# Convergence rates for optimised adaptive importance samplers

arXiv: https://arxiv.org/abs/1903.12044 · HTML: https://mlsanity.com/paper/1903.12044/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2019.

## Built on by (most cited first, 12 of 12)

- [Quantification of Uncertainty with Adversarial Models](https://mlsanity.com/paper/2307.03217/) (2023)
- [Cooperation in the Latent Space: The Benefits of Adding Mixture Components in Variational Autoencoders](https://mlsanity.com/paper/2209.15514/) (2022)
- [Forward chi2 Divergence Based Variational Importance Sampling](https://mlsanity.com/paper/2311.02516/) (2023)
- [MCMC-driven importance samplers](https://mlsanity.com/paper/2105.02579/) (2021)
- [Adaptive importance sampling for heavy-tailed distributions via α-divergence minimization](https://mlsanity.com/paper/2310.16653/) (2023)
- [Learning Latent Variable Models via Jarzynski-adjusted Langevin Algorithm](https://mlsanity.com/paper/2505.18427/) (2025)
- [Global convergence of optimized adaptive importance samplers](https://mlsanity.com/paper/2201.00409/) (2022)
- [Efficient Bayes Inference in Neural Networks through Adaptive Importance Sampling](https://mlsanity.com/paper/2210.00993/) (2022)
- [NEO: Non Equilibrium Sampling on the Orbit of a Deterministic Transform](https://mlsanity.com/paper/2103.10943/) (2021)
- [Adaptively Optimised Adaptive Importance Samplers](https://mlsanity.com/paper/2307.09341/) (2023)
- [Entropic Mirror Monte Carlo](https://mlsanity.com/paper/2602.03165/) (2026)
- [Amortised adaptive importance sampling](https://mlsanity.com/paper/2610.16736/) (2026)

## Abstract (arXiv)

Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which textitadapt themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same O(1/√N) convergence rate as standard importance samplers, where N is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed textitoptimised adaptive importance samplers (OAIS). These samplers rely on the iterative minimisation of the chi2-divergence between an exponential-family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the L2 errors of the optimised samplers converge to the optimal rate of O(1/√N) and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic L2 error increases by a factor √ρ^star > 1, where ρ^star - 1 is the minimum chi2-divergence between the target and an exponential-family proposal.