@article{eastwood2022probable,
  title={Probable domain generalization via quantile risk minimization},
  author={Eastwood, Cian and Robey, Alexander and Singh, Shashank and Von K{\"u}gelgen, Julius and Hassani, Hamed and Pappas, George J and Sch{\"o}lkopf, Bernhard},
  journal={arXiv preprint arXiv:2207.09944},
  year={2022}
}

@article{chen2022iterative,
  title={Iterative feature matching: Toward provable domain generalization with logarithmic environments},
  author={Chen, Yining and Rosenfeld, Elan and Sellke, Mark and Ma, Tengyu and Risteski, Andrej},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={1725--1736},
  year={2022}
}

@inproceedings{
wald2023malign,
title={Malign Overfitting: Interpolation and Invariance are Fundamentally at Odds},
author={Yoav Wald and Gal Yona and Uri Shalit and Yair Carmon},
booktitle={The Eleventh International Conference on Learning Representations },
year={2023},
url={https://openreview.net/forum?id=dQNL7Zsta3}
}

@article{zhang2021adaptive,
  title={Adaptive risk minimization: Learning to adapt to domain shift},
  author={Zhang, Marvin and Marklund, Henrik and Dhawan, Nikita and Gupta, Abhishek and Levine, Sergey and Finn, Chelsea},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={23664--23678},
  year={2021}
}

@misc{buhlmann_invariance_2018,
	title = {Invariance, {Causality} and {Robustness}},
	url = {http://arxiv.org/abs/1812.08233},
	abstract = {We discuss recent work for causal inference and predictive robustness in a unifying way. The key idea relies on a notion of probabilistic invariance or stability: it opens up new insights for formulating causality as a certain risk minimization problem with a corresponding notion of robustness. The invariance itself can be estimated from general heterogeneous or perturbation data which frequently occur with nowadays data collection. The novel methodology is potentially useful in many applications, offering more robustness and better `causal-oriented' interpretation than machine learning or estimation in standard regression or classification frameworks.},
	urldate = {2023-02-15},
	publisher = {arXiv},
	author = {Bühlmann, Peter},
	month = dec,
	year = {2018},
	note = {arXiv:1812.08233 [stat]},
	keywords = {62J99, Statistics - Methodology},
	annote = {Comment: 36 pages, 12 figures},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\K8P2RRPM\\Bühlmann - 2018 - Invariance, Causality and Robustness.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\X6IYKIRC\\1812.html:text/html},
}

@misc{rojas-carulla_invariant_2018,
	title = {Invariant {Models} for {Causal} {Transfer} {Learning}},
	url = {http://arxiv.org/abs/1507.05333},
	doi = {10.48550/arXiv.1507.05333},
	abstract = {Methods of transfer learning try to combine knowledge from several related tasks (or domains) to improve performance on a test task. Inspired by causal methodology, we relax the usual covariate shift assumption and assume that it holds true for a subset of predictor variables: the conditional distribution of the target variable given this subset of predictors is invariant over all tasks. We show how this assumption can be motivated from ideas in the field of causality. We focus on the problem of Domain Generalization, in which no examples from the test task are observed. We prove that in an adversarial setting using this subset for prediction is optimal in Domain Generalization; we further provide examples, in which the tasks are sufficiently diverse and the estimator therefore outperforms pooling the data, even on average. If examples from the test task are available, we also provide a method to transfer knowledge from the training tasks and exploit all available features for prediction. However, we provide no guarantees for this method. We introduce a practical method which allows for automatic inference of the above subset and provide corresponding code. We present results on synthetic data sets and a gene deletion data set.},
	urldate = {2023-02-15},
	publisher = {arXiv},
	author = {Rojas-Carulla, Mateo and Schölkopf, Bernhard and Turner, Richard and Peters, Jonas},
	month = sep,
	year = {2018},
	note = {arXiv:1507.05333 [stat]},
	keywords = {Statistics - Machine Learning},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\894FUV3W\\Rojas-Carulla et al. - 2018 - Invariant Models for Causal Transfer Learning.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\F4FVFKGQ\\1507.html:text/html},
}


