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Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization

Kartik Ahuja, Ethan Caballero, Dinghuai Zhang, Jean-Christophe Gagnon-Audet, Yoshua Bengio, Ioannis Mitliagkas, Irina Rish

arXiv:2106.06607v2cs.LGstat.ML

TL;DR

The paper asks why invariance-based methods fail in classification when invariant features fully determine labels. It proves stronger shift restrictions are needed, then combines invariance with information bottleneck constraints to address the failures while retaining prior guarantees.

  • Problem

    Invariance-based methods fail in common classification settings where invariant features capture all information about the label.

  • Method

    The paper studies linear classification under distribution-shift restrictions and combines information bottleneck constraints with ERM or IRM.

  • Results

    The analysis shows that linear classification needs stronger shift restrictions than linear regression, while information bottleneck plus invariance addresses key failures and retains existing guarantees.

  • Takeaways & Limitations

    Information bottleneck constraints help make invariance-based OOD generalization effective when invariant features are fully informative and preserve success when they are partially informative.

  • Takeaways & Limitations

    The theoretical analysis focuses on linear models; nonlinear extensions require strong structural assumptions for nonlinear ICA.

Abstract

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The invariance principle from causality is at the heart of notable approaches such as invariant risk minimization (IRM) that seek to address out-of-distribution (OOD) generalization failures. Despite the promising theory, invariance principle-based approaches fail in common classification tasks, where invariant (causal) features capture all the information about the label. Are these failures due to the methods failing to capture the invariance? Or is the invariance principle itself insufficient? To answer these questions, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. In contrast to the linear regression tasks, we show that for linear classification tasks we need much stronger restrictions on the distribution shifts, or otherwise OOD generalization is impossible. Furthermore, even with appropriate restrictions on distribution shifts in place, we show that the invariance principle alone is insufficient. We prove that a form of the information bottleneck constraint along with invariance helps address key failures when invariant features capture all the information about the label and also retains the existing success when they do not. We propose an approach that incorporates both of these principles and demonstrate its effectiveness in several experiments.

1 Introduction

The paper examines why invariance-based methods can fail on OOD classification despite their causal motivation and proposes combining invariance with information bottleneck constraints.

  • Motivation: Deep models can exploit spurious features, such as background color instead of animal shape, causing OOD failures.The paper highlights camel-versus-cow classification and COVID-19 detection as concerning real-world examples.
  • Motivation: Invariance principle-based approaches distinguish predictors relying on label causes from predictors that do not.IRM and related methods use this causal principle to target invariant predictors.
  • Contributions: Invariant approaches fail when invariant features capture all information about the label, motivating a reassessment of linear-regression assumptions for classification.The paper contrasts fully informative invariant features with settings where spurious features may carry label information.
  • Contributions: The paper argues that information bottleneck constraints combined with invariance address these failures while retaining existing successes.The proposed principle combination is evaluated in experiments.
  • Contributions: Table 1 summarizes new and existing OOD generalization results for ERM, IRM, and their information-bottleneck variants.IB-ERM and IB-IRM denote information bottleneck versions of ERM and IRM.

2 OOD generalization and invariance: background & failures

This section establishes the OOD-generalization setting, reviews invariance-based regression guarantees, and explains why fully informative invariant features expose failures of IRM and related methods.

  • Problem setting: OOD generalization seeks predictors trained on Etr that perform well across unseen environments Eall.The environments may have different joint distributions over inputs and labels.
  • Problem setting: Without restrictions on Eall, binary-classification OOD generalization cannot be guaranteed because an unseen environment can reverse a method's predicted labels.The resulting error can reach one under 0-1 loss.
  • Invariance principle: The invariance objective searches for representations whose predictor simultaneously minimizes risk in every training environment.Dropping the invariance constraints yields standard ERM.
  • Regression background: For linear regression, invariant predictors can achieve OOD generalization under Assumption 1, rank and environment-count conditions, and linear general position.Theorem 1 states that each solution to the invariant objective solves the OOD target.
  • Failure analysis: IRM has nevertheless been shown to fail in several cases, including linear structural equation models.The paper distinguishes fully informative from partially informative invariant features to analyze these failures.
  • Failure analysis: In CS-CMNIST and many human-labeled datasets, invariant features can contain all label information while spurious features do not affect the label.This setting motivates focusing on classification tasks with fully informative invariant features.

