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An Investigation of Why Overparameterization Exacerbates Spurious Correlations

Shiori Sagawa, Aditi Raghunathan, Pang Wei Koh, Percy Liang

arXiv:2005.04345v3cs.LGcs.CVstat.ML

TL;DR

The paper asks why overparameterized models can improve average error while harming minority-group error under spurious correlations. Using image experiments, simulations, and a linear analysis, it identifies group proportions, spurious-signal informativeness, and memorization bias as relevant factors. It finds that subsampling majority groups can achieve low worst-group error in the overparameterized regime, unlike standard minority upweighting.

  • Problem

    The paper investigates why overparameterization worsens minority-group test error despite improving average test error when spurious correlations exist.

  • Method

    The paper combines experiments on two image datasets, synthetic simulations, and a linear theoretical analysis of memorization and feature selection.

  • Results

    Subsampling majority groups is more effective than upweighting minority groups for improving worst-group error in overparameterized models.

  • Takeaways & Limitations

    Achieving low worst-group error may require choosing between overparameterized models and using all training data, rather than combining both under the studied methods.

  • Takeaways & Limitations

    The synthetic settings assume that models prefer spurious features because they are less noisy than core features, an assumption that need not always apply.

Abstract

from arXiv · show

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.

1. Introduction

The paper investigates why overparameterization can improve average test error while worsening worst-group error under spurious correlations. Experiments and analysis identify data imbalance, spurious-feature informativeness, and memorization bias as key factors, motivating majority-group subsampling.

  • Overparameterization can improve average test error yet worsen minority-group performance when spurious correlations are present.
  • On two image datasets, increasing model size hurts worst-group error under minority upweighting, while underparameterized models perform better on the worst group but worse on average.
  • Empirical risk minimization produces poor worst-group test error regardless of whether models are under- or overparameterized.
  • Synthetic simulations identify majority–minority group proportions and the relative informativeness of spurious versus core features as factors that modulate overparameterization’s effect.
  • The analysis motivates subsampling majority groups, which can improve worst-group error in overparameterized models more effectively than upweighting minority groups.

2. Setup

The paper studies group-structured classification where labels correlate with spurious attributes in training data. It evaluates worst-group error alongside standard empirical-risk objectives and minority-focused reweighting.

  • Each example belongs to a group defined by its label and spurious attribute, whose training-set correlation creates the spurious-correlation setting.
  • The experiments use CelebA for hair-color classification with gender as a spurious attribute and Waterbirds for bird classification with image background as a spurious attribute.
  • Worst-group error measures the error within each group and reports the highest group error.
  • ERM minimizes average training loss, whereas group DRO and reweighting modify training to focus more directly on worst-group loss.
  • Reweighting gives groups equal objective weight by upweighting minority groups and downweighting majority groups, requiring group identities during training.

3. Overparameterization hurts worst-group error

Across CelebA and Waterbirds, overparameterization improves average test error but worsens worst-group error under reweighted training. The paper links this pattern to minority memorization and spurious-feature reliance.

  • Directly increasing model size worsens worst-group error even while improving average error.
  • CelebA’s width-1 model has 25.6% worst-group test error, whereas overparameterized zero-training-error models exceed 60%.
  • Waterbirds’ underparameterized 90-feature model achieves 26.6% worst-group test error, compared with 42.4% at best for overparameterized zero-training-error models.
  • Stronger regularization improves worst-group error but hurts average error in overparameterized models.
  • In the overparameterized regime, the smallest groups have the highest test error despite zero training error, consistent with memorizing minority points through non-generalizing patterns.
  • The paper studies data-distribution properties through simulations and analyzes a simplified linear setting to connect fewer-point memorization bias with spurious-feature selection.

4. Simulation studies

Simulations show that overparameterization worsens worst-group error when majority imbalance and spurious-feature informativeness are sufficiently high, while improving average error. The resulting minority failures arise because overparameterized models rely on spurious features and memorize minority examples.

