arxiv: 2604.26128 · v1 · submitted 2026-04-28 · 📊 stat.ML · cs.LG

Recognition: unknown

Robust Representation Learning through Explicit Environment Modeling

Yuli Slavutsky , David M. Blei

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:17 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords robust representation learningenvironment modelingrandom-intercept modelsdomain generalizationinvariant representationscausal invariance

0 comments

The pith

Representations from modeling and marginalizing environment variation achieve better average robustness on unseen environments than invariant methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines supervised learning from data gathered across multiple environments whose distributions differ and where the environment can directly influence the target label. Standard causal approaches seek invariant features that ignore environment signals, but this discards useful information when those signals affect the outcome. The authors instead propose to model environment-specific variation explicitly inside a predictor and then average over it at test time. This yields features whose predictions remain accurate on average for entirely new environments. A reader would care because many practical datasets exhibit direct environment effects on labels, so average robustness across shifts is often more useful than strict invariance.

Core claim

By using generalized random-intercept models that explicitly capture environment-specific effects and marginalizing those effects out, the resulting representations support superior average prediction on previously unseen environments compared with representations learned by causal invariant-representation methods, even when the environment directly affects the target.

What carries the argument

Generalized random-intercept models, a class of predictors that treat environment-specific effects as random terms whose distribution can be integrated out to obtain marginal predictions.

Load-bearing premise

Generalized random-intercept models can capture enough of the relevant environment-induced variation that averaging over it produces better average robustness than discarding the variation.

What would settle it

A dataset with known direct environment effects on the target where a generalized random-intercept model trained on multiple environments shows no improvement or worse average accuracy on held-out environments than a standard invariant-representation baseline.

Figures

Figures reproduced from arXiv: 2604.26128 by David M. Blei, Yuli Slavutsky.

**Figure 1.** Figure 1: Probabilistic graphical model. Observed variables are colored in gray. The environment e influences spurious features s, and may also affect causal features c. In (a) the label y depends only on the causal features c, so y ⊥ e | c. In (b) an additional direct effect from the environment e to the label y is added. spond to countries, regions, or time periods (Gelman et al., 2007). Similar structure appears… view at source ↗

**Figure 2.** Figure 2: Schematic comparison between the true data-generating process and approximation within the random-intercept family. Left: the true joint distribution p(e, x, y) induces environment-specific conditionals p(y | x, e), shown here as labeled data across environments. Top: the conditionals differ only by intercept shift. Bottom: the environment effect is not of random-intercept form. Right: in both cases we ap… view at source ↗

**Figure 3.** Figure 3: Environment-average risk in the tradeoff simulation for the well-specified and misspecified settings. In the well-specified case, NGMM nearly overlaps with the Bayes-optimal predictor across all values of α. In the misspecified case environment effect is no longer of random-intercept form. NGMM still achieves the lowest risk among the learned methods and moves closer to the optimum as α increases and the … view at source ↗

**Figure 4.** Figure 4: Results for the Colored MNIST experiment, reported as the gap between each method’s accuracy and the Bayes-optimal accuracy in per test environment. Across all settings, including the in-distribution no-color environment, and environments where the sign of the color effect is reversed, NGMM achieves the smallest discrepancy from the Bayes predictor. 5.3 OGB-MolPCBA The OGB-MolPCBA dataset provides data fo… view at source ↗

**Figure 5.** Figure 5: Results for the Camelyon-17 experiment on held-out hospitals. NGMM achieves the highest accuracy and the lowest negative log-likelihood among all methods, indicating the best predictive performance in unseen environments. 6 Comparison of robust and invariant representations Having discussed robustness through marginalization of environment effects, we now turn to a comparison between this approach and inva… view at source ↗

