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The Limitations of Model Retraining in the Face of Performativity

Anmol Kabra, Kumar Kshitij Patel

TL;DR

The paper investigates how performativity—data distribution changes induced by deployed models—undermines naive retraining strategies. It formalizes Performative Risk $PR(\theta)$ and analyzes fixed-point notions: $\Theta_{\text{PS}}$ (Performatively Stable) and $\Theta_{\text{PO}}$ (Performatively Optimal), highlighting that even convex $PR$ can yield a gap between these solutions under simple linear shifts with covariance components. The authors propose Regularized Repeated Risk Minimization (Reg-R-RM) and Regularized Repeated Empirical Risk Minimization (Reg-R-ERM) to fix fixed-point discrepancies and combat finite-sample errors, showing that appropriately chosen regularization can drive convergence to $\Theta_{\text{PO}}$ and that Reg-R-ERM achieves convergence under reasonable sample schedules. These results suggest rethinking retraining in the presence of performativity, balancing data collection with regularization to obtain performatively optimal outcomes in practical settings.

Abstract

We study stochastic optimization in the context of performative shifts, where the data distribution changes in response to the deployed model. We demonstrate that naive retraining can be provably suboptimal even for simple distribution shifts. The issue worsens when models are retrained given a finite number of samples at each retraining step. We show that adding regularization to retraining corrects both of these issues, attaining provably optimal models in the face of distribution shifts. Our work advocates rethinking how machine learning models are retrained in the presence of performative effects.

The Limitations of Model Retraining in the Face of Performativity

TL;DR

The paper investigates how performativity—data distribution changes induced by deployed models—undermines naive retraining strategies. It formalizes Performative Risk and analyzes fixed-point notions: (Performatively Stable) and (Performatively Optimal), highlighting that even convex can yield a gap between these solutions under simple linear shifts with covariance components. The authors propose Regularized Repeated Risk Minimization (Reg-R-RM) and Regularized Repeated Empirical Risk Minimization (Reg-R-ERM) to fix fixed-point discrepancies and combat finite-sample errors, showing that appropriately chosen regularization can drive convergence to and that Reg-R-ERM achieves convergence under reasonable sample schedules. These results suggest rethinking retraining in the presence of performativity, balancing data collection with regularization to obtain performatively optimal outcomes in practical settings.

Abstract

We study stochastic optimization in the context of performative shifts, where the data distribution changes in response to the deployed model. We demonstrate that naive retraining can be provably suboptimal even for simple distribution shifts. The issue worsens when models are retrained given a finite number of samples at each retraining step. We show that adding regularization to retraining corrects both of these issues, attaining provably optimal models in the face of distribution shifts. Our work advocates rethinking how machine learning models are retrained in the presence of performative effects.
Paper Structure (13 sections, 6 theorems, 34 equations)

This paper contains 13 sections, 6 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Retraining Fails for Simple Distribution Shifts
  3. Empirical Retraining Fails to Converge even with Infinite Samples
  4. Regularization Mitigates the Perils of Retraining
  5. Related Work
  6. Performative Prediction
  7. Strategic Classification
  8. Conclusion
  9. Proof of Theorem \ref{['thm:prob_where_ps_po_same']}
  10. Proof of Theorem \ref{['thm:prob_where_ps_stat_no_intersection']}
  11. Proof of Theorem \ref{['thm:Repeated-0-ERM__diverges_from_PS__constant_samples']}
  12. Proof of Theorem \ref{['thm:prob_where_ps_stat_no_intersection_regularized']}
  13. Proof of Theorem \ref{['thm:Repeated-0-ERM__diverges_from_PS__constant_samples_regularized']}

Key Result

Theorem 2.2

Let $\calD(\bm{\theta})$ be a linear shift over $\bfz \in \bbR^d$ with $\bm{\Sigma}(\bm{\theta}) = \bf0 \in \bbR^{d \times d}$ and $\left\lVert \bm{\mu} \right\rVert_\ast < 1$ (the largest singular value of linear map $\bm{\mu}$). Let $\bfA$ be a positive-definite matrix defining the loss function $

Theorems & Definitions (12)

  • Definition 2.1: Linear shifts miller2021outside
  • Theorem 2.2: Mean shifts
  • Theorem 2.3: Linear shifts, scalar setting
  • Theorem 3.1: perdomo2020performative
  • Theorem 3.2: Mean shifts, finite samples
  • Theorem 4.1: Covariance shift, with regularization
  • Theorem 4.2: Mean shifts, with regularization
  • proof
  • proof : Proof of \ref{['thm:prob_where_ps_stat_no_intersection']}
  • proof
  • ...and 2 more