3 OOD generalization theory for linear classification tasks

For linear classification, separated invariant-feature supports make guaranteed OOD generalization impossible; with support overlap, ERM and IRM can still rely on spurious features, motivating stronger constraints.

  • ERM and IRM: In the two-dimensional example, infinitely many invariant predictors attain zero training error, but only predictors with zero spurious weight generalize OOD.Increasing the number of training environments does not resolve this classification ambiguity.
  • Fully informative features: In the fully informative setting, only the classifier matching the labeling function solves OOD generalization when spurious features can change arbitrarily.That classifier achieves error q, the noise level, in all environments.
  • Impossibility: Theorem 2 states that every deterministic algorithm fails when training invariant features are bounded and strictly separable.For every algorithm, some allowed environment has error exceeding q.
  • Impossibility: The impossibility arises because separated training supports admit multiple equally plausible labeling hyperplanes that algorithms cannot distinguish.The hyperplanes disagree in regions that may occur in unseen environments.
  • Support assumptions: Unlike linear regression, linear classification requires invariant-feature support overlap; without it, OOD generalization is impossible.The paper identifies this as the key contrast between the two settings.
  • ERM and IRM: With joint support overlap, both ERM and IRM solve OOD generalization, and some successful classifiers may rely on spurious features.If spurious-feature support overlap fails, both methods fail and no spurious-reliant classifier succeeds.

4 Information bottleneck principle meets invariance principle

The information bottleneck selects low-entropy representations while invariance filters for predictors stable across environments. Together, IB-IRM and IB-ERM address complementary failures depending on whether invariant features are fully or partially informative, while IB can also accelerate learning.

  • Information bottleneck principle: The information bottleneck compresses X while preserving label-relevant information, using representation entropy or mutual information to measure compression.For deterministic representations, entropy of Φ(X) can measure compression.
  • Information bottleneck principle: In the two-dimensional example, the invariant-only representation has the least entropy among representations that achieve zero error.The identity representation has entropy H(p_e) + log(2), whereas invariant-only and spurious-only representations have entropy log(2).
  • Why invariance and information bottleneck: Invariance rejects representations using the spurious feature, while the information bottleneck further selects invariant-only representations when invariance alone leaves spurious solutions.The combined constraints are needed when invariant and spurious features have different predictive and entropy properties.
  • Theorem 4: Under fully informative invariant features, every IB-IRM and IB-ERM solution solves OOD generalization, whereas ERM and IRM fail.This result does not require support overlap for spurious features.
  • Theorem 4: Under partially informative invariant features, IB-IRM and IRM solve OOD generalization, whereas IB-ERM and ERM fail.The partially informative result assumes more than 2d training environments in linear general position.
  • Learning speed: For p = 0.9, the IB-ERM spurious-to-invariant weight ratio is bounded by 0.001, while ERM's ratio is bounded below by 0.09.The analysis concerns continuous-time gradient descent in the two-dimensional example.
  • Learning speed: The joint convergence analysis for the information bottleneck and invariance penalties remains an open challenge, although experiments compare gradient descent on all objectives.The authors point to Figure 3 for these comparisons.

5 Experiments

Experiments compare IB-ERM and IB-IRM with ERM, IRM, and an oracle across linear unit tests and image datasets. Results align with the theory: IRM-based methods help in partially informative settings, while IB-ERM is strongest in fully informative settings.

