# Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectives

arXiv: https://arxiv.org/abs/2201.10989 · HTML: https://mlsanity.com/paper/2201.10989/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 67 references that are in the map.
- First posted on arXiv 26 Jan 2022.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Filtering Variational Objectives](https://mlsanity.com/paper/1705.09279/) (2017)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [On Nesting Monte Carlo Estimators](https://mlsanity.com/paper/1709.06181/) (2017)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Auto-Encoding Sequential Monte Carlo](https://mlsanity.com/paper/1705.10306/) (2017)
- [not-MIWAE: Deep Generative Modelling with Missing not at Random Data](https://mlsanity.com/paper/2006.12871/) (2020)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [Pareto Smoothed Importance Sampling](https://mlsanity.com/paper/1507.02646/) (2015)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [Deep Verifier Networks: Verification of Deep Discriminative Models with Deep Generative Models](https://mlsanity.com/paper/1911.07421/) (2019)
- [Monte Carlo with Determinantal Point Processes](https://mlsanity.com/paper/1605.00361/) (2016)
- [Deep Gaussian Processes with Importance-Weighted Variational Inference](https://mlsanity.com/paper/1905.05435/) (2019)
- [Learning Discrete Distributions by Dequantization](https://mlsanity.com/paper/2001.11235/) (2020)
- [Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation](https://mlsanity.com/paper/1906.10115/) (2019)
- [Learning Wake-Sleep Recurrent Attention Models](https://mlsanity.com/paper/1509.06812/) (2015)
- [Reweighted Expectation Maximization](https://mlsanity.com/paper/1906.05850/) (2019)
- [MissDeepCausal: Causal Inference from Incomplete Data Using Deep Latent Variable Models](https://mlsanity.com/paper/2002.10837/) (2020)
- [Optimal Variance Control of the Score Function Gradient Estimator for Importance Weighted Bounds](https://mlsanity.com/paper/2008.01998/) (2020)
- [Fast amortized inference of neural activity from calcium imaging data with variational autoencoders](https://mlsanity.com/paper/1711.01846/) (2017)
- [Differentiable Antithetic Sampling for Variance Reduction in Stochastic Variational Inference](https://mlsanity.com/paper/1810.02555/) (2018)
- [Note on the bias and variance of variational inference](https://mlsanity.com/paper/1906.03708/) (2019)
- [Hierarchical Importance Weighted Autoencoders](https://mlsanity.com/paper/1905.04866/) (2019)

## Built on by (most cited first, 3 of 3)

- [Are Ensembles Getting Better all the Time?](https://mlsanity.com/paper/2311.17885/) (2023)
- [U-Statistics for Importance-Weighted Variational Inference](https://mlsanity.com/paper/2302.13918/) (2023)
- [Adaptive importance sampling for heavy-tailed distributions via α-divergence minimization](https://mlsanity.com/paper/2310.16653/) (2023)

## Abstract (arXiv)

We revisit the theory of importance weighted variational inference (IWVI), a promising strategy for learning latent variable models. IWVI uses new variational bounds, known as Monte Carlo objectives (MCOs), obtained by replacing intractable integrals by Monte Carlo estimates -- usually simply obtained via importance sampling. Burda, Grosse and Salakhutdinov (2016) showed that increasing the number of importance samples provably tightens the gap between the bound and the likelihood. Inspired by this simple monotonicity theorem, we present a series of nonasymptotic results that link properties of Monte Carlo estimates to tightness of MCOs. We challenge the rationale that smaller Monte Carlo variance leads to better bounds. We confirm theoretically the empirical findings of several recent papers by showing that, in a precise sense, negative correlation reduces the variational gap. We also generalise the original monotonicity theorem by considering non-uniform weights. We discuss several practical consequences of our theoretical results. Our work borrows many ideas and results from the theory of stochastic orders.