# Quantum Speedup of Natural Gradient for Variational Bayes

arXiv: https://arxiv.org/abs/2106.05807 · HTML: https://mlsanity.com/paper/2106.05807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 89 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Recurrent Quantum Neural Networks](https://mlsanity.com/paper/2006.14619/) (2020)
- [Quantum Recommendation Systems](https://mlsanity.com/paper/1603.08675/) (2016)
- [Optimal ridge penalty for real-world high-dimensional data can be zero or negative due to the implicit ridge regularization](https://mlsanity.com/paper/1805.10939/) (2018)
- [Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection](https://mlsanity.com/paper/1506.00552/) (2015)
- [SLANG: Fast Structured Covariance Approximations for Bayesian Deep Learning with Natural Gradient](https://mlsanity.com/paper/1811.04504/) (2018)
- [Quantum Deep Learning](https://mlsanity.com/paper/1412.3489/) (2014)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Quantum Inference on Bayesian Networks](https://mlsanity.com/paper/1402.7359/) (2014)
- [Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension](https://mlsanity.com/paper/1811.04909/) (2018)
- [Adiabatic Quantum Linear Regression](https://mlsanity.com/paper/2008.02355/) (2020)
- [Quantum computing for energy systems optimization: Challenges and opportunities](https://mlsanity.com/paper/2003.00254/) (2020)
- [Geometry of learning neural quantum states](https://mlsanity.com/paper/1910.11163/) (2019)
- [Variational Bayes with Intractable Likelihood](https://mlsanity.com/paper/1503.08621/) (2015)
- [Bayesian Deep Learning on a Quantum Computer](https://mlsanity.com/paper/1806.11463/) (2018)
- [A practical tutorial on Variational Bayes](https://mlsanity.com/paper/2103.01327/) (2021)
- [Variational Bayes on Manifolds](https://mlsanity.com/paper/1908.03097/) (2019)
- [A Quantum Extension of Variational Bayes Inference](https://mlsanity.com/paper/1712.04709/) (2017)
- [Quantum Algorithm for Linear Regression](https://mlsanity.com/paper/1402.0660/) (2014)

## Built on by (most cited first, 1 of 1)

- [Quantum Natural Policy Gradients: Towards Sample-Efficient Reinforcement Learning](https://mlsanity.com/paper/2304.13571/) (2023)

## Abstract (arXiv)

Variational Bayes (VB) is a critical method in machine learning and statistics, underpinning the recent success of Bayesian deep learning. The natural gradient is an essential component of efficient VB estimation, but it is prohibitively computationally expensive in high dimensions. We propose a computationally efficient regression-based method for natural gradient estimation, with convergence guarantees under standard assumptions. The method enables the use of quantum matrix inversion to further speed up VB. We demonstrate that the problem setup fulfills the conditions required for quantum matrix inversion to deliver computational efficiency. The method works with a broad range of statistical models and does not require special-purpose or simplified variational distributions.