  • Methods, datasets & metrics: Experiments compare IB-ERM and IB-IRM with ERM, IRM, and an oracle using linear unit tests, colored MNIST variants, Terra Incognita, and COCO.The oracle is trained after spurious correlations are removed by permutation.
  • Results by informativeness: On partially informative datasets, IRM and IB-IRM outperform ERM and IB-ERM, although all methods perform poorly on Example 3/3S with three environments.The reported partially informative cases include Example 1/1S, Example 3/3S, and AC-CMNIST.
  • Results by informativeness: On fully informative datasets, IB-ERM consistently performs well close to the oracle, and IB-IRM also performs well in some cases.The fully informative cases include Example 2/2S, CS-CMNIST, Terra Incognita, and COCO under the stated labeling assumption.
  • Results by informativeness: The experiments support a crucial role for the information bottleneck in fully informative settings and for the IRMv1 penalty in partially informative settings.This conclusion is based on the reported comparisons and ablations.
  • Experimental variation: As training environments increase from three to six, Example 1/1S exhibits larger MSE variance because newly sampled environments have higher label-noise levels.The data-generation procedure follows Aubin et al. (2021).

6 Extensions, limitations, and future work

The analysis focuses on linear models and binary classification, with multi-class extensions available under a natural SEM extension. Extending the approach to nonlinear models requires strong assumptions for nonlinear ICA, while penalty approximations remain an open area for improvement.

  • Extension to non-linear models and multi-class classification: For nonlinear bijective feature maps, the proposed approach would require first recovering invariant and spurious features through nonlinear ICA.Current state-of-the-art methods for this recovery require strong structural assumptions on dependencies among latent components.
  • Extension to non-linear models and multi-class classification: The paper’s main theoretical discussion concerns linear models, binary classification, and regression, while its results extend to multi-class classification through a natural SEM extension.The authors state that their main results continue to hold in multi-class settings under this extension.
  • On the choice for IB penalty and IRMv1 penalty: The information bottleneck uses an entropy approximation that serves as an effective proxy, but tighter high-dimensional approximations remain future work.The approximation is reported to work in moderately high dimensions, including a 256-dimensional bottleneck layer in CS-CMNIST.
  • On the choice for IB penalty and IRMv1 penalty: Because the proposed loss uses the IRMv1 penalty, existing criticisms of IRMv1—including nonlinear-model failures and penalty tuning—also apply to this objective.The authors identify these limitations as relevant to their proposed loss and note that alternative invariance approximations remain unexplored.

7 Conclusion

The paper argues that linear classification requires stronger distribution-shift assumptions than linear regression for OOD generalization. It finds that invariant features need support overlap and that combining invariance with an information bottleneck addresses failures while retaining provable guarantees.

  • 7 Conclusion: Linear classification tasks require much stronger assumptions on distribution shifts than linear regression tasks for OOD generalization.Without sufficient restrictions, OOD generalization can be impossible.
  • 7 Conclusion: Support overlap of invariant features is necessary; without it, OOD generalization is impossible.The paper characterizes ERM and IRM performance under different support-overlap assumptions for invariant and spurious features.
  • 7 Conclusion: An information bottleneck constraint combined with invariance overcomes key ERM and IRM failures while retaining existing provable guarantees.This conclusion concerns settings where invariant features capture all information about the label.

Funding disclosure

The paper acknowledges financial and institutional support from Samsung Electronics and multiple fellowship, chair, grant, and collaboration programs.

  • Funding disclosure: Samsung Electronics Co., Ltd. funded the research.The acknowledgments also name support for individual authors from IVADO, CIFAR, IBM, NSERC, Samsung, and Microsoft Research.
  • Funding disclosure: The authors acknowledge support from IVADO, CIFAR, IBM, NSERC, Samsung, Canada CIFAR AI Chair programs, and Microsoft Research.These sources supported individual authors through fellowships, chairs, discovery grants, and collaborative research grants.
  • Funding disclosure: The acknowledgments distinguish institutional research funding from author-specific fellowship and grant support.The passage attributes different forms of support to different authors and programs.

