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Out of Distribution Generalization in Machine Learning

Martin Arjovsky

arXiv:2103.02667v1stat.MLcs.LG

TL;DR

The thesis examines how machine learning can generalize when test data differs from training data, reviewing assumptions and approaches for OOD problems. It highlights failures of empirical risk minimization under changing environments and argues that context-stable features are central to prediction in perceptual settings.

  • Problem

    The thesis addresses how machine learning can generalize beyond training distributions, in contrast to human flexibility across new contexts and the narrow scope of current AI systems.

    Source: Introduction · p. 12

  • Method

    It reviews and clarifies OOD assumptions and methods, with a focus on sensory prediction problems where invariant features may need to be learned at the latent level.

    Source: Introduction · p. 13, §3.1 · p. 30

  • Results

    ERM can yield catastrophic OOD generalization when training environments have large positive β(e) but test distributions have negative β(e).

    Source: §2.3 · p. 21

  • Takeaways & Limitations

    In perceptual AI problems, invariance and causality must operate at the latent-variable level because fixed subsets of pixels need not have stable label correlations across images and datasets.

    Source: §3.1 · p. 30

    The theory considers potentially arbitrary or adversarial spurious correlations, while structured variation in those correlations remains an area for improvement.

    Source: §3.6.2 · p. 57

Abstract

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Machine learning has achieved tremendous success in a variety of domains in recent years. However, a lot of these success stories have been in places where the training and the testing distributions are extremely similar to each other. In everyday situations when models are tested in slightly different data than they were trained on, ML algorithms can fail spectacularly. This research attempts to formally define this problem, what sets of assumptions are reasonable to make in our data and what kind of guarantees we hope to obtain from them. Then, we focus on a certain class of out of distribution problems, their assumptions, and introduce simple algorithms that follow from these assumptions that are able to provide more reliable generalization. A central topic in the thesis is the strong link between discovering the causal structure of the data, finding features that are reliable (when using them to predict) regardless of their context, and out of distribution generalization.

Introduction

The thesis frames out-of-distribution generalization as performance on test situations generated differently from training data, and surveys assumptions and approaches for addressing it. It focuses on when those assumptions are useful, especially for sensory problems where reliable prediction may depend on context-invariant features.

Introduction

  • OOD generalization concerns performance on test data whose generating process differs from training, a gap that matters for adapting to new environments and contexts.The thesis notes that practical relevance depends on the available data and the intended test setting; when training and test share a generating process, OOD may not be the primary problem.Source: Introduction · pp. 11–12
  • The thesis reviews OOD assumptions and methods, clarifies their similarities and differences, and examines where they fall short rather than proposing one universally suitable method.It seeks assumptions that are approximately satisfied in real problems yet yield meaningful generalization improvements when paired with suitable algorithms.Source: Introduction · p. 13, §1.1 · p. 13

1.1 Outline

  • The thesis focuses on predictive tasks where OOD difficulty involves distinguishing spurious correlations from relationships that represent the phenomenon of interest.It adds theoretical insights on nonlinear predictors and the number of training distributions, while later chapters describe applied domains and research directions for agents.Source: §1.1 · pp. 13–14

1.2 Definitions and Terminology

  • The exposition mainly uses supervised learning for concrete examples, while noting that many ideas also apply to some self-supervised and reinforcement-learning settings.It defines predictors, losses, environments, and risk to formalize expected loss across distributions.Source: §1.2 · pp. 14–15

2.1 A Framework for Out of Distribution Generalization

  • The framework quantifies OOD performance by minimizing worst-case risk over possible test distributions, assumed to belong to a family related to the training distributions.The text emphasizes that training and test environment families must be related; otherwise, generalization is futile.Source: §2.1 · pp. 17–18
  • Worst-case risk can be dominated by the hardest or most stochastic environments, obscuring differences between predictors on easier environments.The text identifies understanding algorithms that yield Pareto-optimal OOD predictors as a largely unexplored avenue.Source: §2.1 · p. 18

2.2 A Few Motivating Examples

The examples show how predictive correlations can vary across environments, making methods that rely on them fail out of distribution. ERM and robust optimization provide useful baselines, but neither alone ensures generalization beyond the training environments.