@article{blanchard2011generalizing,
  title={Generalizing from several related classification tasks to a new unlabeled sample},
  author={Blanchard, Gilles and Lee, Gyemin and Scott, Clayton},
  journal={Advances in neural information processing systems},
  volume={24},
  year={2011}
}

@inproceedings{mccoy2020right,
  title={Right for the wrong reasons: Diagnosing syntactic heuristics in natural language inference},
  author={McCoy, R Thomas and Pavlick, Ellie and Linzen, Tal},
  booktitle={57th Annual Meeting of the Association for Computational Linguistics, ACL 2019},
  pages={3428--3448},
  year={2020},
  organization={Association for Computational Linguistics (ACL)}
}

@article{sagawa_distributionally_2020,
	title = {Distributionally {Robust} {Neural} {Networks} for {Group} {Shifts}: {On} the {Importance} of {Regularization} for {Worst}-{Case} {Generalization}},
	shorttitle = {Distributionally {Robust} {Neural} {Networks} for {Group} {Shifts}},
	url = {http://arxiv.org/abs/1911.08731},
	abstract = {Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-defined groups. However, we find that naively applying group DRO to overparameterized neural networks fails: these models can perfectly fit the training data, and any model with vanishing average training loss also already has vanishing worst-case training loss. Instead, the poor worst-case performance arises from poor generalization on some groups. By coupling group DRO models with increased regularization---a stronger-than-typical L2 penalty or early stopping---we achieve substantially higher worst-group accuracies, with 10-40 percentage point improvements on a natural language inference task and two image tasks, while maintaining high average accuracies. Our results suggest that regularization is important for worst-group generalization in the overparameterized regime, even if it is not needed for average generalization. Finally, we introduce a stochastic optimization algorithm, with convergence guarantees, to efficiently train group DRO models.},
	urldate = {2020-10-29},
	journal = {arXiv:1911.08731 [cs, stat]},
	author = {Sagawa, Shiori and Koh, Pang Wei and Hashimoto, Tatsunori B. and Liang, Percy},
	month = apr,
	year = {2020},
	note = {arXiv: 1911.08731},
	keywords = {Causal/Invariant},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\TBP8NPRF\\Sagawa et al. - 2020 - Distributionally Robust Neural Networks for Group .pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\66HSSSE7\\1911.html:text/html},
}


@book{quinonero2008dataset,
  title={Dataset shift in machine learning},
  author={Quinonero-Candela, Joaquin and Sugiyama, Masashi and Schwaighofer, Anton and Lawrence, Neil D},
  year={2008},
  publisher={Mit Press}
}

@inproceedings{li2017deeper,
  title={Deeper, broader and artier domain generalization},
  author={Li, Da and Yang, Yongxin and Song, Yi-Zhe and Hospedales, Timothy M},
  booktitle={Proceedings of the IEEE international conference on computer vision},
  pages={5542--5550},
  year={2017}
}

@article{peters2016causal,
  title={Causal inference by using invariant prediction: identification and confidence intervals},
  author={Peters, Jonas and B{\"u}hlmann, Peter and Meinshausen, Nicolai},
  journal={Journal of the Royal Statistical Society. Series B (Statistical Methodology)},
  pages={947--1012},
  year={2016},
  publisher={JSTOR}
}

@article{zhu2014regbayes,
  title={Bayesian Inference with Posterior Regularization
and Applications to Infinite Latent SVMs},
  author={Zhu, Jun and Chen, Ning and Xing, Eric},
  journal={Journal of Machine Learning Research},
  volume = {15},
  pages={1799--1847},
  year={2014}
}

@article{degrave_ai_2021,
	title = {{AI} for radiographic {COVID}-19 detection selects shortcuts over signal},
	volume = {3},
	issn = {2522-5839},
	url = {https://www.nature.com/articles/s42256-021-00338-7},
	doi = {10.1038/s42256-021-00338-7},
	language = {en},
	number = {7},
	urldate = {2023-02-13},
	journal = {Nature Machine Intelligence},
	author = {DeGrave, Alex J. and Janizek, Joseph D. and Lee, Su-In},
	month = may,
	year = {2021},
	pages = {610--619},
	file = {DeGrave et al. - 2021 - AI for radiographic COVID-19 detection selects sho.pdf:C\:\\Users\\w\\Zotero\\storage\\KQR9WBCZ\\DeGrave et al. - 2021 - AI for radiographic COVID-19 detection selects sho.pdf:application/pdf},
}


