# Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff

arXiv: https://arxiv.org/abs/2110.15590 · HTML: https://mlsanity.com/paper/2110.15590/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 56 references that are in the map.
- First posted on arXiv 29 Oct 2021.

## Builds on (most cited first, 16 of 16)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Not All Samples Are Created Equal: Deep Learning with Importance Sampling](https://mlsanity.com/paper/1803.00942/) (2018)
- [Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm](https://mlsanity.com/paper/1310.5715/) (2013)
- [Stochastic Optimization with Importance Sampling](https://mlsanity.com/paper/1401.2753/) (2014)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Understanding the Curse of Horizon in Off-Policy Evaluation via Conditional Importance Sampling](https://mlsanity.com/paper/1910.06508/) (2019)
- [Importance Sampling for Minibatches](https://mlsanity.com/paper/1602.02283/) (2016)
- [Provable Bayesian Inference via Particle Mirror Descent](https://mlsanity.com/paper/1506.03101/) (2015)
- [Safe Adaptive Importance Sampling](https://mlsanity.com/paper/1711.02637/) (2017)
- [From Importance Sampling to Doubly Robust Policy Gradient](https://mlsanity.com/paper/1910.09066/) (2019)
- [Adaptive Importance Sampling for Finite-Sum Optimization and Sampling with Decreasing Step-Sizes](https://mlsanity.com/paper/2103.12243/) (2021)
- [Asymptotic optimality of adaptive importance sampling](https://mlsanity.com/paper/1806.00989/) (2018)
- [Sinkhorn Algorithm as a Special Case of Stochastic Mirror Descent](https://mlsanity.com/paper/1909.06918/) (2019)

## Built on by (most cited first, 13 of 13)

- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Exponential Smoothing for Off-Policy Learning](https://mlsanity.com/paper/2305.15877/) (2023)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [Adaptive importance sampling for heavy-tailed distributions via α-divergence minimization](https://mlsanity.com/paper/2310.16653/) (2023)
- [A Quadrature Rule combining Control Variates and Adaptive Importance Sampling](https://mlsanity.com/paper/2205.11890/) (2022)
- [Information-Theoretic Generative Clustering of Documents](https://mlsanity.com/paper/2412.13534/) (2024)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)
- [How to Approximate Inference with Subtractive Mixture Models](https://mlsanity.com/paper/2604.16714/) (2026)
- [Even More Guarantees for Variational Inference in the Presence of Symmetries](https://mlsanity.com/paper/2604.21407/) (2026)
- [Scalable Expectation Estimation with Subtractive Mixture Models](https://mlsanity.com/paper/2503.21346/) (2025)
- [Entropic Mirror Monte Carlo](https://mlsanity.com/paper/2602.03165/) (2026)

## Abstract (arXiv)

Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically.