# General and Estimable Learning Bound Unifying Covariate and Concept Shifts

arXiv: https://arxiv.org/abs/2506.12829 · HTML: https://mlsanity.com/paper/2506.12829/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 40 references that are in the map.
- First posted on arXiv 15 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [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)
- [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)
- [Bridging Theory and Algorithm for Domain Adaptation](https://mlsanity.com/paper/1904.05801/) (2019)
- [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)
- [Towards a Theoretical Framework of Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.04496/) (2021)
- [NICO++: Towards Better Benchmarking for Domain Generalization](https://mlsanity.com/paper/2204.08040/) (2022)
- [Unsupervised Multi-Class Domain Adaptation: Theory, Algorithms, and Practice](https://mlsanity.com/paper/2002.08681/) (2020)
- [Diagnosing Model Performance Under Distribution Shift](https://mlsanity.com/paper/2303.02011/) (2023)
- [Convergence rate of general entropic optimal transport costs](https://mlsanity.com/paper/2206.03347/) (2022)
- [Algorithm-Dependent Bounds for Representation Learning of Multi-Source Domain Adaptation](https://mlsanity.com/paper/2304.02064/) (2023)

## 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 bounds become loose and non-estimable because their concept shift definition breaks when the source and target supports mismatch. Leveraging entropic optimal transport, we propose new support-agnostic definitions for covariate and concept shifts, and derive a novel unified error bound that 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.