PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models

Banerjee, Imon; Rao, Vinayak; Honnappa, Harsha

doi:10.48550/arxiv.2101.05197

Cited by 1 publication

(1 citation statement)

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“…Theoretical results around variational inference have mostly centered around its statistical properties, including asymptotic properties (Alquier and Ridgway, 2017;Alquier et al, 2016;Banerjee et al, 2021;Bhattacharya et al, 2020;Bhattacharya and Maiti, 2021;Bickel et al, 2013;Campbell and Li, 2019;Celisse et al, 2012;Chen and Ryzhov, 2020;Chérief-Abdellatif, 2019, 2020Chérief-Abdellatif et al, 2018;Guha et al, 2020;Hajargasht, 2019;Hall et al, 2011a,b;Han and Yang, 2019;Jaiswal et al, 2020;Knoblauch, 2019;Pati et al, 2017;Wang and Titterington, 2004;Wang et al, 2006;Blei, 2019, 2018;Westling and McCormick, 2015;Womack et al, 2013;Yang et al, 2017;You et al, 2014;Zhang and Gao, 2017), nite sample approximation error (Chen et al, 2017;Giordano et al, 2017;Huggins et al, 2020Huggins et al, , 2018Sheth and Khardon, 2017), robustness to model misspeci cation (Alquier and Ridgway, 2017;Chérief-Abdellatif et al, 2018;Medina et al, 2021;Wang and Blei, 2019), and properties in high-dimensional settings (Mukherjee and Sen, 2021;Ray and Szabó, 2021;Ray et al, 2020;…”

Section: Main Ideasmentioning

confidence: 99%

Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection

Bhatia¹,

Kuang²,

Ma³

et al. 2022

Preprint

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Variational inference has recently emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference. The core idea of variational inference is to trade statistical accuracy for computational e ciency. It aims to approximate the posterior, reducing computation costs but potentially compromising its statistical accuracy. In this work, we study this statistical and computational trade-o in variational inference via a case study in inferential model selection. Focusing on Gaussian inferential models (also known as variational approximating families) with diagonal plus low-rank precision matrices, we initiate a theoretical study of the trade-o s in two aspects, Bayesian posterior inference error and frequentist uncertainty quanti cation error. From the Bayesian posterior inference perspective, we characterize the error of the variational posterior relative to the exact posterior. We prove that, given a xed computation budget, a lower-rank inferential model produces variational posteriors with a higher statistical approximation error, but a lower computational error; it reduces variances in stochastic optimization and, in turn, accelerates convergence. From the frequentist uncertainty quanti cation perspective, we consider the precision matrix of the variational posterior as an uncertainty estimate. We nd that, relative to the true asymptotic precision, the variational approximation su ers from an additional statistical error originating from the sampling uncertainty of the data. Moreover, this statistical error becomes the dominant factor as the computation budget increases. As a consequence, for small datasets, the inferential model need not be full-rank to achieve optimal estimation error (even with unlimited computation budget). We nally demonstrate these statistical and computational trade-o s inference across empirical studies, corroborating the theoretical ndings.

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