2.2 A Few Motivating Examples

  • In the structural-equation example, X1 is the only linear predictor with finite worst-case OOD risk and is also the causal explanation of Y.Any nonzero linear weight on X2 yields unbounded risk as β(e) grows.Source: §2.2 · p. 19
  • Although X2 predicts Y better in both training environments, its changing correlation can be discarded to obtain the example’s optimal OOD predictor.The training environments provide a hint through the correlation’s variation, even though test environments can make X2 harmful.Source: §2.2 · p. 19
  • Cow-image classification illustrates how background context can become a spurious cue, while animal shape is the phenomenon of interest across environments.With infinite data from one environment, ERM may prefer cow shape; the discussed failure concerns finite data, especially for overparameterized models.Source: §2.2 · p. 20
  • Adversarial generalization varies the test distribution through an attack on training samples, so assumptions about possible attacks are necessary.Training on the original distribution can generalize poorly to small, human-imperceptible perturbations.Source: §2.2 · p. 21

2.3 Empirical Risk Minimization

  • ERM can rely on X2 when training correlations are positive, causing catastrophic OOD generalization when test environments have negative β(e).ERM is optimal for average error when training and test environments are sampled from the same meta-distribution and no further structure is assumed.Source: §2.3 · pp. 21–22

2.4 Robust Optimization

  • Under KKT differentiability and qualification conditions, robust-learning minimizers are first-order stationary points of a weighted average of training-environment errors.The environment-specific weights are nonnegative; the proposition connects robust learning to weighted ERM.Source: §2.4 · p. 22
  • Robust optimization minimizes maximum training-environment error, yet in example 2.1 it can retain spurious X2 weight and fail on test environments.The example motivates extrapolation: discard correlations that vary across training environments rather than interpolate among them.Source: §2.4 · pp. 22–23

2.5 Wasserstein Robustness and Adversarial Examples

The chapter compares distributional assumptions for out-of-distribution generalization: Wasserstein closeness can support robustness to small adversarial perturbations, while f-divergence closeness does not improve on ERM. It then motivates invariant prediction, where features with stable label conditionals across environments can support generalization, and gives conditions linking that invariance to test-error guarantees.

2.5 Wasserstein Robustness and Adversarial Examples

  • For norm-bounded adversarial attacks, Wasserstein closeness places the test distribution in a Wasserstein ball where prior work shows low ROOD; this assumption fails for other examples, including cows on beaches.The chapter concludes that different OOD tasks require different assumptions and algorithms.Source: §2.5 · pp. 23–24

2.6 KL Robustness

  • With an f-divergence ball around the training distribution, ROOD predictor rankings match training-risk rankings, so no algorithm can improve over ERM under this assumption.The chapter attributes this negative result to the ball allowing broad changes in example probabilities without imposing structure.Source: §2.6 · p. 24

2.7 Domain Adaptation

  • Domain adaptation can fail when useful feature marginals vary across environments: enforcing equal feature distributions may erase predictive information, whereas stable label conditionals can identify useful features.In the refined example, X1 varies in distribution but retains a stable conditional relationship with Y; X2’s relationship changes.Source: §2.7 · pp. 25–26

Causality, Invariance, and Generalization

  • When environments represent interventions, features with stable conditionals can support reliable prediction across contexts; the chapter says causality, invariance, and OOD generalization are often equivalent when causality is well-defined.The chapter also presents invariance as useful even when traditional causal structures, such as graphs between observed variables, are not mathematically defined.Source: Causality, Invariance, and… · p. 28

3.1 Invariance and Generalization

  • Across environments, the regression on X1 has fixed coefficients, while regressions involving X2 vary; the chapter proposes learning features with stable label correlations rather than relying on fixed pixel subsets.It notes that fixed subsets of pixels are unsuitable for many perceptual problems because relevant objects can appear in different locations.Source: §3.1 · p. 30
  • Invariant prediction across Eall plus low training error is sufficient for a test-error bound, but invariance across Eall is unverifiable without assumptions and must be connected to training environments.The chapter separates finding invariance and low error across training environments from establishing when that implies invariance across Eall.Source: §3.1 · pp. 30–31

3.2 Invariant Risk Minimization

Invariant Risk Minimization (IRM) seeks representations that predict well while eliciting predictors invariant across training environments. Its practical form uses a gradient penalty to measure invariance, with linear-invariance guarantees depending on the classifier family and loss.