  • Synthetic results: 28.5% is the best synthetic worst-group test error, achieved by an underparameterized model, while highly overparameterized models plateau around 55%.Average test error is better for overparameterized models.
  • Spurious-feature reliance: Overparameterized models perform well on majority groups but poorly on minority groups despite zero training error, indicating reliance on spurious features and minority memorization.Underparameterized models instead show low training and test errors across groups, consistent with greater reliance on core features.
  • Spurious-feature reliance: Removing spurious features eliminates overparameterization’s detrimental effect, with the best worst-group test error then achieved by an overparameterized model.The spurious features are replaced with zero-centered noise.
  • Distributional properties: The synthetic dataset varies majority fraction pmaj and spurious-core information ratio rs:c to study when overparameterization harms worst-group error.Groups combine labels and spurious attributes; higher rs:c means stronger relative signal about the spurious attribute.
  • Distributional properties: When pmaj = 0.5, overparameterization no longer hurts worst-group error and instead outperforms all underparameterized models on that metric.Increasing pmaj makes overparameterization hurt worst-group error more.
  • Distributional properties: As rs:c increases, overparameterized models rely more on spurious features, and underparameterized models outperform them only for sufficiently large rs:c ≥1.Increasing rs:c does not significantly affect underparameterized worst-group error because core features are unchanged.

5. Theoretical analysis

The theoretical analysis replaces nonlinear random projections with a linear model containing explicit noise features for memorization. It shows that minimum-norm inductive bias favors using spurious features when they require memorizing fewer examples, producing high worst-group error under group imbalance and strong spurious signal.

  • Analytical setting: The paper introduces a linear explicit-memorization setting with noise features, making memorization precisely definable and analytically tractable.Memorization capacity is varied through the number of explicit noise features.
  • Analytical setting: In the overparameterized regime, reweighted, ERM, and gradient-descent logistic-regression solutions converge to the max-margin classifier when the data is linearly separable.The max-margin classifier therefore subsumes both reweighted and ERM estimators for the analysis.
  • Theorem: A sufficiently overparameterized model can have worst-group error greater than 1/2 under specified conditions on σ^2_core, nmaj, and nmin, whereas underparameterized models achieve reasonable worst-group error.The theorem applies with high probability in the overparameterized regime.
  • Minimum-norm bias: The max-margin solution is equivalent, up to scaling, to the minimum-norm separator among all separators that perfectly fit the training data.Because scaling does not affect 0-1 error, the analysis studies the minimum-norm separator directly.
  • Memorization mechanism: Using the spurious feature requires memorizing only minority points, while using the noisier core feature requires memorizing a constant fraction of all training points.The corresponding separator norms scale with nmin and n, respectively.
  • Memorization mechanism: When pmaj is sufficiently large, the spurious-feature separator has lower norm than any core-feature separator, so the minimum-norm bias favors spurious reliance and minority memorization.High rs:c and pmaj jointly make this comparison unfavorable to worst-group performance.

6. Subsampling

Subsampling reduces the majority fraction to balance groups, changing the memorization cost that drives overparameterized models toward spurious features. Across studied datasets, this makes overparameterization improve worst-group error and can outperform full-data reweighting.

  • Reducing the majority fraction lowers the memorization cost of using core features, weakening the inductive preference for spurious features.
  • Subsampling randomly removes points from other groups until all groups match the smallest group, then minimizes average loss on this balanced dataset.
  • Across CelebA, Waterbirds, and synthetic datasets, subsampling makes increasing overparameterization improve both average and worst-group test error.
  • Figure 7 reports that overparameterized models achieve low worst-group error when trained on group-balanced subsampled data.
  • Using 3.4% of CelebA data, subsampling reaches 11.1% average and 15.1% worst-group error, versus 11.3% and 25.6% with underparameterized reweighting.
  • In the overparameterized regime, subsampling decreases worst-group error as model size increases, unlike reweighting, which increases it.

7. Related work

Related work establishes that overparameterization can improve average test error and robustness in some settings, while prior worst-group methods largely target underparameterized models. The paper emphasizes that overparameterization’s effect depends on data properties, shift type, and training procedure.

  • The effect of overparameterization: Double descent describes decreasing test error when model size increases beyond zero training error, despite conventional expectations about overfitting.
  • The effect of overparameterization: Prior work has found that overparameterization can improve robustness to some distributional shifts.
  • The effect of overparameterization: The effect of overparameterization on robustness depends heavily on dataset properties such as pmaj and rs:c, the shift type, and the training procedure.
  • Worst-group error: Earlier approaches to worst-group error mainly used weighting, sampling, distributionally robust optimization, or fair algorithms in the underparameterized regime.

8. Discussion

The discussion attributes worsened worst-group error to an inductive bias toward solutions that memorize fewer examples and exploit spurious correlations. It also reports that subsampling can preserve low errors with overparameterized models while discarding much of the data, but notes simplifying assumptions and an unresolved full-data trade-off.