Checklist

The checklist reports affirmative answers for theoretical assumptions and proofs, reproducibility materials, training details, error bars, compute disclosure, asset licensing, and limitations; several human-subject and privacy items are marked not applicable.

  • Claims, limitations, and impacts: The paper states that its claims reflect the paper’s information, describes limitations, and discusses potential negative societal impacts.The checklist directs readers to Sections 4.1, 6, and Appendix Section A.1.
  • Theoretical results: The paper states that it provides assumptions for theoretical results and complete proofs in the supplementary appendix.The checklist points to Sections 2–4 for assumptions and the supplementary appendix for proofs.
  • Reproducibility: The authors report that code, data, reproduction instructions, training details, error bars, and compute information are included or documented.The checklist refers readers to the public repository and Appendix Section A.2.
  • Assets and data: Existing code assets are cited and their MIT licenses are identified, while new experiment code is included in the supplementary material.The checklist also states that consent and personally identifiable information discussions were not applicable.
  • Human subjects: Human-subject checklist items concerning participant instructions, risks, institutional review, wages, and compensation are marked not applicable.These entries appear under the checklist’s crowdsourcing and human-subject research section.

A Appendix

The appendix organizes supplementary material into societal impact, experimental details, structural-equation-model background, notation, technical remarks, and proofs.

  • The appendix covers societal impact, experiment details, structural equation models, proof notation, technical remarks, and an impossibility-result proof.

A.1 Societal impact

The work is motivated by deploying trustworthy machine-learning systems beyond their training-data regime, especially in safety-critical applications.

  • Safety-critical deployments require models whose decisions remain trustworthy beyond the training-data regime.The passage gives self-driving cars and healthcare as examples.
  • Current models can exploit spurious correlations or shortcuts, making their decisions unreliable.

A.2 Experiments details

The experiments are documented with reproducible code available through the authors’ repository and an added DomainBed implementation.

  • The authors provide code to reproduce the experiments at the IB-IRM GitHub repository.
  • The authors also added the experimental code to DomainBed.

A.2.1 Datasets

The appendix evaluates linear unit tests and image benchmarks spanning regression, classification, invariant-feature settings, and spurious-feature shifts across environments and domains.

  • Linear unit tests: The linear unit tests comprise Examples 1/1S, 2/2S, and 3/3S, with results reported in Table 3.
  • Linear unit tests: Example 1/1S follows the linear regression SEM, varying environment-specific noise variances and optionally rotating latents with a random unitary scrambling matrix.The first three variances are 0.1, 1.5, and 2.0; later variances are sampled from Uniform(10^-2, 10).
  • Linear unit tests: Example 2/2S follows a zero-noise linear classification SEM modeling cow-versus-camel labels with animal and background features.The invariant features are causal and carry full information about the label; environments vary background and animal parameters, with optional latent rotation.
  • Linear unit tests: Example 3/3S is a linear classification version of the spiral task, with anti-causal invariant features and spurious features carrying additional label information.Example 3 uses identity scrambling, whereas Example 3S uses a random unitary rotation.
  • Image benchmarks: AC-CMNIST is a binary digit-classification task with two training environments and one test environment, where color-label flipping probabilities are 0.2, 0.1, and 0.9.The colored digit’s spurious feature contains label information not present in the uncolored image.
  • Image benchmarks: CS-CMNIST, Terra Incognita, and COCO evaluate ten-class image classification under color or domain shifts, including randomized test backgrounds and camera locations.CS-CMNIST randomizes all test colors, Terra Incognita uses four camera locations, and COCO randomizes object backgrounds at test time.

A.2.2 Training and evaluation procedure

The experiments use linear models for unit tests, MLP or convolutional architectures for CMNIST, and a pretrained ResNet-50 featurizer for Terra Incognita, with method-specific hyperparameter tuning and repeated accuracy evaluation.

