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The Stability of Online Algorithms in Performative Prediction

Gabriele Farina, Juan Carlos Perdomo

TL;DR

The 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.

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.

The Stability of Online Algorithms in Performative Prediction

TL;DR

The 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.

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.
Paper Structure (13 sections, 8 theorems, 31 equations, 2 figures, 1 table)

This paper contains 13 sections, 8 theorems, 31 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Performative Prediction.
  5. The Power of No-Regret.
  6. Preliminaries and Technical Background
  7. Main Results
  8. Comparison to Prior Work.
  9. Implications of Main Result
  10. Finite-Sample Guarantees for Retraining.
  11. Improved Guarantees for Gradient Descent.
  12. Guaranteeing Performative Stability in New Regimes.
  13. Discussion and Future Work

Key Result

Theorem 1

Let $\theta_1, \dots, \theta_T$ be models produced by an online algorithm that achieves no regret on the sequence of losses $\ell_t(\theta) = \ell(z_t, \theta)$ where $z_t \sim \mathcal{D}(\theta_t)$ for $t=1$ to $T$. And, define $\mu$ to be the uniform distribution over $\theta_1, \dots, \theta_T$.

Figures (2)

  • Figure 1: Visualizing the performative stability guarantee. A mixture $\mu$ over models $\theta$ is stable if it shapes the data in a way that its own predictions are risk minimizing. Our notion, \ref{['def:sensitivity']}, generalizes the prior notion where $\mu$ is a point mass on a single model.
  • Figure 2: Our main result shows that any no-regret algorithm yields a sequence of models whose uniform mixture is performatively stable. The idea is simple and surprisingly strong. It sidesteps recent hardness results, and establishes the first proofs of stability that do not require any continuity assumptions on $\mathcal{D}(\cdot)$ or curvature assumptions on $\ell$.

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1: Informal
  • Definition 2
  • Definition 3
  • Theorem 2: perdomo2020performativemendler2020stochastic
  • Example 1: perdomo2020performative
  • Example 1: Continued
  • Definition 4
  • Theorem 3
  • Lemma 1: shalev2008mindhazan2007logarithmicabernethy2008optimal
  • ...and 4 more