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Invariant Risk Minimization Is A Total Variation Model
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Shows that invariant risk minimization (IRM) and related variants (V-REx, ZIN, TIVA) can be written as total-variation-based models, and proposes TV-ℓ1 variants (IRM-TV-ℓ1, Minimax-TV-ℓ1) evaluated on simulated temporal data, house prices, CelebA, Landcover and Adult.
Problem
IRM's mathematical essence is unclear, and its gradient-norm penalty is a surrogate for a bi-level problem; the paper asks what IRM really optimizes and how to get more robust, blocky invariant risks for out-of-distribution generalization.
Approach
Reinterpret IRM (and V-REx, ZIN, TIVA) as a TV-ℓ2 model of the risk with respect to the classifier variable, and propose TV-ℓ1 variants whose coarea-formula property encourages piecewise-constant (blocky) risks that are robust to environment change.
Key results
- On the simulation study with environment partitions, IRM-TV-l1 reaches 88.03% mean / 86.40% worst accuracy at (0.999,0.7), p_v=0.9, versus 87.84% / 86.20% for IRM.
- Minimax-TV-l1 achieves the best mean and worst accuracy on CelebA without environment partitions (83.68% test, 81.45% worst) versus ZIN at 76.73% test.
- On House Prices without environment partitions, Minimax-TV-l1 reports the lowest test MSE (0.3395) and worst-case MSE (0.4983).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace ZIN with Minmax-TV-ℓ1 | simulation without environment partitions · Simulation (0.999, 0.7), p_v(t)=0.9 | mean accuracy: 87.70 -> 88.67 | Helped |
| replace IRM with IRM-TV-ℓ1 | simulation with environment partitions · Simulation (0.999, 0.7), p_v(t)=0.9 | mean accuracy: 87.84 -> 88.03 | Helped |
| replace EIIL with Minmax-TV-ℓ1 | house price prediction without environment partitions · House Prices test | MSE: 0.3939 -> 0.3395 | Helped |
| replace groupDRO with IRM-TV-ℓ1 | house price prediction with environment partitions · House Prices test | MSE: 0.7358 -> 0.4420 | Helped |
| replace ZIN with Minmax-TV-ℓ1 | CelebA without environment partitions · CelebA test | accuracy: 76.73 -> 83.68 | Helped |
| replace IRM with IRM-TV-ℓ1 | CelebA with environment partitions · CelebA test | accuracy: 82.54 -> 83.47 | Helped |
| replace ZIN with Minmax-TV-ℓ1 | Landcover · Landcover OOD Test | accuracy: 62.22 -> 63.77 | Helped |
| replace ZIN with Minmax-TV-ℓ1 | Adult Income without environment partitions · Adult Income test | accuracy: 82.26 -> 83.33 | Helped |
| replace IRM with IRM-TV-ℓ1 | Adult Income with environment partitions · Adult Income test | accuracy: 82.32 -> 83.31 | Helped |
Surprising findings
- Theory shows lambda must vary with the feature extractor Phi for the TV-l1 global objective to match the minimax OOD objective; a fixed lambda fails even in a toy fitting problem.
Limitations
- Experiments use shallow models on pre-extracted features (ResNet18 features, MLPs, 1D-CNN), not deep end-to-end TV-l1 models.
- Penalty parameter lambda is fixed in experiments; adaptive lambda is left to future work.
- Non-differentiable TV term requires custom subgradient computation rather than standard backpropagation.
Experimental setup
Datasets: Simulated temporal heterogeneity data House Prices CelebA Landcover Adult
Benchmarks: Simulation study (temporal spurious correlation) House Prices CelebA Landcover Adult
Metrics: accuracy worst-environment accuracy mean squared error
Baselines: ERM IRM groupDRO EIIL LfF HRM ZIN TIVA
Builds on: IRM REx ZIN TIVA total variation (ROF/TV-l1 denoising)
Closest papers by meaning
- The Risks of Invariant Risk Minimization (ICLR 2021 · similarity 0.92)
- Out-of-distribution Generalization for Total Variation based Invariant Risk Minimization (ICLR 2025 · similarity 0.88)
- Sufficient Invariant Learning for Distribution Shift (CVPR 2025 · similarity 0.85)
- On the Efficiency of ERM in Feature Learning (NeurIPS 2024 · similarity 0.85)
- Dual Risk Minimization: Towards Next-Level Robustness in Fine-tuning Zero-Shot Models (NeurIPS 2024 · similarity 0.84)
- Understanding Domain Generalization: A Noise Robustness Perspective (ICLR 2024 · similarity 0.83)
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