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Optimal Classification under Performative Distribution Shift
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Models performative effects as push-forward measures, derives a reparameterization-based performative gradient estimator with lower variance than score-function estimators, and proves convexity of the performative risk for linear classification under PSD shift matrices of arbitrary strength.
Problem
Performative prediction requires optimizing the performative risk, but existing gradient estimators need the full parametric distribution and convexity results only hold for small performative effects.
Approach
Represent performative effects as push-forward maps Z = phi(U; theta) so the performative gradient can be estimated by reparameterization (pathwise) without knowing the density; for linear shifts in classification, the performative risk is convex when the shift matrix is PSD regardless of its magnitude, and it equals an adversarially robust min-max problem.
Key results
- Proves the performative risk is convex for linear classifiers and linear shift operators when Pi is positive semidefinite, for quadratic loss or convex non-increasing surrogate losses, with no bound on the shift magnitude.
- For Gaussian mean estimation, the reparameterization estimator's covariance is sigma^2 Pi^T Pi / n, independent of dimension d, whereas the score-function estimator's covariance grows with d^2.
- On the Housing dataset, RPPerfGD outperforms RGD as the magnitude of the performative shift increases, and converges faster than the non-performative approach.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Account for the performative effect with RPPerfGD instead of ignoring it | Logistic regression on two Gaussian distributions · Synthetic Gaussian classification with performative shift | accuracy | Helped |
| Use RPPerfGD rather than SFPerfGD’s score-function estimator | Performative gradient estimation · 7-dimensional Gaussian classification | convergence stability | Helped |
| Estimate the performative shift and use it in RPPerfGD | RPPerfGD · Synthetic Gaussian classification with learned shift matrix | performance | No clear effect |
| Account for performative shifts with RPPerfGD instead of RGD | Logistic regression on the Housing dataset · Housing | accuracy | Helped |
| Use Repeated Risk Minimization when performative effects are non-negligible | Repeated Risk Minimization (RRM) · Synthetic classification tasks | convergence | Hurt |
| Use any compared method when there is no performative effect | Compared learning methods · Synthetic Gaussian classification, γ = 0 | accuracy | No clear effect |
Surprising findings
- Repeated risk minimization is unstable and diverges once performative effects are non-negligible in the experiments.
Limitations
- Experiments use small synthetic and single real dataset, with simulated performative shifts rather than true performative data.
- Convexity results restricted to linear-in-parameters shifts and linear classifiers; class-1 distribution is assumed unaffected by performative effects.
- Plug-in estimation of Pi via ridge regression is not theoretically guaranteed to converge.
Experimental setup
Datasets: Housing (OpenML 823)
Benchmarks: Housing
Metrics: accuracy parameter trajectory estimator variance distance between estimated and true Pi
Baselines: Repeated Risk Minimization Repeated Gradient Descent Regularized Repeated Gradient Descent Score Function Performative Gradient Descent
Builds on: performative prediction repeated risk minimization score function gradient estimator reparameterization trick
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