3.2 Invariant Risk Minimization

  • IRM learns representations that are predictive and elicit invariant predictors across training environments.The formulation poses these goals as a constrained optimization problem.Source: §3.2 · p. 31
  • IRM’s constraints create a bi-level optimization problem, so the authors instantiate a practical version, IRMv1.The ideal objective is described as challenging because each constraint calls an inner optimization routine.Source: §3.2 · p. 31, §3.2.1 · p. 32

Phrasing the constraints as a penalty

  • The Ddist discrepancy is discontinuous and remains numerically problematic under severe regularization, whereas Dlin is smooth and reaches a minimum at the invariant representation.Dlin is non-convex for general representations, and the resulting objective remains non-convex and sensitive to initialization.Source: Choosing a penalty D for linear… · p. 33, Figure 3.1 · p. 33, Scalar fixed classifiers ˜w are… · p. 35
  • A scalar fixed classifier is sufficient to monitor invariance because reparameterization can make environment-optimal classifiers equal to a chosen fixed vector.The linear-predictor characterization permits restricting the representation search to rank-1 matrices with fixed scalar classifier 1.0.Source: Fixing the linear classifier w · p. 34, Scalar fixed classifiers ˜w are… · pp. 34–35
  • For convex differentiable environment risks, a linear predictor is simultaneously optimal exactly when v⊤∇Re(v) = 0 in every environment.The corresponding representation matrices have nullspaces orthogonal to v and containing all environment gradients; squared-loss conditions form ellipsoids.Source: Scalar fixed classifiers ˜w are… · p. 34, Figure 3.2 · p. 35
  • High-rank representation decompositions matter for out-of-distribution generalization even when IRM searches only over rank-1 representations.The linear characterization also shows that the penalty landscape can have multiple connected components, including the trivial solution.Source: Scalar fixed classifiers ˜w are… · p. 35

Extending to general losses and multivariate outputs

  • IRMv1 measures invariance with the squared gradient norm at the fixed scalar classifier w = 1.0, balancing it against predictive risk with λ.The penalty measures whether that fixed classifier is optimal at each environment and extends to convex losses such as cross-entropy and multivariate outputs.Source: §3.2 · p. 32, Extending to general losses and… · p. 35

3.2.2 Implementation details

  • The implementation estimates the squared gradient norm with two random mini-batches from each environment, yielding an unbiased estimate.The batches have size b.Source: §3.2.2 · p. 36

3.3 Generalizing with Invariance

The section connects invariant prediction to out-of-distribution generalization, using causal structure to motivate assumptions under which training-environment invariance transfers to unseen environments. It develops linear and nonlinear guarantees and a setting where two training environments suffice.

3.3 Generalizing with Invariance

  • The section frames low error and training-environment invariance as necessary ingredients, alongside an underlying invariance and sufficiently diverse training environments that respect it.Different ways of formalizing environment coverage can yield drastically different sample complexities.Source: §3.3 · p. 37

3.3.1 Causal graphs: a starting point for invariance and generalization

  • The causal framework is a starting point, but expressing invariance statistically extends the account to settings beyond causal graphs and to latent variables in nonlinear perceptual inputs.The section also notes that its causal-graph equivalence depends on the intervention set; environment-specific baselines for ROOD are left for future work.Source: §3.3 · p. 37, §3.3.1 · p. 39
  • Finding Y’s causal parents, finding an invariant predictor across Eall(C), and achieving optimal out-of-distribution risk across Eall(C) are equivalent under the stated causal setup.The equivalence is only unique to the causal predictor when all interventions that do not affect Y are allowed; restricting interventions permits additional invariant predictors.Source: §3.3.1 · p. 39