**Figure 6.** Figure 6: Summary of the comparison between robustness and invariance. view at source ↗

read the original abstract

We consider learning from labeled data collected across multiple environments, where the data distribution may vary across these environments. This problem is commonly approached from a causal perspective, seeking invariant representations that retain causal factors while discarding spurious ones. However, this framework assumes that the environment has no direct effect on the target. In contrast, we consider settings in which this assumption fails, but still aim to learn representations that support robust prediction on average across previously unseen environments. To this end, we study representations learned by explicitly modeling variation across environments and then marginalizing that variation out. We analyze the resulting representations and characterize when they are preferable to those learned by causal invariant-representation methods. We propose a concrete method based on generalized random-intercept models, a class of predictors in which such marginalization is possible, and study their generalization properties. Empirically, we show that these models outperform invariant-learning methods across a range of challenging settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper relaxes the no-direct-effect assumption in invariant representation learning by modeling environment variation explicitly with generalized random-intercept models and marginalizing it out.

read the letter

The core contribution is a practical alternative for multi-environment robustness when the environment directly influences the target. Instead of forcing invariance that discards useful signal, the approach fits generalized random-intercept models to capture that variation and then marginalizes over it for predictions on new environments. They characterize the conditions where this beats standard invariant methods and provide generalization analysis plus empirical comparisons in challenging settings. That extension of random-intercept ideas to representation learning is the actual novelty here, and it addresses a concrete limitation in the causal robustness literature without obvious circularity. The proposal is concrete enough that someone could implement the marginalization step and test it. Experiments are claimed to show outperformance, which is the right kind of evidence to include. The main soft spot is that the abstract gives little detail on the exact marginalization procedure, the form of the generalization bounds, or how environments were sampled and excluded in the trials. If the full paper supplies those derivations and reports error bars with clear data splits, the claims hold up better; otherwise the empirical edge remains hard to assess. This work is aimed at people building predictors under realistic distribution shifts, such as in healthcare or sensor data where site or device effects matter. It is worth a serious referee because the problem is well-motivated, the method is grounded in existing statistical tools, and the comparison to invariant baselines is a useful contribution even if revisions are needed on the analysis details. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript considers multi-environment supervised learning where the environment can directly affect the target. It proposes to learn representations by fitting generalized random-intercept models that explicitly capture environment variation and then marginalizing that variation out. The authors analyze the resulting representations, characterize conditions under which they are preferable to those obtained from causal invariant-representation methods, derive generalization bounds, and report empirical comparisons in which the proposed models outperform invariant baselines across several challenging regimes.

Significance. If the theoretical characterization and generalization analysis hold, the work supplies a concrete, non-causal alternative for robust prediction precisely when the standard invariant-learning assumption (no direct environment effect on the target) is violated. The use of random-intercept models to enable explicit marginalization is a technically clean contribution that could be useful in domains with group-level effects. The empirical claim of consistent outperformance, if substantiated with proper controls, would strengthen the practical case for the approach.

major comments (2)

[Abstract] Abstract: the claim that the authors 'analyze the resulting representations and characterize when they are preferable' and 'study their generalization properties' is central to the contribution, yet the abstract supplies no indication of the specific conditions, theorems, or proof strategy. Without these details the soundness of the theoretical component cannot be assessed from the provided text.
[Empirical evaluation] Empirical section: the abstract states that the models 'outperform invariant-learning methods across a range of challenging settings,' but the description contains no reference to error bars, statistical tests, or data-exclusion rules. This information is load-bearing for evaluating whether the reported outperformance supports the practical recommendation.

minor comments (2)

[Introduction] The relationship between the proposed marginalization and existing random-intercept literature could be stated more explicitly to clarify the incremental contribution.
[Method] Notation for the generalized random-intercept predictor and the marginalization operation should be introduced with a clear running example to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We are pleased that the referee recognizes the potential of our approach as a non-causal alternative for robust prediction when the invariant-learning assumption is violated. Below, we provide point-by-point responses to the major comments and describe the revisions we intend to make.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the authors 'analyze the resulting representations and characterize when they are preferable' and 'study their generalization properties' is central to the contribution, yet the abstract supplies no indication of the specific conditions, theorems, or proof strategy. Without these details the soundness of the theoretical component cannot be assessed from the provided text.

