[Paper Review] The Stability of Online Algorithms in Performative Prediction

Gabriele Farina, Juan C. Perdomo|arXiv (Cornell University)|Feb 27, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper proves that any no-regret online algorithm deployed in performative settings converges to a performatively stable mixture of models, without assuming Lipschitz continuity of the distribution map, and extends stability guarantees to weakly convex or non-smooth losses.

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.

Motivation & Objective

• Motivate and formalize performativity in predictive decision-making where deployed models influence future data.
• Show that any no-regret online algorithm yields a performatively stable mixture of models.
• Provide stability guarantees that do not require Lipschitz or smoothness assumptions on the data-generating map.
• Demonstrate that common algorithms like gradient descent attain stability when their iterates are mixed.
• Extend stability results to weakly convex or non-smooth loss functions and finite-sample retraining regimes.

Proposed method

• Model the data-generating process as z_t ~ D(θ_t) where θ_t is the current model.
• Consider a sequence of models produced by a no-regret online algorithm and form the uniform mixture μ over {θ_1,...,θ_T}.
• Prove that the uniform mixture μ is Regret(T)/T-performatively stable by a martingale-based online-to-batch argument.
• Show that this reduction does not require continuity of D(·) or strong convexity of the loss, unlike prior work.
• Derive finite-sample guarantees for retraining and corollaries for SGD/gradient-descent under broad conditions.
• Discuss implications for stability under weakly convex or non-smooth losses.

Experimental results

Research questions

• RQ1 Does any no-regret online algorithm, coupled with an induced distribution via D(·), converge to a performatively stable mixture?
• RQ2 Can stability guarantees be obtained without Lipschitz/continuity assumptions on the distribution map D(·)?
• RQ3 Do common optimization procedures (e.g., gradient descent, retraining) inherit performative stability through mixture-based reductions?
• RQ4 What finite-sample guarantees can be derived for retraining in performative settings with non-smooth or weakly convex losses?
• RQ5 How do these results relate to and extend prior performative prediction work under stricter smoothness assumptions?

Key findings

• A uniform mixture over iterates of any no-regret online algorithm is Regret(T)/T-performatively stable.
• Stability holds for arbitrary distribution maps D(·), without continuity or Lipschitz assumptions.
• Corollaries show gradient descent and retraining converge to stability under weaker loss/DT conditions, including non-smooth/weakly convex losses.
• Finite-sample results are obtained for smoothed retraining, matching or improving prior rates under broad assumptions.
• The framework explains why classical algorithms naturally resist runaway feedback loops in dynamic, performative environments.
• The results provide a unifying online-learning perspective for performativity, extending stability guarantees to new regimes.