3.3.2 Generalization theory for IRM

  • A rank-r representation invariant across training environments is invariant across all environments when the environments satisfy linear general position and the theorem’s assumptions hold.The section notes that higher-rank representations need fewer training environments, but the general theorem’s environment requirement scales with representation parameters.Source: §3.3.2 · p. 41, §3.3.3 · p. 43

3.3.3 On the nonlinear case

  • For nonlinear predictors, invariance transfers to test environments when the function class contains an invariant predictor and enough environments satisfy general position, but the condition is not yet characterized for real data.The theorem requires at least q/p training environments for a class with q parameters and p-dimensional features; the authors describe the result as illustrative pending a better characterization of nonlinear general position.Source: §3.3.3 · p. 42

3.3.4 Generalizing with fewer environments

  • With two training environments, invariant prediction in the specified latent cause-and-effect setting recovers the cause and yields the optimal invariant predictor across all environments.Outside a measure-zero set of environment parameters, any representation producing an invariant predictor discards noncausal information; a minimum-error invariant representation recovers the cause up to an invertible function.Source: §3.3.4 · pp. 43–44

3.4 Related Work

The related-work discussion contrasts causal discovery and invariant prediction approaches by their assumptions, supervision, and computational trade-offs. It presents multiple environments as a middle ground and relates invariance to laws that remain predictive across experimental conditions.

3.4 Related Work

  • Rubin’s strong-ignorability setting lets ERM solve the problem asymptotically when training data has full support and the input conditional is invariant, though finite, skewed data remains important.The cited methods reweight examples for that setting.Source: §3.4 · pp. 44–45
  • Multiple-environment ICP avoids requiring known interventions or the stringent assumptions of additive-noise models, but its hypothesis tests, subset search, and observed-variable graph assumption limit robustness and representation learning.The discussion also notes that nonlinear ICP relaxes linear regression but still tests residual distributions for invariance.Source: §3.4 · pp. 45–46
  • The ID-based method offers guarantees and can handle mild model misspecification and latent or absent causal graphs, but requires labeled matching IDs and more supervision than multiple environments.In the d-variable causal-graph setup, it needs d + 1 datapoints with matching IDs, whereas IRM needs a bit more than d environments.Source: §3.4 · p. 47
  • Invariant correlations can support prediction across conditions even when a structural equation model does not clearly describe the relationship, as illustrated by gas and gravitation laws.The passage gives PV = nRT and Newton’s universal gravitation as examples.Source: §3.4 · p. 48

3.5 Experiments

The experiments test whether IRM can recover invariant predictors across varied synthetic environments and on Colored MNIST, where color is spuriously correlated with labels. Across both settings, IRM generalizes better than ERM, while its invariance on Colored MNIST remains approximate.

3.5.1 Synthetic data

  • IRM estimates causal and non-causal weights most accurately across the synthetic setups, usually outperforming ERM by orders of magnitude and ICP by a large margin.ICP’s conservative tendency to reject covariates yields large causal-weight errors and small non-causal-weight errors.Source: §3.5.1 · p. 49, Figure 3.4 · p. 50

3.5.2 Colored MNIST

  • In Colored MNIST, labels are more strongly correlated with color than digit content, so training-error minimizers tend to exploit color and fail at test time when correlation direction reverses.Color id is sampled by flipping the final label with probability 0.2 and 0.1 in the two training environments, versus 0.9 at test.Source: §3.5.2 · p. 51
  • On Colored MNIST, IRM sacrifices training performance but relies less on color and generalizes better than ERM; a grayscale-only oracle outperforms IRM only slightly.The table reports accuracy across algorithms, and ERM falls below chance in the test environment by relying mainly on color.Source: §3.5.2 · p. 51, Table 3.1 · p. 51
  • IRM’s Colored MNIST predictor is closer to invariance than ERM, but does not achieve perfect invariance, particularly in the tails of the logit distribution.The authors suspect finite-sample difficulty at the tails, where few samples make small differences between training environments hard to estimate.Source: §3.5.2 · p. 52, Figure 3.5 · p. 52

3.6 Looking Forward: Invariance and Generalization

This section examines when invariance can help with spurious correlations and where its assumptions or methods remain limited. Its discussion spans realizable problems, disjoint environments, finite-sample and capacity constraints, and nonrealizable settings.

