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$H$-Consistency Guarantees for Regression
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Generalizes H-consistency bound tools to regression and proves H-consistency bounds for Huber, l_p, and squared epsilon-insensitive surrogates of the squared loss under symmetric bounded settings, then applies them to smooth adversarial regression.
Problem
Bayes-consistency of regression surrogate losses is asymptotic and ignores the hypothesis set; no prior H-consistency or finite-sample guarantees existed for common regression surrogates such as Huber, l_p, and squared epsilon-insensitive losses relative to the squared loss.
Approach
Generalized convex/concave-function H-consistency theorems to allow non-constant alpha(h,x) weights, then characterize the best-in-class conditional error under symmetric distributions to derive H-consistency bounds for regression surrogates, and use the bounds to design smooth adversarial regression losses (surrogate plus a local smoothness penalty).
Key results
- Smooth adversarial regression with Huber loss (delta=0.2) on MAGIC wheat at gamma=0.01 reduces clean error to 0.97 ± 0.02 and robust error to 1.01 ± 0.02, versus 1.30 and 1.78 for Adv-sq.
- On Diabetes at gamma=0.001, the L=l2 smooth variant reduces clean error to 1.24 ± 0.21 from 2.53 ± 0.48 for Adv-sq.
- The paper proves H-consistency bounds for Huber, all l_p (p>=1), and squared epsilon-insensitive losses, and a negative result that the Huber and squared epsilon-insensitive losses are not H-consistent when p_min is zero, and that the epsilon-insensitive loss is not H-consistent.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use smooth adversarial regression with L = ℓ₂ instead of Adv-sq | Linear regression model · Diabetes, γ=0.001 | Clean error: 2.53 ± 0.48 -> 1.24 ± 0.21 | Helped |
| Use smooth adversarial regression with L = ℓ₂ instead of Adv-sq | Linear regression model · Diverse MAGIC wheat, γ=0.001 | Clean error: 1.28 ± 0.10 -> 1.28 ± 0.09 | No clear effect |
| Use smooth adversarial regression with L = ℓδ instead of Adv-sq | Linear regression model · Diverse MAGIC wheat, γ=0.005 | Robust error: 1.53 ± 0.10 -> 1.12 ± 0.10 | Helped |
| Use smooth adversarial regression with L = ℓ₂ instead of Adv-sq | Linear regression model · Diverse MAGIC wheat, γ=0.005 | Robust error: 1.53 ± 0.10 -> 1.46 ± 0.10 | Helped |
| Use smooth adversarial regression with L = ℓδ instead of Adv-sq | Linear regression model · Diabetes, γ=0.01 | Robust error: 0.87 ± 0.05 -> 0.82 ± 0.05 | Helped |
| Use smooth adversarial regression with L = ℓ₂ instead of Adv-sq | Linear regression model · Diabetes, γ=0.01 | Robust error: 0.87 ± 0.05 -> 0.76 ± 0.05 | Helped |
| Use Huber loss (L = ℓδ) rather than squared loss (L = ℓ₂) in the smooth adversarial method | Linear regression model · Diabetes, γ=0.01 | Robust error: 0.76 ± 0.05 -> 0.82 ± 0.05 | Hurt |
| Use Huber loss (L = ℓδ) rather than squared loss (L = ℓ₂) in the smooth adversarial method | Linear regression model · Diverse MAGIC wheat, γ=0.01 | Robust error: 1.62 ± 0.14 -> 1.01 ± 0.02 | Helped |
Surprising findings
- The Huber-loss variant of the smooth adversarial method sometimes outperforms the squared-loss variant.
Limitations
- Bounds require symmetric conditional distributions and bounded conditional distributions and hypothesis sets.
- Huber and squared epsilon-insensitive bounds are uninformative when p_min is zero, and those losses are not H-consistent in that case.
- Experiments are limited to two small datasets with a linear hypothesis set.
Experimental setup
Datasets: Diabetes Diverse MAGIC wheat
Benchmarks: Diabetes Diverse MAGIC wheat
Metrics: clean MSE robust adversarial squared loss
Baselines: Adv-sq (adversarial squared loss training)
Builds on: H-consistency bounds for classification Bayes-consistency of regression losses support vector regression
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