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On the Impact of Performative Risk Minimization for Binary Random Variables
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Analyzes the broader impact of performative risk minimization (PRM) on predictions and data distributions in a sequential mean-estimation problem for binary variables with linear performative shifts, deriving closed-form bias and mean-shift results under full and partial information.
Problem
Performative risk minimization preserves test accuracy under performative distribution shift, but its broader side effects on the model's predictions (bias) and the underlying data distribution (mean shift) are not understood, especially over sequential deployments.
Approach
Studies a tractable sequential performative mean-estimation problem for binary random variables with linear response, derives explicit PRM paths and two impact measures (bias and mean shift), and compares them with a naive risk-minimization path, showing PRM can be more biased and push means to extremes, with the error-uncertainty tradeoff as the driver.
Key results
- In the one-period slow case, the PRM prediction is generally biased (bias = alpha(1-|alpha|)s_1/(1-2alpha)) even with perfect information, pushing the mean toward extremes under positive feedback.
- Under negative feedback with rapid model updates and high performativity, the naive path can have larger bias and shift than PRM, while under positive feedback the naive path has zero bias and shift but a worse loss by alpha^2 s_1^2/(1-2alpha).
- With unknown initial mean, the performative estimator's expected loss converges to the optimal loss as the sample size m grows, and RL simulations converge to theoretical equilibria.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PRM instead of the naive path | Binary mean estimation, one-period slow deployment with positive feedback · T=1, slow deployment, α > 0 | bias | Hurt |
| Use PRM instead of the naive path | Binary mean estimation, one-period slow deployment with positive feedback · T=1, slow deployment, α > 0 | loss | Helped |
| Use PRM instead of the naive path | Binary mean estimation, one-period slow deployment with negative feedback · T=1, slow deployment, α ≤ 0 | bias | Helped |
| Use PRM instead of the naive path | Binary mean estimation, one-period slow deployment with negative feedback · T=1, slow deployment, α ≤ 0 | loss | Helped |
| Use PRM instead of the naive path | Binary mean estimation, infinite-horizon slow deployment with positive feedback · T=∞, slow deployment, α > 0 | bias | Hurt |
| Use PRM instead of the naive path | Binary mean estimation, infinite-horizon slow deployment with negative feedback · T=∞, slow deployment, α < 0 | mean shift | Helped |
| Use PRM instead of the naive path | Binary mean estimation, infinite-horizon rapid deployment with negative feedback · T=∞, rapid deployment, α < 0 | mean shift | Helped |
| Use PRM rather than RM | Binary mean estimation with performative distribution shifts · Sequential performative prediction | bias | Hurt |
| Use PRM rather than RM | Binary mean estimation with performative distribution shifts · Sequential performative prediction | mean shift | Hurt |
| Increase the deployment horizon | PRM for binary mean estimation · Slow deployment, multi-period compared with one period | bias | Hurt |
| Use rapid rather than slow deployment | PRM for binary mean estimation with negative feedback · Sequential performative prediction | mean shift | Mixed |
| Use more samples to estimate the initial mean | Performative estimator · T=1, slow deployment, unknown initial mean | loss | Helped |
Surprising findings
- PRM is biased even with full information, because the provider trades MSE for reduced aleatoric uncertainty of the future distribution.
- Under positive feedback, the naive path has zero bias and shift, yet PRM can still be preferred for loss.
Limitations
- Restricted to mean estimation of binary variables under a linear response model.
- Results may only transfer qualitatively to more general settings; regression and other models of response left to future work.
- Group extension assumes groups evolve independently, which may fail for traffic prediction.
Experimental setup
Metrics: bias mean shift expected loss (MSE with aleatoric uncertainty)
Baselines: naive risk minimization path (ERM-style, predicts previously observable mean)
Builds on: performative prediction (Perdomo et al., 2020) stateful performative prediction PRM optimization algorithms (repeated gradient descent / episodic RL algorithm)
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