# Learning Conditional Invariances through Non-Commutativity

arXiv: https://arxiv.org/abs/2402.11682 · HTML: https://mlsanity.com/paper/2402.11682/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 57 references that are in the map.
- First posted on arXiv 18 Feb 2024.

## Builds on (most cited first, 30 of 42)

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Domain Randomization for Transferring Deep Neural Networks from Simulation to the Real World](https://mlsanity.com/paper/1703.06907/) (2017)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Moment Matching for Multi-Source Domain Adaptation](https://mlsanity.com/paper/1812.01754/) (2018)
- [Deep Hashing Network for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1706.07522/) (2017)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [Learning Robust Global Representations by Penalizing Local Predictive Power](https://mlsanity.com/paper/1905.13549/) (2019)
- [Deeper, Broader and Artier Domain Generalization](https://mlsanity.com/paper/1710.03077/) (2017)
- [Learning to Adapt Structured Output Space for Semantic Segmentation](https://mlsanity.com/paper/1802.10349/) (2018)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Generalizing to Unseen Domains via Adversarial Data Augmentation](https://mlsanity.com/paper/1805.12018/) (2018)
- [Bridging Theory and Algorithm for Domain Adaptation](https://mlsanity.com/paper/1904.05801/) (2019)
- [Bidirectional Learning for Domain Adaptation of Semantic Segmentation](https://mlsanity.com/paper/1904.10620/) (2019)
- [Learning to Learn Single Domain Generalization](https://mlsanity.com/paper/2003.13216/) (2020)
- [The Risks of Invariant Risk Minimization](https://mlsanity.com/paper/2010.05761/) (2020)
- [ThreeDWorld: A Platform for Interactive Multi-Modal Physical Simulation](https://mlsanity.com/paper/2007.04954/) (2020)
- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [Learning Robust Representations via Multi-View Information Bottleneck](https://mlsanity.com/paper/2002.07017/) (2020)
- [Fishr: Invariant Gradient Variances for Out-of-Distribution Generalization](https://mlsanity.com/paper/2109.02934/) (2021)
- [MIC: Masked Image Consistency for Context-Enhanced Domain Adaptation](https://mlsanity.com/paper/2212.01322/) (2022)
- [Domain Adaptation for Structured Output via Discriminative Patch Representations](https://mlsanity.com/paper/1901.05427/) (2019)
- [Using Simulation and Domain Adaptation to Improve Efficiency of Deep Robotic Grasping](https://mlsanity.com/paper/1709.07857/) (2017)
- [Delivering Arbitrary-Modal Semantic Segmentation](https://mlsanity.com/paper/2303.01480/) (2023)
- [CrDoCo: Pixel-level Domain Transfer with Cross-Domain Consistency](https://mlsanity.com/paper/2001.03182/) (2020)
- [Structured Domain Randomization: Bridging the Reality Gap by Context-Aware Synthetic Data](https://mlsanity.com/paper/1810.10093/) (2018)
- [A Closer Look at Smoothness in Domain Adversarial Training](https://mlsanity.com/paper/2206.08213/) (2022)
- [T2Net: Synthetic-to-Realistic Translation for Solving Single-Image Depth Estimation Tasks](https://mlsanity.com/paper/1808.01454/) (2018)
- [Probable Domain Generalization via Quantile Risk Minimization](https://mlsanity.com/paper/2207.09944/) (2022)

## Built on by (most cited first, 1 of 1)

- [A Closer Look at Multimodal Representation Collapse](https://mlsanity.com/paper/2505.22483/) (2025)

## Abstract (arXiv)

Invariance learning algorithms that conditionally filter out domain-specific random variables as distractors, do so based only on the data semantics, and not the target domain under evaluation. We show that a provably optimal and sample-efficient way of learning conditional invariances is by relaxing the invariance criterion to be non-commutatively directed towards the target domain. Under domain asymmetry, i.e., when the target domain contains semantically relevant information absent in the source, the risk of the encoder varphi^* that is optimal on average across domains is strictly lower-bounded by the risk of the target-specific optimal encoder Phi^*τ. We prove that non-commutativity steers the optimization towards Phi^*τ instead of varphi^*, bringing the H-divergence between domains down to zero, leading to a stricter bound on the target risk. Both our theory and experiments demonstrate that non-commutative invariance (NCI) can leverage source domain samples to meet the sample complexity needs of learning Phi^*τ, surpassing SOTA invariance learning algorithms for domain adaptation, at times by over 2%, approaching the performance of an oracle. Implementation is available at https://github.com/abhrac/nci.