3.6.1 Realizable problems

  • In realizable problems, ERM can recover the invariant conditional expectation with shared supports, infinite data, and sufficient model capacity; departures from these conditions can explain spurious-feature learning.The section highlights that invariance may help address such biases when one or more of these conditions fails.Source: §3.6.1 · p. 53

Disjoint environments

  • With disjoint feature supports, matching environment-specific predictors is not directly meaningful; linear conditional expectations permit extrapolation across supports, but this structure can fail, as in Colored MNIST.Domain adaptation to align supports is another option, but excessive weight on that term can create problems discussed in Appendix C.Source: Disjoint environments · pp. 53–54

Finite sample and overparameterized models

  • In overparameterized finite-sample settings, SGD may favor low-capacity spurious correlations, while invariance can help detect and discard them when they require less data to estimate than the phenomenon of interest.After nuisance variables are discarded, the phenomenon of interest can be estimated more reasonably, improving OOD generalization; the required sample-complexity reduction remains open.Source: Finite sample and overparameterized… · pp. 54–56

Finite capacity

  • With infinite data but finite capacity, invariance may discard useful predictors when the model cannot represent the phenomenon of interest; IRM permits a trade-off between invariance and predictive power.For a quadratic target with a linear model, the only linearly invariant solution may be to discard everything.Source: Finite capacity · p. 56

3.6.2 Nonrealizable problems

  • In nonrealizable problems, invariant prediction still faces disjoint-support and insufficient-capacity issues, while the number of training environments and optimization remain challenges.The section also flags the linear conditional-expectation assumption as sometimes unrealistic and the IRMv1 gradient penalty as potentially suboptimal.Source: §3.6.2 · pp. 56–57

Domain-Specific Case Studies

The case studies describe how OOD problems arise in applied domains and what structures may be available to address them. Robotics illustrates these issues through sim-to-real transfer and learning robot policies from human demonstrations.

Domain-Specific Case Studies

  • The domain-specific case studies focus on the structures available in applied fields and the forms their OOD problems take, rather than reviewing potential solutions.The chapter describes its domain examples as settings where solutions remain nontrivial research tasks.Source: Domain-Specific Case Studies · p. 59, §4.1 · p. 60

4.1 Robotics

  • Robotics faces severe sample-complexity constraints: physical robots may have 4 to 6 orders of magnitude fewer samples than simulated agents, and simulation, sensors, and human demonstrations differ from deployment.Sim-to-real transfer and learning from human demonstrations are examples of OOD problems in this setting.Source: §4.1 · pp. 59–60

4.1.1 Sim-to-Real

  • Domain randomization varies simulation parameters across training runs, expanding the training support that can inform a policy, but simulated and real-world supports may still be disjoint.The authors identify stable policies under small distribution changes and compositional representations as possible routes for generalizing beyond support.Source: §4.1.1 · pp. 60–61

4.1.2 Imitation Learning from Humans

  • Human-demonstration imitation learning maps videos to robot policies, but sparse examples per environment risk overfitting, and invariant mappings may require nonlinear representations and losses.The discussion suggests searching across groups of humans, objects, and scenes, while noting that scaling the overparameterized realizable approach may require substantial work.Source: §4.1.2 · pp. 61–62

4.2 Computer Vision

The examples show that distribution shifts can undermine vision and language generalization, while fairness criteria raise questions about which behaviors are acceptable. The section connects these problems to invariance and causal structure.

4.2 Computer Vision

  • The section surveys how datasets both enable quantitative evaluation and can act as imperfect proxies for real-world vision tasks.The “Name That Dataset” experiment illustrates that dataset identity can be recovered from images.Source: §4.2 · p. 63
  • Models trained on one object-recognition dataset generalize poorly to another, suggesting they learn dataset-specific cues rather than what an object is.Computer-vision datasets can also be identified by humans and classifiers, and disjoint supports leave predictors no a priori reason to generalize.Source: §4.2 · p. 63

4.3 Natural Language Understanding

  • On compositional language tasks, generic attention models fail systematic generalization, while tree-structured Neural Module Networks generalize almost perfectly.Empirical-risk-minimization methods largely fail to learn the tree structure; the authors identify invariant prediction as a possible route to discovering it.Source: §4.3 · pp. 66–67

