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A robust assessment for invariant representations

Wenlu Tang, Zicheng Liu

TL;DR

The paper addresses evaluating invariant representations under covariate-shifted environments in invariant risk minimization (IRM). It introduces the Covariate-shift Representation Invariance Criterion (CRIC), defined via a likelihood-ratio weighted invariance condition that compares environment-specific expectations and summarizes this with a normalized statistic Q_Phi. The authors provide an empirical estimator for CRIC, establish consistency results, and validate the approach through synthetic SEM experiments and real financial data, showing IRM-based methods achieve lower CRIC than ERM, with REx-V often outperforming IRMv1 in complex settings. They also discuss integrating CRIC with prediction accuracy as a multi-objective framework, highlighting CRIC as a robust, complementary criterion for evaluating invariant representations and guiding domain-generalization research.

Abstract

The performance of machine learning models can be impacted by changes in data over time. A promising approach to address this challenge is invariant learning, with a particular focus on a method known as invariant risk minimization (IRM). This technique aims to identify a stable data representation that remains effective with out-of-distribution (OOD) data. While numerous studies have developed IRM-based methods adaptive to data augmentation scenarios, there has been limited attention on directly assessing how well these representations preserve their invariant performance under varying conditions. In our paper, we propose a novel method to evaluate invariant performance, specifically tailored for IRM-based methods. We establish a bridge between the conditional expectation of an invariant predictor across different environments through the likelihood ratio. Our proposed criterion offers a robust basis for evaluating invariant performance. We validate our approach with theoretical support and demonstrate its effectiveness through extensive numerical studies.These experiments illustrate how our method can assess the invariant performance of various representation techniques.

A robust assessment for invariant representations

TL;DR

The paper addresses evaluating invariant representations under covariate-shifted environments in invariant risk minimization (IRM). It introduces the Covariate-shift Representation Invariance Criterion (CRIC), defined via a likelihood-ratio weighted invariance condition that compares environment-specific expectations and summarizes this with a normalized statistic Q_Phi. The authors provide an empirical estimator for CRIC, establish consistency results, and validate the approach through synthetic SEM experiments and real financial data, showing IRM-based methods achieve lower CRIC than ERM, with REx-V often outperforming IRMv1 in complex settings. They also discuss integrating CRIC with prediction accuracy as a multi-objective framework, highlighting CRIC as a robust, complementary criterion for evaluating invariant representations and guiding domain-generalization research.

Abstract

The performance of machine learning models can be impacted by changes in data over time. A promising approach to address this challenge is invariant learning, with a particular focus on a method known as invariant risk minimization (IRM). This technique aims to identify a stable data representation that remains effective with out-of-distribution (OOD) data. While numerous studies have developed IRM-based methods adaptive to data augmentation scenarios, there has been limited attention on directly assessing how well these representations preserve their invariant performance under varying conditions. In our paper, we propose a novel method to evaluate invariant performance, specifically tailored for IRM-based methods. We establish a bridge between the conditional expectation of an invariant predictor across different environments through the likelihood ratio. Our proposed criterion offers a robust basis for evaluating invariant performance. We validate our approach with theoretical support and demonstrate its effectiveness through extensive numerical studies.These experiments illustrate how our method can assess the invariant performance of various representation techniques.
Paper Structure (14 sections, 2 theorems, 28 equations, 2 figures, 2 tables)

This paper contains 14 sections, 2 theorems, 28 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Invariant Risk Minimization
  4. Covariate shift and likelihood ratio
  5. Methodology
  6. From invariance property to CRIC
  7. Empirical estimation of CRIC
  8. Estimation of the likelihood ratio
  9. Simulation studies
  10. Synthetic structural equation model data
  11. Real data
  12. Integrating CRIC with Prediction Accuracy
  13. Concluding remarks
  14. Proof of Theorem \ref{['sec3:consistency']}

Key Result

Theorem 2.2

If the optimal classifier in any environment of ${\mathcal{E}}$ can be written as a conditional expectation, then a data representation $\Phi$ is invariant if and only if, for all $e, {e^{\prime}} \in {\mathcal{E}}$ and all $h$ in the intersection of the supports of $\Phi(X^e)$ and $\Phi(X^{{e^{\pri

Figures (2)

  • Figure 1: The training and testing sample size is $800$ and $500$ respectively. The $y$-axis presents methods IRMv1 and REx-V on testing and training data. The red line is the baseline $\log(\hat{Q}_{\Phi})=0$ of ERM.
  • Figure 2: The training and testing sample size is $1200$ and $600$ respectively. The $y$-axis presents methods IRMv1 and REx-V on testing and training data. The red line is the baseline $\log(\hat{Q}_{\Phi})=0$ of ERM.

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.2: Invariance property on expectation
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • proof