  • Overparameterization hurts worst-group error on real datasets with spurious correlations, across the paper’s implicit- and explicit-memorization settings.
  • The proposed mechanism is an inductive bias toward solutions that memorize fewer training points, which can favor exploiting spurious correlations.
  • The synthetic settings assume that the spurious feature is less noisy than the core feature, an assumption that need not always hold.
  • Subsampling lets overparameterized models achieve low average and worst-group error despite eliminating many training examples, whereas current full-data methods require underparameterized models for low worst-group error.

Reproducibiltity

The paper provides code, data, and experiments through a GitHub repository and the Codalab platform.

  • Code is available in the paper’s GitHub repository.
  • The paper states that all code, data, and experiments are available on Codalab.
  • The reproducibility links include a Codalab worksheet URL.

A.1. ERM models have poor worst-group error regardless of the degree of overparameterization

ERM models improve average test error as model size grows, but their worst-group error remains poor across model sizes and datasets.

  • ERM models obtain poor worst-group test error regardless of whether they are underparameterized or overparameterized.This pattern holds on the CelebA, Waterbirds, and synthetic datasets.
  • 4.6% for CelebA at width 96 and 4.2% for Waterbirds at 6,000 random features are the best average test errors reported for highly overparameterized ERM models.
  • Worst-group error is consistently worse than random (>50%) for CelebA and nearly random (44%) for Waterbirds across model sizes.These values are substantially worse than the 25.6% and 26.6% worst-group errors obtained by reweighted underparameterized models.
  • Increasing model size helps average test error but does not resolve ERM models’ poor worst-group error.
  • On the synthetic dataset, the best average test error is 9.0% at 9,000 random features, while worst-group error remains nearly random or worse (>48%).

A.2. Stronger L2 regularization improves worst-group error in overparameterized reweighted models

Stronger L2 regularization improves worst-group error for overparameterized reweighted models, but can hurt average error and does not eliminate dataset-dependent behavior.

  • Increasing L2 regularization improves worst-group error but hurts average error in overparameterized reweighted models.In the underparameterized regime, regularization has little effect on either metric.
  • Strongly-regularized reweighted models achieve much lower overparameterized worst-group error than weakly or unregularized counterparts across three datasets.
  • Overparameterization can still hurt worst-group error under strong regularization on CelebA and synthetic data, although it appears helpful on Waterbirds over the studied model sizes.
  • ERM models remain poor on worst-group error across regularization strengths, while subsampled models remain relatively insensitive and achieve reasonable worst-group error at most settings.
  • For reweighted models, the best worst-group error requires strong regularization that prevents zero training error, whereas nearly zero regularization gives the best average error.
  • The regularization trend is not uniform: with sufficiently large λ, overparameterized models appear to outperform underparameterized models over the studied sizes, but this does not hold clearly on CelebA.

A.3. Overparameterization helps average test error on the synthetic data regardless of pmaj and rs:c

On synthetic data, overparameterization improves average test error across majority proportions and spurious-core ratios, while the analysis links worst-group degradation to minimum-norm models’ reduced memorization.

  • Overparameterization improves average test error across all tested majority fractions pmaj and spurious-core ratios rs:c.
  • Subsampling creates group-balanced data and enables overparameterized models to achieve low worst-group error while improving both average and worst-group test error across three datasets.Subsampled overparameterized models are comparable to or better than the best full-data underparameterized reweighted models.
  • Under specified conditions, the minimum-norm separator memorizes only a small fraction of majority points and therefore incurs high worst-group error with high probability.
  • When the core feature is noisier than the spurious feature, using the spurious feature requires memorizing fewer majority examples but increases worst-group error.

B.2.6. PROOF OF PROPOSITION 1

Proposition 1 establishes the existence of a separator using the spurious feature under the theorem’s variance conditions.

  • Under the conditions in Theorem 1, there exists, with high probability for sufficiently large N, a separator in Wuse-spu that uses the spurious feature.

B.2.7. PROOF OF PROPOSITION 2

The analysis characterizes reweighted logistic regression in underparameterized and overparameterized regimes. In the underparameterized regime, the population minimizer uses the core feature, whereas overparameterization can leave reweighting with high worst-group error.

  • Proof strategy: The proof computes the population minimizer and bounds the asymptotic variance of the reweighted estimator.The stated steps use the population first-order condition and the sandwich-form asymptotic variance.
  • Underparameterized regime: The population minimizer has the best possible worst-group error by using the core feature rather than the spurious feature.
  • Underparameterized regime: In the underparameterized regime, reweighted logistic regression achieves worst-group error below 1/4 under the theorem’s parameter settings.
  • Overparameterized regime: In contrast, when N ≫ n, the reweighted estimator has high worst-group error even for very large n.
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