# Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

arXiv: https://arxiv.org/abs/2307.06424 · HTML: https://mlsanity.com/paper/2307.06424/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 74 references that are in the map.
- First posted on arXiv 12 Jul 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Online Structured Laplace Approximations For Overcoming Catastrophic Forgetting](https://mlsanity.com/paper/1805.07810/) (2018)
- [Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning](https://mlsanity.com/paper/2104.04975/) (2021)
- [Approximate Inference Turns Deep Networks into Gaussian Processes](https://mlsanity.com/paper/1906.01930/) (2019)
- [Bayesian Deep Learning via Subnetwork Inference](https://mlsanity.com/paper/2010.14689/) (2020)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Rethinking Parameter Counting in Deep Models: Effective Dimensionality Revisited](https://mlsanity.com/paper/2003.02139/) (2020)
- [Variational Boosting: Iteratively Refining Posterior Approximations](https://mlsanity.com/paper/1611.06585/) (2016)
- [Estimating Model Uncertainty of Neural Networks in Sparse Information Form](https://mlsanity.com/paper/2006.11631/) (2020)
- [Stacking for Non-mixing Bayesian Computations: The Curse and Blessing of Multimodal Posteriors](https://mlsanity.com/paper/2006.12335/) (2020)
- [Boosting Variational Inference](https://mlsanity.com/paper/1611.05559/) (2016)
- [Estimating Uncertainty in Neural Networks for Cardiac MRI Segmentation: A Benchmark Study](https://mlsanity.com/paper/2012.15772/) (2020)
- [Stochastic Backpropagation through Mixture Density Distributions](https://mlsanity.com/paper/1607.05690/) (2016)
- [Good Initializations of Variational Bayes for Deep Models](https://mlsanity.com/paper/1810.08083/) (2018)
- [Statistical Inference in Mean-Field Variational Bayes](https://mlsanity.com/paper/1911.01525/) (2019)

## Abstract (arXiv)

For predictive modeling relying on Bayesian inversion, fully independent, or ``mean-field'', Gaussian distributions are often used as approximate probability density functions in variational inference since the number of variational parameters is twice the number of unknown model parameters. The resulting diagonal covariance structure coupled with unimodal behavior can be too restrictive when dealing with highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors in the form of Gaussian mixtures can capture any distribution to an arbitrary degree of accuracy while maintaining some analytical tractability. Variational inference with Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. Coupled with the existence of multiple local minima due to nonconvex trends in the loss functions often associated with variational inference, these challenges motivate the need for robust initialization procedures to improve the performance and scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with an optimization stage in model parameter space in which local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace method. Finally, the mixture weights are determined through constrained least squares regression. Robustness and scalability are demonstrated using synthetic tests. The methodology is applied to an inversion problem in structural dynamics involving unknown viscous damping coefficients.