Authors: We agree with the referee that the abstract could more clearly convey the theoretical contributions to allow readers to assess the work's soundness. In the revised version, we will update the abstract to specify the conditions (when the environment has a direct effect on the target), reference the characterization of preference over invariant methods, and mention the derivation of generalization bounds using analysis of the marginalization procedure in generalized random-intercept models. revision: yes
Referee: [Empirical evaluation] Empirical section: the abstract states that the models 'outperform invariant-learning methods across a range of challenging settings,' but the description contains no reference to error bars, statistical tests, or data-exclusion rules. This information is load-bearing for evaluating whether the reported outperformance supports the practical recommendation.

Authors: We thank the referee for highlighting this. The empirical evaluations in the manuscript do include error bars in the reported figures to indicate variability across multiple runs. However, we acknowledge that explicit references to statistical tests and data-exclusion rules are not detailed in the text. We will revise the experimental section to include a paragraph describing the statistical significance testing (e.g., paired t-tests), the number of independent runs, and any preprocessing or exclusion criteria applied to the datasets. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central proposal relies on generalized random-intercept models drawn from established statistical literature to explicitly model and marginalize environment variation. No equations or steps in the provided abstract or description reduce the robustness claim to a fitted parameter by construction, a self-definition, or a load-bearing self-citation chain. Generalization properties are analyzed separately, and empirical comparisons are presented as independent validation rather than tautological outputs. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the modeling power of generalized random-intercept models for environment effects and the validity of marginalization for average robustness; no free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption Generalized random-intercept models can capture and allow marginalization of environment variation
Invoked when the paper states that marginalization is possible within this predictor class.

pith-pipeline@v0.9.0 · 5450 in / 1165 out tokens · 52114 ms · 2026-05-07T14:17:30.299068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Invariant risk minimization games

Kartik Ahuja, Karthikeyan Shanmugam, Kush Varshney, and Amit Dhurandhar. Invariant risk minimization games. In International Conference on Machine Learning, pages 145--155. PMLR, 2020

2020
[2]

Invariant Risk Minimization

Martin Arjovsky, L \'e on Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019

work page internal anchor Pith review arXiv 1907
[3]

Mixed-effects modeling with crossed random effects for subjects and items

R Harald Baayen, Douglas J Davidson, and Douglas M Bates. Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59 0 (4): 0 390--412, 2008

2008
[4]

Robust solutions of optimization problems affected by uncertain probabilities

Aharon Ben-Tal, Dick Den Hertog, Anja De Waegenaere, Bertrand Melenberg, and Gijs Rennen. Robust solutions of optimization problems affected by uncertain probabilities. Management Science, 59 0 (2): 0 341--357, 2013

2013
[5]

Statistical analysis of longitudinal neuroimage data with linear mixed effects models

Jorge L Bernal-Rusiel, Douglas N Greve, Martin Reuter, Bruce Fischl, Mert R Sabuncu, Alzheimer's Disease Neuroimaging Initiative, et al. Statistical analysis of longitudinal neuroimage data with linear mixed effects models. NeuroImage, 66: 0 249--260, 2013

2013
[6]

Hierarchical Linear Models: Applications and Data Analysis Methods

Anthony S Bryk and Stephen W Raudenbush. Hierarchical Linear Models: Applications and Data Analysis Methods. Sage Publications, Inc, 1992

1992
[7]

Invariant rationalization

Shiyu Chang, Yang Zhang, Mo Yu, and Tommi Jaakkola. Invariant rationalization. In International Conference on Machine Learning, pages 1448--1458. PMLR, 2020

2020
[8]

Learning models with uniform performance via distributionally robust optimization

John C Duchi and Hongseok Namkoong. Learning models with uniform performance via distributionally robust optimization. The Annals of Statistics, 49 0 (3): 0 1378--1406, 2021

