On the Computational Complexity of Performative Prediction

TL;DR

This work studies the computational complexity of finding performatively stable points in performative prediction, where deploying a model shifts the data distribution. It identifies a sharp phase transition governed by ρ = Lβ/α, showing that computing an ε-performatively stable point is PPAD-complete when ρ ≤ 1 + O(ε), even with a simple quadratic loss and affine shifts; however, a poly-time algorithm exists for ρ = 1 + Oε(ε^4) achieving an ε^{1/4}-approximation, and there are unconditional exponential query lower bounds. The hardness results extend to general convex domains and demonstrate PPAD-hardness, while strategic classification exhibits PLS-hardness for computing strategic local optima and hardness with endogenous costs. Together, these findings reveal a sharply delineated complexity boundary for performative stability and highlight fundamental limits on retraining-based approaches in the presence of strong performative effects. These results have implications for designing robust deployment and evaluation protocols in settings where predictive models influence the very data they aim to predict.

Abstract

Performative prediction captures the phenomenon where deploying a predictive model shifts the underlying data distribution. While simple retraining dynamics are known to converge linearly when the performative effects are weak ($ρ< 1$), the complexity in the regime $ρ> 1$ was hitherto open. In this paper, we establish a sharp phase transition: computing an $ε$-performatively stable point is PPAD-complete -- and thus polynomial-time equivalent to Nash equilibria in general-sum games -- even when $ρ= 1 + O(ε)$. This intractability persists even in the ostensibly simple setting with a quadratic loss function and linear distribution shifts. One of our key technical contributions is to extend this PPAD-hardness result to general convex domains, which is of broader interest in the complexity of variational inequalities. Finally, we address the special case of strategic classification, showing that computing a strategic local optimum is PLS-hard.

Figures (3)

• Figure 1: The complexity landscape for computing $\epsilon$-performatively stable points.
• Figure 2: Left: Illustration of the proof of \ref{['theorem:PPAD_hardness_convex_set']}. Regions outside $\triangle {A}_1 {A}_2 {A}_3$ are mapped with corresponding colors. The shaded bands represents the $\epsilon$-thickness strips used in the construction. Inside $\triangle {A}_1 {A}_2 {A}_3$ the domain forms a triangular grid with colors assigned to the vertices. Trichromatic triangles correspond to VI solutions, and are highlighted by shading. Right: Directional vector mapping induced by the coloring. The directions are chosen so that any point outside $\triangle {A}_1 {A}_2 {A}_3$ is pushed toward the interior of the triangle. As a result, all VI solutions must lie inside $\triangle {A}_1 {A}_2 {A}_3$.
• Figure 3: Our basic edge gadget for edge $(u, v)$. The Jury would like to classify $x_{(u, v)^+}$ as 1, but $x_{(u, v)^-}$ would then strategically deviate to $x_{(u, v)^+}$. This forces the Jury to pick a classifier such that $f(x_{(u, v)^+}) = 0 = f(x_{(u, v)^-})$. Furthermore, if the Jury switches the label of $x_{u^-}$ from $0$ to $1$, all edges incident to $u$ in the graph that were previously classified as $0$ can profitably deviate to $x_{u^-}$. The change in the Jury's utility reflects the change in the weight of the cut induced by $f$.

Theorems & Definitions (60)

• Definition 1.1: perdomo2021performativeprediction
• Theorem 1.2
• Definition 2.1: Performative optimality; perdomo2021performativeprediction
• Definition 2.2: Performative stability; perdomo2021performativeprediction
• Definition 2.4
• Proposition 3.1: From VIs to performative stability
• Proposition 3.2: From fixed points to performative stability
• Lemma 3.3: bernasconi2024role
• Theorem 3.4
• Theorem 3.5