Hessian-Free Laplace Approximation in Bayesian Deep Learning

Core Concepts

Hessian-free Laplace (HFL) provides an efficient alternative to traditional Laplace approximation in Bayesian deep learning, avoiding the need for explicit Hessian calculation and inversion.

Abstract

In Bayesian deep learning, the Laplace approximation of the posterior distribution is commonly used to quantify uncertainty post-hoc. However, calculating and inverting the Hessian matrix can be computationally intensive. The Hessian-free Laplace (HFL) method proposed in this paper estimates predictive variance without explicitly computing the Hessian. By using a regularized form of the maximum a posteriori parameter, HFL targets similar variance as traditional Laplace approximation. Experimental results show comparable performance to exact and approximate Hessians, with good coverage for in-between uncertainty scenarios. The method scales efficiently and offers potential benefits for uncertainty quantification in deep neural networks.

Stats

PICP: 0.9375
CRPS: 0.002347
NLL: -23.899
PICP: 0.9333
CRPS: 0.006976
NLL: -38.440
PICP: 0.8750
CRPS: 0.001782
NLL: -22.571

Quotes

"We propose an alternative framework that sidesteps Hessian calculation and inversion."
"Under standard assumptions of Laplace approximation in Bayesian deep learning, HFL targets the same variance as LA."
"HFL performs commensurately with other approximations, achieving best out-of-distribution performance."

Key Insights Distilled From

Hessian-Free Laplace in Bayesian Deep Learning

by James McIner... at arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.10671.pdf

Deeper Inquiries

How does the computational efficiency of HFL impact its scalability to larger neural network architectures

Hessian-Free Laplace (HFL) offers computational efficiency by sidestepping the explicit calculation and inversion of the Hessian matrix, which is a significant bottleneck in traditional Laplace approximation methods. This efficiency directly impacts its scalability to larger neural network architectures. In larger networks with high-dimensional parameters, calculating and inverting the full Hessian becomes increasingly computationally expensive and impractical. By using only two point estimates - the standard maximum a posteriori parameter and an optimal parameter under a loss regularized by the network prediction - HFL significantly reduces computational burden while still targeting similar predictive variance as Laplace approximation. This streamlined approach allows for more efficient uncertainty quantification post-hoc without sacrificing accuracy or reliability.

What are the implications of assuming local optimality for uncertainty quantification in Bayesian deep learning

Assuming local optimality for uncertainty quantification in Bayesian deep learning has several implications. Firstly, it restricts the analysis to a specific region around the optimum of the joint distribution where curvature is evaluated at that particular location. This means that uncertainties are characterized based on this local information, potentially overlooking global properties of the model's behavior or missing out on multiple modes present in complex distributions. Additionally, relying solely on local optima may lead to underestimation or overestimation of uncertainties if there are significant deviations from optimality elsewhere in parameter space. Therefore, while assuming local optimality simplifies computations and provides insights into immediate surroundings of optimal solutions, it may not capture all nuances of uncertainty across broader contexts.

How might pre-trained HFL be extended to handle more complex tasks or datasets beyond those evaluated in this study

Pre-trained Hessian-Free Laplace (HFL) can be extended to handle more complex tasks or datasets beyond those evaluated in this study by incorporating additional regularization strategies tailored to specific challenges presented by these tasks. For instance:

Adaptive Regularization: Implementing adaptive regularization techniques that adjust regularization strength based on task complexity or dataset characteristics.
Task-Specific Regularizers: Introducing task-specific regularizers that capture unique patterns or structures relevant to different types of data.
Ensemble Approaches: Leveraging ensemble methods within pre-training frameworks to enhance robustness and generalization capabilities.
By customizing pre-trained HFL with these advanced strategies, it can effectively address diverse scenarios requiring nuanced uncertainty quantification while maintaining computational efficiency and scalability across varied applications in Bayesian deep learning settings.