4.4.1 Oblivious and Causal Criteria for Fairness

  • Oblivious fairness measures cannot distinguish settings with the same joint distribution that differ in whether a feature is causally acceptable to use.Causal criteria can distinguish such cases, but unresolved-discrimination tests require causal knowledge and interventions that may be inaccessible.Source: §4.4.1 · pp. 68–69

4.4.2 Invariance, Fairness, and Out of distribution Generalization

  • The section frames fairness as low error across protected groups, treating spurious correlations that cause group-specific OOD errors as fairness violations.It proposes evaluating worst-case OOD error over environments formed from protected-group distributions.Source: §4.4.2 · p. 70

Looking Forward

The section proposes viewing learning across policies as an OOD generalization problem and outlines model-free and model-based approaches. It identifies distributional and predictive limitations, while suggesting that more generalizable Q-functions could support policy improvement.

5.1 Reinforcement Learning and Out of Distribution Generalization

  • Estimating a Q-function for a new policy from data generated by past policies is an OOD regression problem over changed future-state distributions.Without assumptions linking those distributions, importance sampling gives the best estimates described here.Source: §5.1 · pp. 72–73
  • The model-free approach predicts future rewards, whereas the model-based approach predicts future states and uses a known reward function to calculate returns.The state-prediction approach requires samples from the future-state distribution, not only its expected state.Source: §5.1 · p. 73
  • Predicting only the expected future state is insufficient to recover expected returns; the model-based approach needs the full future-state distribution.Adapting the chapter’s ideas to conditional-distribution prediction is left for future work.Source: §5.1 · p. 73
  • Invariant Q-estimates across past policies or groups of states are proposed as research directions, though state distributions across policies may be disjoint.The section suggests better-generalizing Q-functions could also support policy iteration or SARSA.Source: §5.1 · pp. 73–74

5.2 Exploration, Uncertainty, and Out of Distribution Generalization

The section connects efficient exploration to uncertainty and OOD generalization. It contrasts count-based bonuses with density-model pseudocounts and proposes reducing the generalization gap in Q-function estimation to support smaller exploration bonuses.

5.2 Exploration, Uncertainty, and Out of Distribution Generalization

  • Count-based exploration can reduce a chain task from an expected 2^n episodes under random actions to n episodes by rewarding untried state-action pairs.The section describes tabular methods as assigning intrinsic rewards based on visit counts and observed rewards.Source: §5.2 · p. 74
  • Tabular exploration bonuses upper-bound the optimal Q-function while decaying with visits; tighter concentration-based bonuses achieve regret optimal up to log factors.The exploration bonus must balance fast decay against maintaining the upper bound.Source: §5.2 · p. 75
  • Density-model pseudocounts can overgeneralize and miss novelty in important states, or undergeneralize and fail to transfer similarity.These issues depend on the density model’s capacity and inductive bias.Source: §5.2 · pp. 75–76
  • The proposed alternative treats learning an optimistic Q-function as an OOD generalization problem and seeks regularizers or bonuses that bound its error.Reducing the gap to the distributionally robust objective can permit smaller bonuses and better approximate the optimal policy.Source: §5.2 · p. 77

Appendix A

Appendix A relates invariant prediction to causal structure and optimal out-of-distribution generalization, and gives conditions under which invariant features identify a predictor that generalizes across environments. In linear structural models, the same relationship also permits direct recovery of causal parents.

Additional Theorems and Assumptions

  • In the structural-equation setting, invariance, recovery of causal parents with non-causal information discarded, and optimal out-of-distribution generalization are equivalent.The result assumes each causal parent can affect the outcome for some input; the predictor is invariant across environments arising from interventions outside the outcome equation.Source: Additional Theorems and Assumptions · p. 89
  • An invariant predictor can be optimal across all environments and estimated from any environment where the induced feature distribution has full support.The result states that its final predictor minimizes risk over predictors on top of the features for every environment in Eall.Source: Additional Theorems and Assumptions · pp. 89–90
  • For linear structural equation models with linear interventions, the equivalence also holds with linear featurizers and predictors, and causal parents are the variables with nonzero predictor entries.This gives a direct parent-recovery rule from an invariant predictor.Source: Additional Theorems and Assumptions · p. 90
  • If an invariant predictor uses a featurizer in the specified smooth family, general-position training environments with |Etrain| > q/p identify Φ* and yield an invariant predictor across Eall.The theorem assumes full-rank feature Jacobians and a unique Φ* and w*; its general-position condition also requires an overdetermined system with isolated solutions.Source: Additional Theorems and Assumptions · p. 90

