Provable Bayesian Inference via Particle Mirror Descent

By Bo Dai, Niao He, Hanjun Dai, Le Song. 2015

77 citations recorded by the source corpus.

Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes' rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose a simple yet provable algorithm, \emph{Particle Mirror Descent} (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, with $m$ particles, PMD provides a posterior density estimator that converges in terms of $KL$-divergence to the true posterior in rate $O(1/\sqrt{m})$. We demonstrate competitive empirical performances of PMD compared to several approximate inference algorithms in mixture models, logistic regression, sparse Gaussian processes and latent Dirichlet allocation on large scale datasets.

Subfields: Artificial Intelligence, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Sergey Levine
2. Qiang Liu
3. Bo Dai
4. Ofir Nachum
5. Jun Zhu
6. Le Song
7. George Tucker
8. Dale Schuurmans

In Statistics and Probability

1. Qiang Liu
2. Jun Zhu
3. Jiaxin Shi
4. Arthur Gretton
5. Bo Dai
6. Dilin Wang
7. Le Song
8. Bernhard Schölkopf

Citations in the corpus

• Approximate Inference Turns Deep Networks into Gaussian Processes (2019, background): “The GP update can be seen as a mirror-descent step in the function space [4], but the kernel in our update changes at every time step.”
• Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm (2016, methodology): “[7], and the doubly stochastic variational inference (DSVI) by Titsias and Lázaro-Gredilla [19].”
• Stein Variational Gradient Descent as Gradient Flow (2017, background): “There are also particle algorithms based on optimization, or variational principles, with theoretical guarantees [e.g., 3–5], but they often do not use the gradient information effectively and do not scale well in high dimensions.”
• Exponential Family Estimation via Adversarial Dynamics Embedding (2019, methodology): “The kernel bandwidth is chosen using median trick (Dai et al., 2016).”
• Kernel Implicit Variational Inference (2017, methodology): “Moreover, KIVI is directly applicable to both global and local latent variable models (LVMs), which is an advantage over nonparametric VI methods like particle mirror descent (Dai et al., 2015) and Stein variational gradient descent (SVGD) (Liu & Wang, 2016).”
• A Spectral Approach to Gradient Estimation for Implicit Distributions (2018, methodology): “Nonparametric Inference Nonparametric variational inference (VI) methods such as PMD (Dai et al., 2016) and SVGD (Liu & Wang, 2016) remove the need of parametric families, they keep a set of particles and gradually adjust them towards the true posterior.”
• Learning Deep Hidden Nonlinear Dynamics from Aggregate Data (2018, methodology): “Dai et al. (2016) use a probability q ( U ) to approximate the posterior probability p ( U | V ) by minimizing (11) over the space of all valid densities P .”
• Offline Policy Selection under Uncertainty (2020, background): “3This tuple-based representation of the dataset is for notational and theoretical convenience, following Dai et al. (2020); Kostrikov and Nachum (2020), among others.”
• Wild Variational Approximations (2016, methodology): “MCMC and particle-based approximate inference methods [46, 47], though very accurate, become inefficient when inference from multiple different distributions is repeatedly required.”