@article{bachem2017practical,
  title={Practical coreset constructions for machine learning},
  author={Bachem, Olivier and Lucic, Mario and Krause, Andreas},
  journal={arXiv preprint arXiv:1703.06476},
  year={2017}
}

@article{zhang2022sampling,
  title={Sampling in Constrained Domains with Orthogonal-Space Variational Gradient Descent},
  author={Zhang, Ruqi and Liu, Qiang and Tong, Xin T},
  journal={arXiv preprint arXiv:2210.06447},
  year={2022}
}

@article{sato2022convergence,
  title={Convergence error analysis of reflected gradient langevin dynamics for globally optimizing non-convex constrained problems},
  author={Sato, Kanji and Takeda, Akiko and Kawai, Reiichiro and Suzuki, Taiji},
  journal={arXiv preprint arXiv:2203.10215},
  year={2022}
}

@article{bubeck2018sampling,
  title={Sampling from a log-concave distribution with projected langevin monte carlo},
  author={Bubeck, S{\'e}bastien and Eldan, Ronen and Lehec, Joseph},
  journal={Discrete \& Computational Geometry},
  volume={59},
  pages={757--783},
  year={2018},
  publisher={Springer}
}

@article{lions1984stochastic,
  title={Stochastic differential equations with reflecting boundary conditions},
  author={Lions, Pierre-Louis and Sznitman, Alain-Sol},
  journal={Communications on pure and applied Mathematics},
  volume={37},
  number={4},
  pages={511--537},
  year={1984},
  publisher={Wiley Online Library}
}

@article{nagarajan2020understanding,
  title={Understanding the failure modes of out-of-distribution generalization},
  author={Nagarajan, Vaishnavh and Andreassen, Anders and Neyshabur, Behnam},
  journal={arXiv preprint arXiv:2010.15775},
  year={2020}
}

@misc{lee_diversify_2022,
	title = {Diversify and {Disambiguate}: {Learning} {From} {Underspecified} {Data}},
	shorttitle = {Diversify and {Disambiguate}},
	url = {http://arxiv.org/abs/2202.03418},
	abstract = {Many datasets are underspecified: there exist multiple equally viable solutions to a given task. Underspecification can be problematic for methods that learn a single hypothesis because different functions that achieve low training loss can focus on different predictive features and thus produce widely varying predictions on out-of-distribution data. We propose DivDis, a simple two-stage framework that first learns a diverse collection of hypotheses for a task by leveraging unlabeled data from the test distribution. We then disambiguate by selecting one of the discovered hypotheses using minimal additional supervision, in the form of additional labels or inspection of function visualization. We demonstrate the ability of DivDis to find hypotheses that use robust features in image classification and natural language processing problems with underspecification.},
	urldate = {2023-02-11},
	publisher = {arXiv},
	author = {Lee, Yoonho and Yao, Huaxiu and Finn, Chelsea},
	month = jun,
	year = {2022},
	note = {arXiv:2202.03418 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	annote = {Comment: Code is available at https://github.com/yoonholee/DivDis},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\6ZCFIKVG\\Lee et al. - 2022 - Diversify and Disambiguate Learning From Underspe.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\MXS4QZF5\\2202.html:text/html},
}

@inproceedings{lin2022bayesian,
  title={Bayesian invariant risk minimization},
  author={Lin, Yong and Dong, Hanze and Wang, Hao and Zhang, Tong},
  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  pages={16021--16030},
  year={2022}
}

