The Stability of Online Algorithms in Performative Prediction

TL;DR

This paper proves that any no-regret online algorithm in performative prediction settings converges to a stable equilibrium where models influence data distributions to make their predictions appear optimal, using a martingale approach.

Contribution

It introduces an unconditional convergence result for no-regret algorithms in performative prediction, relaxing previous restrictive assumptions and explaining the stabilizing nature of common algorithms.

Findings

Any no-regret algorithm converges to a performatively stable equilibrium.

Martingale arguments enable analysis without restrictive assumptions.

Common algorithms like gradient descent are inherently stabilizing.

Abstract

The use of algorithmic predictions in decision-making leads to a feedback loop where the models we deploy actively influence the data distributions we see, and later use to retrain on. This dynamic was formalized by Perdomo et al. 2020 in their work on performative prediction. Our main result is an unconditional reduction showing that any no-regret algorithm deployed in performative settings converges to a (mixed) performatively stable equilibrium: a solution in which models actively shape data distributions in ways that their own predictions look optimal in hindsight. Prior to our work, all positive results in this area made strong restrictions on how models influenced distributions. By using a martingale argument and allowing randomization, we avoid any such assumption and sidestep recent hardness results for finding stable models. Lastly, on a more conceptual note, our connection sheds light on why common algorithms, like gradient descent, are naturally stabilizing and prevent runaway feedback loops. We hope our work enables future technical transfer of ideas between online optimization and performativity.