2021
[9]

Statistics of robust optimization: A generalized empirical likelihood approach

John C Duchi, Peter W Glynn, and Hongseok Namkoong. Statistics of robust optimization: A generalized empirical likelihood approach. Mathematics of Operations Research, 46 0 (3): 0 946--969, 2021

2021
[10]

Domain-adversarial training of neural networks

Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, Fran c ois Laviolette, Mario March, and Victor Lempitsky. Domain-adversarial training of neural networks. Journal of Machine Learning Research, 17 0 (59): 0 1--35, 2016

2016
[11]

stop-and-frisk

Andrew Gelman, Jeffrey Fagan, and Alex Kiss. An analysis of the new york city police department's “stop-and-frisk” policy in the context of claims of racial bias. Journal of the American Statistical Association, 102 0 (479): 0 813--823, 2007

2007
[12]

Simple data balancing achieves competitive worst-group-accuracy

Badr Youbi Idrissi, Martin Arjovsky, Mohammad Pezeshki, and David Lopez-Paz. Simple data balancing achieves competitive worst-group-accuracy. In Conference on Causal Learning and Reasoning, pages 336--351. PMLR, 2022

2022
[13]

Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem

Charles M Judd, Jacob Westfall, and David A Kenny. Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103 0 (1): 0 54, 2012

2012
[14]

Out-of-distribution generalization via risk extrapolation ( REx )

David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapolation ( REx ). In International Conference on Machine Learning, pages 5815--5826. PMLR, 2021

2021
[15]

Bayesian invariant risk minimization

Yong Lin, Hanze Dong, Hao Wang, and Tong Zhang. Bayesian invariant risk minimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16021--16030, 2022

2022
[16]

Correction for hidden confounders in the genetic analysis of gene expression

Jennifer Listgarten, Carl Kadie, Eric E Schadt, and David Heckerman. Correction for hidden confounders in the genetic analysis of gene expression. Proceedings of the National Academy of Sciences, 107 0 (38): 0 16465--16470, 2010

2010
[17]

Just train twice: Improving group robustness without training group information

Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training group information. In International Conference on Machine Learning, pages 6781--6792. PMLR, 2021

2021
[18]

Invariant causal representation learning for out-of-distribution generalization

Chaochao Lu, Yuhuai Wu, Jos \'e Miguel Hern \'a ndez-Lobato, and Bernhard Sch \"o lkopf. Invariant causal representation learning for out-of-distribution generalization. In International Conference on Learning Representations, 2021

2021
[19]

Estimating heterogeneous treatment effects within latent class multilevel models: A bayesian approach

Weicong Lyu, Jee-Seon Kim, and Youmi Suk. Estimating heterogeneous treatment effects within latent class multilevel models: A bayesian approach. Journal of Educational and Behavioral Statistics, 48 0 (1): 0 3--36, 2023

2023
[20]

Domain generalization using causal matching

Divyat Mahajan, Shruti Tople, and Amit Sharma. Domain generalization using causal matching. In International Conference on Machine Learning, pages 7313--7324. PMLR, 2021

2021
[21]

The effects of small classes on academic achievement: The results of the tennessee class size experiment

Barbara Nye, Larry V Hedges, and Spyros Konstantopoulos. The effects of small classes on academic achievement: The results of the tennessee class size experiment. American Educational Research Journal, 37 0 (1): 0 123--151, 2000

2000
[22]

Revisiting the incremental effects of context on word processing: Evidence from single-word event-related brain potentials

Brennan R Payne, Chia-Lin Lee, and Kara D Federmeier. Revisiting the incremental effects of context on word processing: Evidence from single-word event-related brain potentials. Psychophysiology, 52 0 (11): 0 1456--1469, 2015

2015
[23]

Causal inference by using invariant prediction: Identification and confidence intervals

Jonas Peters, Peter B \"u hlmann, and Nicolai Meinshausen. Causal inference by using invariant prediction: Identification and confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology, 78 0 (5): 0 947--1012, 2016