B.1 Proof of Proposition 2.1

  • Under the stated KKT differentiability and qualification conditions, the proof of Proposition 2.1 obtains nonnegative environment multipliers with zero Lagrangian gradient.The passage gives the stationarity condition for the constrained optimization problem.Source: §B.1 · p. 91

B.2 Proof of Theorem 3.2

These proof fragments establish risk comparisons under bounded loss and distributional closeness, characterize predictors representable by a featurizer through subspace orthogonality, and use rank constraints in the proof of Theorem 3.8. The Theorem 3.9 proof also partitions feature preimages by their dependence on the binary variable.

B.2 Proof of Theorem 3.2

  • With bounded loss, the proof uses total-variation closeness of conditional distributions to compare training- and test-environment risks.It applies the closeness condition to a training environment and a test environment, both drawn from Eall.Source: §B.2 · p. 92

B.3 Proof of Theorem 3.3

  • For a linear featurizer Φ, predictors v = Φᵀw are exactly the vectors orthogonal to Ker(Φ), and minimization over w yields Φ∇Re(v) = 0 in each environment.Conversely, the proof constructs such a featurizer from the orthogonality conditions on the environment risk gradients.Source: §B.3 · p. 92

B.4 Proof of Theorem 3.8

  • The Theorem 3.8 proof rules out Φᵀw differing from the target by showing the resulting kernel dimension would exceed the bound implied by rank(Φ).The contradiction uses the assumed lower bound on the span of the training-environment vectors.Source: §B.4 · p. 93

B.5 Proof of Theorem 3.9

  • In the Theorem 3.9 proof, the preimages of a feature value are partitioned by whether the two binary-variable inputs map to that value, yielding a representation independent of that variable.The proof concludes that GΦ(h, z) = G(h).Source: §B.5 · pp. 93–94

B.6 Proof of Theorem A.1

The proof of Theorem A.1 derives the causal-parent conditional predictor from invariance, then establishes its optimality over all environments and the equivalence with optimal prediction on the learned features.

B.6 Proof of Theorem A.1

  • Invariance across all intervention environments forces w ◦ Φ to equal the conditional expectation of the outcome given its causal parents.The proof considers interventions fixing X to arbitrary values and uses the resulting equality for every x.Source: §B.6 · p. 95
  • The causal-parent conditional predictor minimizes worst-case risk over Eall because an environment can fix noncausal variables while leaving the causal prediction optimal.The proof argues by contradiction against a predictor with strictly lower worst-case risk, using an environment where the noncausal parents are constant and independent of the outcome.Source: §B.6 · p. 96
  • Once Φ discards noncausal variables, w ◦ Φ is optimal on top of Φ in every environment; optimality on top of Φ across Eall also implies invariance.The proof uses the conditional expectation given causal variables for the per-environment optimality step.Source: §B.6 · pp. 96–97

B.7 Proof of Theorem A.2

The proof of Theorem A.2 uses transversality and dimension counting to exclude low-rank intersections for generic training-environment parameters. It establishes transversality by showing the relevant map is surjective, using symmetric matrices in that argument.

B.7 Proof of Theorem A.2

  • For generic training-environment parameters, transversality and dimension counting rule out intersections with rank-k matrices when k < d − r under the stated environment-count condition.The proof uses the codimension of the rank-k matrix set to show the intersection would have negative dimension.Source: §B.7 · pp. 97–98
  • The proof establishes generic transversality by showing the parameter map is surjective, constructing symmetric matrices to realize the needed vectors.The surjectivity argument is applied to the map from covariance parameters to the matrix-valued constraint.Source: §B.7 · pp. 97–98

B.8 Proof of Theorem A.4

The proof establishes uniqueness under its overdetermined-system and full-rank assumptions. The section also shows that domain-adaptation objectives can reject invariant predictors, while the usefulness of meta-learning depends on the structure shared across environments.

