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\icmltitlerunning{Optimal Regularization for Performative Learning}

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  \icmltitle{Optimal Regularization for Performative Learning}



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  \begin{icmlauthorlist}
    \icmlauthor{Edwige Cyffers}{yyy}
    \icmlauthor{Alireza Mirrokni}{sch}
    \icmlauthor{Marco Mondelli}{sch}
  \end{icmlauthorlist}

  \icmlaffiliation{yyy}{CNRS, LAMSADE, Dauphine PSL}
  \icmlaffiliation{sch}{Institute of Science and Technology Austria,
Klosterneuburg, Austria}

  \icmlcorrespondingauthor{Edwige Cyffers}{edwige.cyffers@cnrs.fr}
  \icmlcorrespondingauthor{Marco Mondelli}{marco.mondelli@ist.ac.at}

  \icmlkeywords{Performative Learning, Regularization, High Dimension Linear Regression}

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\begin{abstract}
In performative learning, the data distribution reacts to the deployed model—for example, because strategic users adapt their features to game it—which creates a more complex dynamic than in classical supervised learning. One should thus not only optimize the model for the current data but also take into account that the model might steer the distribution in a new direction, without knowing the exact nature of the potential shift. We explore how regularization can help cope with performative effects by studying its impact in high-dimensional ridge regression. 
We show that, while performative effects worsen the test risk in the population setting, when moving to the over-parameterized regime where the number of features exceeds the number of samples, the optimal regularization in the presence of performativity helps reduce the variance in the estimated parameters, thereby improving performance.
Furthermore, we prove that the optimal regularization scales with the overall strength of the performative effect, making it possible to set the regularization in anticipation of this effect. We illustrate this finding through empirical evaluations of the optimal regularization parameter on both synthetic and real-world datasets.
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\section*{Impact Statement}


This work studies performative learning, which contributes to the study of the long-term impact of machine learning when deployed. Taking into account the risk of modifying the data distribution is likely to improve the impact of machine learning. However, the two evaluations we propose, namely evaluation on the initial distribution and evaluation on the final distribution, can both lead to unfortunate situations. Evaluating on the initial distribution might amount to turning a blind eye to the evolution of society, while the second might encourage intentionally steering the data distribution toward less diverse distributions that are easier to tackle. We mitigate this risk by deriving results for both regimes, which allows readers to pick the setting that makes sense for a given use case, in the absence of alternative evaluation criteria currently proposed for performative learning.

The choice to use high-dimensional tools to study performative learning should open interesting research directions and contribute to widening the tools available to study performativity. High-dimensional theory has proven useful for better explaining deep learning in classical machine learning, and we hope for similar benefits in the context of performative learning.

The focus on regularization offers rather practical and actionable takeaways from the paper, advocating for regularization as a convenient and agnostic mitigation for performativity. We trust readers to understand that regularization is only one tool among many for building more resilient models.

\section*{Acknowledgments}

This research was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/COE12. For the purpose of open access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission. Alireza Mirrokni was an intern at ISTA while working on this project. Edwige Cyffers is supported by the National Research Agency under France 2030, reference “ANR-23-IACL-0008” and this work was partially done while visiting the Simons Institute for the Theory of Computing.

\section*{Conflict of Interest}
The authors have no conflict of interest.

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