# Variational Inference with Tail-adaptive f-Divergence

arXiv: https://arxiv.org/abs/1810.11943 · HTML: https://mlsanity.com/paper/1810.11943/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2018.

## Built on by (most cited first, 25 of 25)

- [Aligning Language Models with Preferences through f-divergence Minimization](https://mlsanity.com/paper/2302.08215/) (2023)
- [GFlowNets and variational inference](https://mlsanity.com/paper/2210.00580/) (2022)
- [AlphaNet: Improved Training of Supernets with Alpha-Divergence](https://mlsanity.com/paper/2102.07954/) (2021)
- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [On Proximal Policy Optimization's Heavy-tailed Gradients](https://mlsanity.com/paper/2102.10264/) (2021)
- [Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics](https://mlsanity.com/paper/2210.06226/) (2022)
- [Multiple Importance Sampling ELBO and Deep Ensembles of Variational Approximations](https://mlsanity.com/paper/2202.10951/) (2022)
- [Variational Inference MPC using Tsallis Divergence](https://mlsanity.com/paper/2104.00241/) (2021)
- [Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows](https://mlsanity.com/paper/2205.07918/) (2022)
- [Particle-based Variational Inference with Generalized Wasserstein Gradient Flow](https://mlsanity.com/paper/2310.16516/) (2023)
- [Mixture weights optimisation for Alpha-Divergence Variational Inference](https://mlsanity.com/paper/2106.05114/) (2021)
- [Markov Chain Score Ascent: A Unifying Framework of Variational Inference with Markovian Gradients](https://mlsanity.com/paper/2206.06295/) (2022)
- [Monotonic Alpha-divergence Minimisation for Variational Inference](https://mlsanity.com/paper/2103.05684/) (2021)
- [f-Trajectory Balance: A Loss Family for Tuning GFlowNets, Generative Models, and LLMs with Off- and On-Policy Data](https://mlsanity.com/paper/2605.15417/) (2026)
- [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)
- [Optimal Regret Is Achievable with Bounded Approximate Inference Error: An Enhanced Bayesian Upper Confidence Bound Framework](https://mlsanity.com/paper/2201.12955/) (2022)
- [Improved Stein Variational Gradient Descent with Importance Weights](https://mlsanity.com/paper/2210.00462/) (2022)
- [A Heavy-Tailed Algebra for Probabilistic Programming](https://mlsanity.com/paper/2306.09262/) (2023)
- [Text-to-3D Generation using Jensen-Shannon Score Distillation](https://mlsanity.com/paper/2503.10660/) (2025)
- [FlowVAT: Normalizing Flow Variational Inference with Affine-Invariant Tempering](https://mlsanity.com/paper/2505.10466/) (2025)
- [CIM-PPO:Proximal Policy Optimization with Liu-Correntropy Induced Metric](https://mlsanity.com/paper/2110.10522/) (2021)
- [Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence](https://mlsanity.com/paper/2301.11476/) (2023)
- [Adaptive Heterogeneous Mixtures of Normalising Flows for Robust Variational Inference](https://mlsanity.com/paper/2510.02056/) (2025)
- [Trust Regions Sell, But Who's Buying? Overlap Geometry as an Alternative Trust Region for Policy Optimization](https://mlsanity.com/paper/2602.06627/) (2026)

## Abstract (arXiv)

Variational inference with α-divergences has been widely used in modern probabilistic machine learning. Compared to Kullback-Leibler (KL) divergence, a major advantage of using α-divergences (with positive α values) is their mass-covering property. However, estimating and optimizing α-divergences require to use importance sampling, which could have extremely large or infinite variances due to heavy tails of importance weights. In this paper, we propose a new class of tail-adaptive f-divergences that adaptively change the convex function f with the tail of the importance weights, in a way that theoretically guarantees finite moments, while simultaneously achieving mass-covering properties. We test our methods on Bayesian neural networks, as well as deep reinforcement learning in which our method is applied to improve a recent soft actor-critic (SAC) algorithm. Our results show that our approach yields significant advantages compared with existing methods based on classical KL and α-divergences.