# Rethinking Losses for Diffusion Bridge Samplers

arXiv: https://arxiv.org/abs/2506.10982 · HTML: https://mlsanity.com/paper/2506.10982/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 41 references that are in the map.
- First posted on arXiv 12 Jun 2025.

## Builds on (most cited first, 29 of 29)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [GFlowNet Foundations](https://mlsanity.com/paper/2111.09266/) (2021)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [GFlowNets and variational inference](https://mlsanity.com/paper/2210.00580/) (2022)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [VarGrad: A Low-Variance Gradient Estimator for Variational Inference](https://mlsanity.com/paper/2010.10436/) (2020)
- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling](https://mlsanity.com/paper/2406.07423/) (2024)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [Langevin Diffusion Variational Inference](https://mlsanity.com/paper/2208.07743/) (2022)
- [Diffusive Gibbs Sampling](https://mlsanity.com/paper/2402.03008/) (2024)
- [Probabilistic Circuits for Variational Inference in Discrete Graphical Models](https://mlsanity.com/paper/2010.11446/) (2020)

## Built on by (most cited first, 7 of 7)

- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching](https://mlsanity.com/paper/2603.00530/) (2026)
- [Discrete Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2602.08243/) (2026)
- [Diffusion-Augmented Markov Decision Processes for Maximum Entropy Reinforcement Learning](https://mlsanity.com/paper/2512.02019/) (2025)
- [Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spaces](https://mlsanity.com/paper/2602.05961/) (2026)
- [Unsupervised Continual Learning for Amortized Bayesian Inference](https://mlsanity.com/paper/2602.22884/) (2026)
- [Guided Discovery of New Behaviors using Diffusion Policies](https://mlsanity.com/paper/2606.08743/) (2026)

## Abstract (arXiv)

Diffusion bridges are a promising class of deep-learning methods for sampling from unnormalized distributions. Recent works show that the Log Variance (LV) loss consistently outperforms the reverse Kullback-Leibler (rKL) loss when using the reparametrization trick to compute rKL-gradients. While the on-policy LV loss yields identical gradients to the rKL loss when combined with the log-derivative trick for diffusion samplers with non-learnable forward processes, this equivalence does not hold for diffusion bridges or when diffusion coefficients are learned. Based on this insight we argue that for diffusion bridges the LV loss does not represent an optimization objective that can be motivated like the rKL loss via the data processing inequality. Our analysis shows that employing the rKL loss with the log-derivative trick (rKL-LD) does not only avoid these conceptual problems but also consistently outperforms the LV loss. Experimental results with different types of diffusion bridges on challenging benchmarks show that samplers trained with the rKL-LD loss achieve better performance. From a practical perspective we find that rKL-LD requires significantly less hyperparameter optimization and yields more stable training behavior.