@article{wilson_survey_2020,
	title = {A {Survey} of {Unsupervised} {Deep} {Domain} {Adaptation}},
	volume = {11},
	issn = {2157-6904, 2157-6912},
	url = {https://dl.acm.org/doi/10.1145/3400066},
	doi = {10.1145/3400066},
	abstract = {Deep learning has produced state-of-the-art results for a variety of tasks. While such approaches for supervised learning have performed well, they assume that training and testing data are drawn from the same distribution, which may not always be the case. As a complement to this challenge, single-source unsupervised domain adaptation can handle situations where a network is trained on labeled data from a source domain and unlabeled data from a related but different target domain with the goal of performing well at test-time on the target domain. Many single-source and typically homogeneous unsupervised deep domain adaptation approaches have thus been developed, combining the powerful, hierarchical representations from deep learning with domain adaptation to reduce reliance on potentially costly target data labels. This survey will compare these approaches by examining alternative methods, the unique and common elements, results, and theoretical insights. We follow this with a look at application areas and open research directions.},
	language = {en},
	number = {5},
	urldate = {2022-04-24},
	journal = {ACM Transactions on Intelligent Systems and Technology},
	author = {Wilson, Garrett and Cook, Diane J.},
	month = oct,
	year = {2020},
	pages = {1--46},
	file = {Wilson and Cook - 2020 - A Survey of Unsupervised Deep Domain Adaptation.pdf:C\:\\Users\\w\\Zotero\\storage\\N5DVFK8V\\Wilson and Cook - 2020 - A Survey of Unsupervised Deep Domain Adaptation.pdf:application/pdf},
}


@misc{wang_generalizing_2022,
	title = {Generalizing to {Unseen} {Domains}: {A} {Survey} on {Domain} {Generalization}},
	shorttitle = {Generalizing to {Unseen} {Domains}},
	url = {http://arxiv.org/abs/2103.03097},
	abstract = {Machine learning systems generally assume that the training and testing distributions are the same. To this end, a key requirement is to develop models that can generalize to unseen distributions. Domain generalization (DG), i.e., out-of-distribution generalization, has attracted increasing interests in recent years. Domain generalization deals with a challenging setting where one or several different but related domain(s) are given, and the goal is to learn a model that can generalize to an unseen test domain. Great progress has been made in the area of domain generalization for years. This paper presents the first review of recent advances in this area. First, we provide a formal definition of domain generalization and discuss several related fields. We then thoroughly review the theories related to domain generalization and carefully analyze the theory behind generalization. We categorize recent algorithms into three classes: data manipulation, representation learning, and learning strategy, and present several popular algorithms in detail for each category. Third, we introduce the commonly used datasets, applications, and our open-sourced codebase for fair evaluation. Finally, we summarize existing literature and present some potential research topics for the future.},
	urldate = {2023-02-09},
	publisher = {arXiv},
	author = {Wang, Jindong and Lan, Cuiling and Liu, Chang and Ouyang, Yidong and Qin, Tao and Lu, Wang and Chen, Yiqiang and Zeng, Wenjun and Yu, Philip S.},
	month = may,
	year = {2022},
	note = {arXiv:2103.03097 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning},
	annote = {Comment: Accepted by IEEE Trans. on Knowledge and Data Engineering (TKDE); 19 pages; short version (6 pages) is accepted by IJCAI-21 survey track; codebase: https://github.com/jindongwang/transferlearning/tree/master/code/DeepDG},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\WH6GHNX9\\Wang et al. - 2022 - Generalizing to Unseen Domains A Survey on Domain.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\W7PQXYSX\\2103.html:text/html},
}


@misc{jiang_transferability_2022,
	title = {Transferability in {Deep} {Learning}: {A} {Survey}},
	shorttitle = {Transferability in {Deep} {Learning}},
	url = {http://arxiv.org/abs/2201.05867},
	abstract = {The success of deep learning algorithms generally depends on large-scale data, while humans appear to have inherent ability of knowledge transfer, by recognizing and applying relevant knowledge from previous learning experiences when encountering and solving unseen tasks. Such an ability to acquire and reuse knowledge is known as transferability in deep learning. It has formed the long-term quest towards making deep learning as data-efficient as human learning, and has been motivating fruitful design of more powerful deep learning algorithms. We present this survey to connect different isolated areas in deep learning with their relation to transferability, and to provide a unified and complete view to investigating transferability through the whole lifecycle of deep learning. The survey elaborates the fundamental goals and challenges in parallel with the core principles and methods, covering recent cornerstones in deep architectures, pre-training, task adaptation and domain adaptation. This highlights unanswered questions on the appropriate objectives for learning transferable knowledge and for adapting the knowledge to new tasks and domains, avoiding catastrophic forgetting and negative transfer. Finally, we implement a benchmark and an open-source library, enabling a fair evaluation of deep learning methods in terms of transferability.},
	urldate = {2023-02-11},
	publisher = {arXiv},
	author = {Jiang, Junguang and Shu, Yang and Wang, Jianmin and Long, Mingsheng},
	month = jan,
	year = {2022},
	note = {arXiv:2201.05867 [cs]
version: 1},
	keywords = {Computer Science - Machine Learning},
	annote = {Comment: 64 pages. Code at https://github.com/thuml/Transfer-Learning-Library},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\MSNBIGCU\\Jiang et al. - 2022 - Transferability in Deep Learning A Survey.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\WP647D8D\\2201.html:text/html},
}


