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Tight Rates in Supervised Outlier Transfer Learning
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Gives information-theoretic limits and an adaptive algorithm for transfer from imperfect source rare-class data to a target Neyman-Pearson outlier detection task, governed by an outlier transfer exponent.
Problem
Supervised outlier detection framed as Neyman-Pearson classification with scarce target rare-class data, where related but imperfect source outlier data is available; the question is when and how fast knowledge transfers.
Approach
Defines an outlier transfer exponent (adapted from the transfer exponent in traditional classification), proves a minimax lower bound controlled by it and source/target sample sizes, and gives an adaptive procedure that picks between empirical source and target classifiers by comparing their performance on target data, matching the lower bound up to log factors.
Key results
- Minimax lower bound of order min{Delta + (d/n_S)^{1/(2rho)}, (d/n_T)^{1/2}} for target-excess error, with outlier transfer exponent rho.
- Adaptive algorithm attains the lower bound up to logarithmic factors without knowing rho or the source-target discrepancy.
- Seemingly dissimilar sources (e.g., different variance Gaussians or differently shaped uniform densities) can share the same optimal classifiers under the restricted class U*, enabling fast transfer.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use source samples with outlier transfer exponent ρ and target samples, then adaptively select between source- and target-trained classifiers | Finite-sample transfer outlier detection · Finite-sample transfer outlier detection | target-excess error: \min\{\mathcal{E}_{1,T}(h^*_{S,\alpha})+C\cdot A_{n_S}^{\frac{1}{2\rho(r)}},C\cdot A_{n_T}^{\frac{1}{2}}\} | Helped |
| Increase source sample size n_S, with transfer exponent ρ fixed | Source-based transfer learning · Finite-sample transfer outlier detection | target-excess error: (\frac{d_{\mathcal{H}}}{n_S})^{\frac{1}{2\rho}} | Helped |
| Increase the outlier transfer exponent ρ | Source-based transfer learning · Finite-sample transfer outlier detection | target-excess error: (\frac{d_{\mathcal{H}}}{n_S})^{\frac{1}{2\rho}} | Hurt |
| Use source samples when source and target share the same optimal decision rules and ρ=1 | Source-based transfer learning · Finite-sample transfer outlier detection | relative effectiveness of source samples: ρ=1 | Helped |
| Use a source distribution whose optimal classifiers match the target's | Supervised outlier transfer learning · Population Neyman-Pearson classification | target optimality | Helped |
| Allow source and target optimal classifiers to differ by discrepancy Δ | Finite-sample transfer learning · Finite-sample transfer outlier detection | minimax target-excess error: \Delta+(\frac{d_{\mathcal{H}}}{n_S})^{\frac{1}{2\rho}} | Hurt |
Surprising findings
- Seemingly very dissimilar source distributions can share the same universally optimal decision rules and transfer quickly, unlike in balanced classification.
- Even with identical optimal classifiers, a source may give poor transfer rates when the outlier transfer exponent is large.
Limitations
- Theory only; no experiments on real outlier data are reported in the provided text.
- Lower bound and adaptive upper bound match only up to logarithmic factors.
- Minimax results assume finite VC dimension and idealized distribution classes.
Experimental setup
Metrics: target-excess error minimax risk
Builds on: Neyman-Pearson classification transfer exponent (Hanneke and Kpotufe-style classification transfer) Neyman-Pearson lemma
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