# Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2504.02618 · HTML: https://mlsanity.com/paper/2504.02618/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 113 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Builds on (most cited first, 30 of 47)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Rethinking FID: Towards a Better Evaluation Metric for Image Generation](https://mlsanity.com/paper/2401.09603/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Adversarial Latent Autoencoders](https://mlsanity.com/paper/2004.04467/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Exponential Moving Average of Weights in Deep Learning: Dynamics and Benefits](https://mlsanity.com/paper/2411.18704/) (2024)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)

## Built on by (most cited first, 2 of 2)

- [Mirror Descent Using the Tempesta Generalized Multi-parametric Logarithms](https://mlsanity.com/paper/2506.13984/) (2025)
- [Variational Entropic Optimal Transport](https://mlsanity.com/paper/2602.02241/) (2026)

## Abstract (arXiv)

The Schr"odinger bridge (SB) has evolved into a universal class of probabilistic generative models. In practice, however, estimated learning signals are innately uncertain, and the reliability promised by existing methods is often based on speculative optimal case scenarios. Recent studies regarding the Sinkhorn algorithm through mirror descent (MD) have gained attention, revealing geometric insights into solution acquisition of the SB problems. In this paper, we propose a variational online MD (OMD) framework for the SB problems, which provides further stability to SB solvers. We formally prove convergence and a regret bound for the novel OMD formulation of SB acquisition. As a result, we propose a simulation-free SB algorithm called Variational Mirrored Schr"odinger Bridge (VMSB) by utilizing the Wasserstein-Fisher-Rao geometry of the Gaussian mixture parameterization for Schr"odinger potentials. Based on the Wasserstein gradient flow theory, the algorithm offers tractable learning dynamics that precisely approximate each OMD step. In experiments, we validate the performance of the proposed VMSB algorithm across an extensive suite of benchmarks. VMSB consistently outperforms contemporary SB solvers on a wide range of SB problems, demonstrating the robustness as well as generality predicted by our OMD theory.