@misc{ye_freeze_2022,
	title = {Freeze then {Train}: {Towards} {Provable} {Representation} {Learning} under {Spurious} {Correlations} and {Feature} {Noise}},
	shorttitle = {Freeze then {Train}},
	url = {http://arxiv.org/abs/2210.11075},
	abstract = {The existence of spurious correlations such as image backgrounds in the training environment can make empirical risk minimization (ERM) perform badly in the test environment. To address this problem, Kirichenko et al. (2022) empirically found that the core features that are causally related to the outcome can still be learned well even with the presence of spurious correlations. This opens a promising strategy to first train a feature learner rather than a classifier, and then perform linear probing (last layer retraining) in the test environment. However, a theoretical understanding of when and why this approach works is lacking. In this paper, we find that core features are only learned well when they are less noisy than spurious features, which is not necessarily true in practice. We provide both theories and experiments to support this finding and to illustrate the importance of feature noise. Moreover, we propose an algorithm called Freeze then Train (FTT), that first freezes certain salient features and then trains the rest of the features using ERM. We theoretically show that FTT preserves features that are more beneficial to test time probing. Across two commonly used real-world benchmarks, FTT outperforms ERM, JTT and CVaR-DRO, with especially substantial improvement in accuracy (by 4.8\%) when the feature noise is large.},
	urldate = {2023-02-07},
	publisher = {arXiv},
	author = {Ye, Haotian and Zou, James and Zhang, Linjun},
	month = oct,
	year = {2022},
	note = {arXiv:2210.11075 [cs]},
	keywords = {Computer Science - Machine Learning},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\T6CII9VU\\Ye et al. - 2022 - Freeze then Train Towards Provable Representation.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\92UBMA5A\\2210.html:text/html},
}



@misc{kirichenko_last_2022,
	title = {Last {Layer} {Re}-{Training} is {Sufficient} for {Robustness} to {Spurious} {Correlations}},
	url = {http://arxiv.org/abs/2204.02937},
	doi = {10.48550/arXiv.2204.02937},
	abstract = {Neural network classifiers can largely rely on simple spurious features, such as backgrounds, to make predictions. However, even in these cases, we show that they still often learn core features associated with the desired attributes of the data, contrary to recent findings. Inspired by this insight, we demonstrate that simple last layer retraining can match or outperform state-of-the-art approaches on spurious correlation benchmarks, but with profoundly lower complexity and computational expenses. Moreover, we show that last layer retraining on large ImageNet-trained models can also significantly reduce reliance on background and texture information, improving robustness to covariate shift, after only minutes of training on a single GPU.},
	urldate = {2023-02-11},
	publisher = {arXiv},
	author = {Kirichenko, Polina and Izmailov, Pavel and Wilson, Andrew Gordon},
	month = apr,
	year = {2022},
	note = {arXiv:2204.02937 [cs, stat]},
	keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Statistics - Machine Learning},
	annote = {Comment: Code is available at https://github.com/PolinaKirichenko/deep\_feature\_reweighting},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\E9MUZD6D\\Kirichenko et al. - 2022 - Last Layer Re-Training is Sufficient for Robustnes.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\RK2NARTK\\2204.html:text/html},
}


@article{zhang2006varepsilon,
  title={From $\varepsilon$-entropy to KL-entropy: Analysis of minimum information complexity density estimation},
  author={Zhang, Tong},
  journal={Annals of statistics},
  volume={34},
  number={5},
  pages={2180--2210},
  year={2006}
}

