesian neural networks, and trajectories in stochastic gradient descents. However, instead of using each model in the "ensemble" to predict and then aggregating their predictions, we integrate the Gaussian distribution and the softmax outputs of the neural networks. We use a mean-field approximation formula to compute this analytically intractable integral. The proposed approach has several appealing properties: it functions as an ensemble without requiring multiple models, and it enables closed-form approximate inference using only the first and second moments of the Gaussian. Empirically, the proposed approach performs competitively when compared to state-of-the-art methods, including deep ensembles, temperature scaling, dropout and Bayesian NNs, on standard uncertainty estimation tasks. It also outperforms many methods on out-of-distribution detection.