# All in the Exponential Family: Bregman Duality in Thermodynamic Variational Inference

arXiv: https://arxiv.org/abs/2007.00642 · HTML: https://mlsanity.com/paper/2007.00642/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2020.

## Built on by (most cited first, 11 of 11)

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- [Hamiltonian Dynamics with Non-Newtonian Momentum for Rapid Sampling](https://mlsanity.com/paper/2111.02434/) (2021)
- [ItDPDM: Information-Theoretic Discrete Poisson Diffusion Model](https://mlsanity.com/paper/2505.05082/) (2025)
- [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)
- [Optimal Regret Is Achievable with Bounded Approximate Inference Error: An Enhanced Bayesian Upper Confidence Bound Framework](https://mlsanity.com/paper/2201.12955/) (2022)

## Abstract (arXiv)

The recently proposed Thermodynamic Variational Objective (TVO) leverages thermodynamic integration to provide a family of variational inference objectives, which both tighten and generalize the ubiquitous Evidence Lower Bound (ELBO). However, the tightness of TVO bounds was not previously known, an expensive grid search was used to choose a "schedule" of intermediate distributions, and model learning suffered with ostensibly tighter bounds. In this work, we propose an exponential family interpretation of the geometric mixture curve underlying the TVO and various path sampling methods, which allows us to characterize the gap in TVO likelihood bounds as a sum of KL divergences. We propose to choose intermediate distributions using equal spacing in the moment parameters of our exponential family, which matches grid search performance and allows the schedule to adaptively update over the course of training. Finally, we derive a doubly reparameterized gradient estimator which improves model learning and allows the TVO to benefit from more refined bounds. To further contextualize our contributions, we provide a unified framework for understanding thermodynamic integration and the TVO using Taylor series remainders.