@article{bissiri_general_2016,
	title = {A general framework for updating belief distributions},
	volume = {78},
	issn = {1467-9868},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12158},
	doi = {10.1111/rssb.12158},
	abstract = {We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.},
	language = {en},
	number = {5},
	urldate = {2023-01-24},
	journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
	author = {Bissiri, P. G. and Holmes, C. C. and Walker, S. G.},
	year = {2016},
	note = {\_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/rssb.12158},
	keywords = {Decision theory, General Bayesian updating, Generalized estimating equations, Gibbs posteriors, Information, Loss function, Maximum entropy, Provably approximately correct Bayes methods, Self-information loss function},
	pages = {1103--1130},
}


@inproceedings{krueger2021out,
  title={Out-of-distribution generalization via risk extrapolation (rex)},
  author={Krueger, David and Caballero, Ethan and Jacobsen, Joern-Henrik and Zhang, Amy and Binas, Jonathan and Zhang, Dinghuai and Le Priol, Remi and Courville, Aaron},
  booktitle={International Conference on Machine Learning},
  pages={5815--5826},
  year={2021},
  organization={PMLR}
}

@book{greene2003econometric,
  title={Econometric analysis},
  author={Greene, William H},
  year={2003},
  publisher={Pearson Education India}
}

@misc{abramowitz1988handbook,
  title={Handbook of mathematical functions with formulas, graphs, and mathematical tables},
  author={Abramowitz, Milton and Stegun, Irene A and Romer, Robert H},
  year={1988},
  publisher={American Association of Physics Teachers}
}

@article{gulrajani_search_2020,
	title = {In {Search} of {Lost} {Domain} {Generalization}},
	url = {http://arxiv.org/abs/2007.01434},
	abstract = {The goal of domain generalization algorithms is to predict well on distributions different from those seen during training. While a myriad of domain generalization algorithms exist, inconsistencies in experimental conditions -- datasets, architectures, and model selection criteria -- render fair and realistic comparisons difficult. In this paper, we are interested in understanding how useful domain generalization algorithms are in realistic settings. As a first step, we realize that model selection is non-trivial for domain generalization tasks. Contrary to prior work, we argue that domain generalization algorithms without a model selection strategy should be regarded as incomplete. Next, we implement DomainBed, a testbed for domain generalization including seven multi-domain datasets, nine baseline algorithms, and three model selection criteria. We conduct extensive experiments using DomainBed and find that, when carefully implemented, empirical risk minimization shows state-of-the-art performance across all datasets. Looking forward, we hope that the release of DomainBed, along with contributions from fellow researchers, will streamline reproducible and rigorous research in domain generalization.},
	urldate = {2020-11-02},
	journal = {arXiv:2007.01434 [cs, stat]},
	author = {Gulrajani, Ishaan and Lopez-Paz, David},
	month = jul,
	year = {2020},
	note = {arXiv: 2007.01434},
	keywords = {Causal/Invariant},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\SS45H8JF\\Gulrajani and Lopez-Paz - 2020 - In Search of Lost Domain Generalization.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\JVJFICX8\\2007.html:text/html},
}

@inproceedings{rosenfeld_risks_2021,
	title = {The {Risks} of {Invariant} {Risk} {Minimization}},
	url = {https://openreview.net/forum?id=BbNIbVPJ-42},
	booktitle = {International {Conference} on {Learning} {Representations}},
	author = {Rosenfeld, Elan and Ravikumar, Pradeep Kumar and Risteski, Andrej},
	year = {2021},
	file = {Rosenfeld et al. - 2021 - The Risks of Invariant Risk Minimization.pdf:C\:\\Users\\w\\Zotero\\storage\\JFREEYS4\\Rosenfeld et al. - 2021 - The Risks of Invariant Risk Minimization 2.pdf:application/pdf;Rosenfeld et al. - 2021 - The Risks of Invariant Risk Minimization.pdf:C\:\\Users\\w\\Zotero\\storage\\JFREEYS4\\Rosenfeld et al. - 2021 - The Risks of Invariant Risk Minimization.pdf:application/pdf},
}