B.8 Proof of Theorem A.4

  • Under the stated dimension, rank, and uniqueness assumptions, the only feature map yielding an invariant predictor is Φ*.The proof argues that F(Φ)=0 is overdetermined and has at most isolated solutions, then applies assumption A.3.Source: §B.8 · p. 98

Failure Cases of Domain Adaptation

  • Domain adaptation can enforce equal predicted-label distributions across source and target, limiting accuracy to at most 20% even when source training generalizes to the target.This example uses source and target label prevalences of 1/2 and 9/10, respectively.Source: Failure Cases of Domain Adaptation · p. 99
  • Almost surely, conditional domain adaptation disregards every non-zero representation eliciting invariant prediction, including the identity representation.The argument shows that the set of environment-distribution pairs satisfying its matching condition has measure zero for each representation.Source: Failure Cases of Domain Adaptation · pp. 99–100
  • The usual domain-adaptation bound also involves λ⋆, which depends on the representation and creates a trade-off often ignored when motivating domain alignment.λ⋆ is the summed error of the optimal classifier in the hypothesis class on the two domains.Source: Failure Cases of Domain Adaptation · p. 100

Meta-Learning

  • Meta-learning is not uniformly suitable for OOD tasks: ERM and related approaches can make sense in some tasks and fail catastrophically in others.The authors conjecture that reusing i.i.d. machinery is futile for an important class of AI-related problems.Source: Meta-Learning · p. 101

D.1 A Bayesian Meta-Learner

The section frames OOD prediction in a Bayesian meta-learning setup, estimating what is shared across environments alongside what is specific to a test environment. It argues that the meta-prior must encode identifiable shared structure, because likelihood-based i.i.d. machinery alone may be inadequate when only a few environments are observed.

D.1 A Bayesian Meta-Learner

  • The analysis assumes training and test environments are i.i.d. draws from a meta-distribution, a strong limitation when environments depend on prior interactions or decisions.Independently collected hospital datasets are given as a setting where this assumption may be closer to true.Source: §D.1 · p. 102
  • The Bayesian target averages test-environment predictions using the posterior over the test environment given training and test data, while training environments inform shared structure and test data inform environment-specific details.The derivation is presented as a computationally intractable reference for understanding deviations from the Bayes-optimal solution.Source: §D.1 · pp. 101–102
  • The posterior over the meta-distribution captures properties shared across environments, such as causal structure, and is the term the chapter links to zero-shot OOD generalization.The account interprets earlier invariant-prediction assumptions as meta-priors supporting low-error predictors shared across each meta-distribution's environments.Source: §D.1 · p. 103
  • The meta-posterior objective combines a constant, the meta-prior, and data likelihood, but the variational bound is generally strict unless the meta-prior is concentrated on a single distribution.With enough samples per environment, the bound approaches equality, making MAP estimation approximately equivalent to optimizing likelihood and the meta-prior.Source: §D.1 · p. 104
  • Unlike ordinary architectural priors, useful OOD meta-priors must make stable properties identifiable from training environments, not merely constrain environments to a shared support.For the cause-direction example, identifying the invariant conditional requires multiple environments and a meta-prior that specifies which conditional remains invariant.Source: §D.1 · p. 105
  • MAML's common-initialization assumption does not identify invariant prediction in example 2.1, where the initialization can place the learned conditional far from the test environment's conditional.The passage notes that such a meta-prior may still be reasonable for some problems.Source: §D.1 · p. 106
  • Shared causal structure can make optimal OOD predictors identifiable through adaptation speed, though efficiently exploiting this or other shared structures remains open.The cited work establishes this result with two environments.Source: §D.1 · p. 106
  • Combining learned invariances across training environments with fine-tuning on limited test-domain data is identified as a promising future direction.The passage presents this as a direction for future work, not an established result.Source: §D.1 · p. 106

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