2016
[24]

Elements of Causal Inference: Foundations and Learning Algorithms

Jonas Peters, Dominik Janzing, and Bernhard Sch \"o lkopf. Elements of Causal Inference: Foundations and Learning Algorithms. The MIT Press, 2017

2017
[25]

Invariant causal prediction for sequential data

Niklas Pfister, Peter B \"u hlmann, and Jonas Peters. Invariant causal prediction for sequential data. Journal of the American Statistical Association, 114 0 (527): 0 1264--1276, 2019

2019
[26]

Focus on the common good: Group distributional robustness follows

Vihari Piratla, Praneeth Netrapalli, and Sunita Sarawagi. Focus on the common good: Group distributional robustness follows. In International Conference on Learning Representations, 2021

2021
[27]

Synthesizing results from the trial state assessment

Stephen W Raudenbush, Randall P Fotiu, and Yuk Fai Cheong. Synthesizing results from the trial state assessment. Journal of Educational and Behavioral Statistics, 24 0 (4): 0 413--438, 1999

1999
[28]

Invariant models for causal transfer learning

Mateo Rojas-Carulla, Bernhard Sch \"o lkopf, Richard Turner, and Jonas Peters. Invariant models for causal transfer learning. Journal of Machine Learning Research, 19 0 (36): 0 1--34, 2018

2018
[29]

a usler, Nicolai Meinshausen, Peter B \

Dominik Rothenh \"a usler, Nicolai Meinshausen, Peter B \"u hlmann, and Jonas Peters. Anchor regression: Heterogeneous data meet causality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 83 0 (2): 0 215--246, 2021

2021
[30]

Distributionally robust neural networks

Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust neural networks. In International Conference on Learning Representations, 2019

2019
[31]

Certifying

Aman Sinha, Hongseok Namkoong, Riccardo Volpi, and John Duchi. Certifying some distributional robustness with principled adversarial training. arXiv preprint arXiv:1710.10571, 2017

work page arXiv 2017
[32]

Neural Generalized Mixed-Effects Models

Yuli Slavutsky, Sebastian Salazar, and David Blei. Neural generalized mixed-effects models. arXiv preprint arXiv:2604.10976, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[33]

Mixed blocked/event-related designs separate transient and sustained activity in fMRI

Kristina M Visscher, Francis M Miezin, James E Kelly, Randy L Buckner, David I Donaldson, Mark P McAvoy, Vidya M Bhalodia, and Steven E Petersen. Mixed blocked/event-related designs separate transient and sustained activity in fMRI . NeuroImage, 19 0 (4): 0 1694--1708, 2003

2003
[34]

On calibration and out-of-domain generalization

Yoav Wald, Amir Feder, Daniel Greenfeld, and Uri Shalit. On calibration and out-of-domain generalization. Advances in Neural Information Processing Systems, 34: 0 2215--2227, 2021

2021
[35]

Distributionally robust post-hoc classifiers under prior shifts

Jiaheng Wei, Harikrishna Narasimhan, Ehsan Amid, Wen-Sheng Chu, Yang Liu, and Abhishek Kumar. Distributionally robust post-hoc classifiers under prior shifts. In International Conference on Learning Representations, 2023

2023
[36]

Mixed linear model approach adapted for genome-wide association studies

Zhiwu Zhang, Elhan Ersoz, Chao-Qiang Lai, Rory J Todhunter, Hemant K Tiwari, Michael A Gore, Peter J Bradbury, Jianming Yu, Donna K Arnett, Jose M Ordovas, et al. Mixed linear model approach adapted for genome-wide association studies. Nature Genetics, 42 0 (4): 0 355--360, 2010

2010
[37]

Genome-wide efficient mixed-model analysis for association studies

Xiang Zhou and Matthew Stephens. Genome-wide efficient mixed-model analysis for association studies. Nature Genetics, 44 0 (7): 0 821--824, 2012

2012