@article{arjovsky2019invariant,
  title={Invariant risk minimization},
  author={Arjovsky, Martin and Bottou, L{\'e}on and Gulrajani, Ishaan and Lopez-Paz, David},
  journal={arXiv preprint arXiv:1907.02893},
  year={2019}
}

@book{wainwright2019high,
  title={High-Dimensional Statistics: A Non-Asymptotic Viewpoint},
  author={Wainwright, Martin J},
  volume={48},
  year={2019},
  publisher={Cambridge University Press}
}


@misc{guo_out--distribution_2021,
	title = {Out-of-distribution {Prediction} with {Invariant} {Risk} {Minimization}: {The} {Limitation} and {An} {Effective} {Fix}},
	shorttitle = {Out-of-distribution {Prediction} with {Invariant} {Risk} {Minimization}},
	url = {http://arxiv.org/abs/2101.07732},
	abstract = {This work considers the out-of-distribution (OOD) prediction problem where (1){\textasciitilde}the training data are from multiple domains and (2){\textasciitilde}the test domain is unseen in the training. DNNs fail in OOD prediction because they are prone to pick up spurious correlations. Recently, Invariant Risk Minimization (IRM) is proposed to address this issue. Its effectiveness has been demonstrated in the colored MNIST experiment. Nevertheless, we find that the performance of IRM can be dramatically degraded under {\textbackslash}emph\{strong \${\textbackslash}Lambda\$ spuriousness\} -- when the spurious correlation between the spurious features and the class label is strong due to the strong causal influence of their common cause, the domain label, on both of them (see Fig. 1). In this work, we try to answer the questions: why does IRM fail in the aforementioned setting? Why does IRM work for the original colored MNIST dataset? How can we fix this problem of IRM? Then, we propose a simple and effective approach to fix the problem of IRM. We combine IRM with conditional distribution matching to avoid a specific type of spurious correlation under strong \${\textbackslash}Lambda\$ spuriousness. Empirically, we design a series of semi synthetic datasets -- the colored MNIST plus, which exposes the problems of IRM and demonstrates the efficacy of the proposed method.},
	urldate = {2023-02-07},
	publisher = {arXiv},
	author = {Guo, Ruocheng and Zhang, Pengchuan and Liu, Hao and Kiciman, Emre},
	month = feb,
	year = {2021},
	note = {arXiv:2101.07732 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning},
	annote = {Comment: 22 pages},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\PWWN4ZEK\\Guo et al. - 2021 - Out-of-distribution Prediction with Invariant Risk.pdf:application/pdf},
}



@article{sagawa_investigation_2020,
	title = {An {Investigation} of {Why} {Overparameterization} {Exacerbates} {Spurious} {Correlations}},
	url = {http://arxiv.org/abs/2005.04345},
	abstract = {We study why overparameterization -- increasing model size well beyond the point of zero training error -- can hurt test error on minority groups despite improving average test error when there are spurious correlations in the data. Through simulations and experiments on two image datasets, we identify two key properties of the training data that drive this behavior: the proportions of majority versus minority groups, and the signal-to-noise ratio of the spurious correlations. We then analyze a linear setting and theoretically show how the inductive bias of models towards "memorizing" fewer examples can cause overparameterization to hurt. Our analysis leads to a counterintuitive approach of subsampling the majority group, which empirically achieves low minority error in the overparameterized regime, even though the standard approach of upweighting the minority fails. Overall, our results suggest a tension between using overparameterized models versus using all the training data for achieving low worst-group error.},
	urldate = {2020-10-29},
	journal = {arXiv:2005.04345 [cs, stat]},
	author = {Sagawa, Shiori and Raghunathan, Aditi and Koh, Pang Wei and Liang, Percy},
	month = aug,
	year = {2020},
	note = {arXiv: 2005.04345},
	keywords = {Causal/Invariant, Robustness},
	annote = {See Notes.},
	file = {arXiv Fulltext PDF:C\:\\Users\\w\\Zotero\\storage\\MHG8E492\\Sagawa et al. - 2020 - An Investigation of Why Overparameterization Exace.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\w\\Zotero\\storage\\R8XMURW7\\2005.html:text/html},
}
