# Variational Linearized Laplace Approximation for Bayesian Deep Learning

## TL;DR

This paper introduces a variational sparse Gaussian Process approach to efficiently approximate the Linearized Laplace Approximation for Bayesian deep learning, significantly reducing computational costs while improving predictive uncertainty estimates.

## Contribution

It proposes a novel variational sparse GP method for LLA that is computationally efficient and maintains the original DNN output as the predictive mean.

## Key findings

- Outperforms existing LLA variants in predictive quality.
- Reduces training time to be independent of dataset size.
- Demonstrates effectiveness on regression and classification tasks.

## Abstract

The Linearized Laplace Approximation (LLA) has been recently used to perform uncertainty estimation on the predictions of pre-trained deep neural networks (DNNs). However, its widespread application is hindered by significant computational costs, particularly in scenarios with a large number of training points or DNN parameters. Consequently, additional approximations of LLA, such as Kronecker-factored or diagonal approximate GGN matrices, are utilized, potentially compromising the model's performance. To address these challenges, we propose a new method for approximating LLA using a variational sparse Gaussian Process (GP). Our method is based on the dual RKHS formulation of GPs and retains, as the predictive mean, the output of the original DNN. Furthermore, it allows for efficient stochastic optimization, which results in sub-linear training time in the size of the training dataset. Specifically, its training cost is independent of the number of training points. We compare our proposed method against accelerated LLA (ELLA), which relies on the Nystr\"om approximation, as well as other LLA variants employing the sample-then-optimize principle. Experimental results, both on regression and classification datasets, show that our method outperforms these already existing efficient variants of LLA, both in terms of the quality of the predictive distribution and in terms of total computational time.