  • Example 1/1S, 2/2S, 3/3S: All three linear unit-test examples use linear models, with square-error loss for regression and binary cross-entropy for classification.IB-IRM and IB-ERM introduce a γ hyperparameter for the Var(Φ) term, selected through random hyperparameter search.
  • AC-CMNIST: AC-CMNIST uses a two-layer 256-unit ReLU MLP with an output layer of size two, trained with Adam for 500 steps.The learning rate is 1e−3, batch size 256, and grid search tunes λ for IRMv1 and γ for IB.
  • Evaluation: Evaluation reports accuracy and standard deviations averaged over thirty trials, while hyperparameters are selected using train-domain validation across methods.The reported procedure covers the linear tests, CMNIST experiments, and Terra Incognita setup described in the supplied passages.
  • CS-CMNIST: CS-CMNIST uses a three-layer ConvNet with feature-map dimensions 64, 128, and 256, trained with SGD for 2000 steps.Each convolutional layer uses ReLU and batch normalization; grid search tunes the method penalties.
  • Terra Incognita: Terra Incognita uses a pretrained ResNet-50 featurizer producing 2048-dimensional features, followed by a one-layer MLP for classification.Model selection uses 20 random hyperparameter configurations and a train-domain validation set; results use L100 as the test environment.

A.2.3 Supplementary experiments

Supplementary experiments examine validation procedures on AC-CMNIST and CS-CMNIST and ablate the information-bottleneck and invariance penalties on linear unit tests.

  • AC-CMNIST: 70.39±0.47 percent is the AC-CMNIST Oracle model’s test accuracy when background colors have no label correlation.Under train-domain validation tuning, none of the evaluated methods work in this AC-CMNIST setting.
  • Validation procedures: Table 6 compares AC-CMNIST methods under train-domain validation while varying the validation fraction of the 50000-point dataset.The table’s columns indicate the percentage of total data used for validation.
  • Validation procedures: Tables 7–9 compare CS-CMNIST methods under oracle test-domain and train-domain validation procedures, with columns indicating validation-data fractions.The supplied captions identify the validation procedures but do not provide the table entries.
  • CS-CMNIST: 99.03 ± 0.08 percent is the CS-CMNIST Oracle model’s test accuracy, and IB-IRM and IB-ERM perform well under both validation procedures.The supplementary experiments compare train-domain and test-domain validation tuning for CS-CMNIST.
  • Penalty ablation: Figure 4 varies the IB and IRM penalties on six-environment linear unit tests, placing ERM, IRM, IB-ERM, and IB-IRM at four square corners with performance values.The ablation is designed to isolate the roles of the two penalties.

A.2.4 Compute description

The paper shows that linear classification can make OOD generalization impossible even when invariant predictors are identifiable in training, and that ERM and IRM depend critically on support-overlap assumptions. Adding an information bottleneck to invariance or ERM recovers OOD generalization in settings where ERM and IRM fail.

  • Invariance and OOD optimality: Causal-parent predictors solve OOD generalization over environments formed by valid interventions.The predictor depends only on the causal parents of the label and remains OOD-optimal under the specified intervention class.
  • Impossibility: Every deterministic algorithm fails guaranteed OOD generalization for linear classification when invariant features are bounded and strictly separable.The failure means some unseen environment produces error above the noise level q, even under the stated structural assumptions.
  • Impossibility: Multiple equally good candidate hyperplanes can generate identical training data, while only the correct generating hyperplane is OOD-optimal.Consequently, an algorithm cannot distinguish among the equivalent generating models and succeeds only on the appropriate one.
  • ERM and IRM: ERM and IRM succeed under joint support overlap, but both fail when spurious-feature support overlap is violated.Under overlap, some successful solutions may still rely on spurious features; without it, no such spurious-reliant classifier remains OOD-optimal.
  • Information bottleneck: IB-IRM and IB-ERM solve OOD generalization in the stated bounded-noise setting, whereas ERM and IRM fail.The bottleneck selects lower-entropy predictors among highly predictive invariant predictors, distinguishing it from input-space sparsity constraints.
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