# General Quantification of Covariate and Concept Shifts

arXiv: https://arxiv.org/abs/2609.11918 · HTML: https://mlsanity.com/paper/2609.11918/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 27 of 27)

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Towards Out-Of-Distribution Generalization: A Survey](https://mlsanity.com/paper/2108.13624/) (2021)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Bridging Theory and Algorithm for Domain Adaptation](https://mlsanity.com/paper/1904.05801/) (2019)
- [Pretrained Transformers Improve Out-of-Distribution Robustness](https://mlsanity.com/paper/2004.06100/) (2020)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Multi-Adversarial Domain Adaptation](https://mlsanity.com/paper/1809.02176/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Theoretical Analysis of Domain Adaptation with Optimal Transport](https://mlsanity.com/paper/1610.04420/) (2016)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [NICO++: Towards Better Benchmarking for Domain Generalization](https://mlsanity.com/paper/2204.08040/) (2022)
- [How Smooth Is Attention?](https://mlsanity.com/paper/2312.14820/) (2023)
- [Unsupervised Multi-Class Domain Adaptation: Theory, Algorithms, and Practice](https://mlsanity.com/paper/2002.08681/) (2020)
- [On Lipschitz Bounds of General Convolutional Neural Networks](https://mlsanity.com/paper/1808.01415/) (2018)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Flowing Datasets with Wasserstein over Wasserstein Gradient Flows](https://mlsanity.com/paper/2506.07534/) (2025)
- [Convergence rate of general entropic optimal transport costs](https://mlsanity.com/paper/2206.03347/) (2022)
- [Theoretical Guarantees for Domain Adaptation with Hierarchical Optimal Transport](https://mlsanity.com/paper/2210.13331/) (2022)

## Abstract (arXiv)

Generalization under distribution shift remains a core challenge in modern machine learning, yet existing learning bound theory is limited to narrow, idealized settings and is non-estimable from samples. In this paper, we bridge the gap between theory and practical applications. We first show that existing definition of concept shift breaks when the source and target supports mismatch. Leveraging entropic optimal transport, we propose a key notion: γ*!-concept shifts, and derive a general error bound unifying covariate and γ*!-concept shifts, which applies to broad loss functions, label spaces, and stochastic labeling. We further develop estimators for these shifts with concentration guarantees, and the DataShifts algorithm, which can quantify distribution shifts and estimate the error bound in most applications - a rigorous and general tool for analyzing learning error under distribution shift.