# q-Paths: Generalizing the Geometric Annealing Path using Power Means

arXiv: https://arxiv.org/abs/2107.00745 · HTML: https://mlsanity.com/paper/2107.00745/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 56 references that are in the map.
- First posted on arXiv 1 Jul 2021.

## Builds on (most cited first, 12 of 12)

- [On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means](https://mlsanity.com/paper/1904.04017/) (2019)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Pareto Smoothed Importance Sampling](https://mlsanity.com/paper/1507.02646/) (2015)
- [On the Quantitative Analysis of Decoder-Based Generative Models](https://mlsanity.com/paper/1611.04273/) (2016)
- [The Thermodynamic Variational Objective](https://mlsanity.com/paper/1907.00031/) (2019)
- [Sandwiching the marginal likelihood using bidirectional Monte Carlo](https://mlsanity.com/paper/1511.02543/) (2015)
- [Constraining Variational Inference with Geometric Jensen-Shannon Divergence](https://mlsanity.com/paper/2006.10599/) (2020)
- [Two-temperature logistic regression based on the Tsallis divergence](https://mlsanity.com/paper/1705.07210/) (2017)
- [All in the Exponential Family: Bregman Duality in Thermodynamic Variational Inference](https://mlsanity.com/paper/2007.00642/) (2020)
- [q-VAE for Disentangled Representation Learning and Latent Dynamical Systems](https://mlsanity.com/paper/2003.01852/) (2020)
- [Expectation Propagation for t-Exponential Family Using Q-Algebra](https://mlsanity.com/paper/1705.09046/) (2017)
- [Likelihood Ratio Exponential Families](https://mlsanity.com/paper/2012.15480/) (2020)

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

- [Score-Based Diffusion meets Annealed Importance Sampling](https://mlsanity.com/paper/2208.07698/) (2022)
- [Information-Theoretic Diffusion](https://mlsanity.com/paper/2302.03792/) (2023)
- [Accelerated Parallel Tempering via Neural Transports](https://mlsanity.com/paper/2502.10328/) (2025)
- [Provable benefits of annealing for estimating normalizing constants: Importance Sampling, Noise-Contrastive Estimation, and beyond](https://mlsanity.com/paper/2310.03902/) (2023)
- [CREPE: Controlling Diffusion with Replica Exchange](https://mlsanity.com/paper/2509.23265/) (2025)
- [Adaptive Annealed Importance Sampling with Constant Rate Progress](https://mlsanity.com/paper/2306.15283/) (2023)
- [Variational Inference with Holder Bounds](https://mlsanity.com/paper/2111.02947/) (2021)
- [Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding](https://mlsanity.com/paper/2209.07481/) (2022)
- [AMiD: Knowledge Distillation for LLMs with α-mixture Assistant Distribution](https://mlsanity.com/paper/2510.15982/) (2025)
- [Optimization of Annealed Importance Sampling Hyperparameters](https://mlsanity.com/paper/2209.13226/) (2022)
- [Score Operator Newton transport](https://mlsanity.com/paper/2305.09792/) (2023)
- [Two tales for a geometric Jensen--Shannon divergence](https://mlsanity.com/paper/2508.05066/) (2025)

## Abstract (arXiv)

Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce q-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our q-paths as corresponding to a q-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of α-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.