Performative Stability in Adaptive Prediction

• Performative stability is the concept where retraining a deployed model on its self-induced distribution returns the same parameter, forming a fixed-point equilibrium.
• It distinguishes the dynamics of adaptive prediction by contrasting performative risk minimization with stability, emphasizing conditions like contraction that ensure unique convergence.
• Algorithms such as repeated risk minimization and gradient descent exploit these contraction properties, though extensions to multi-agent and nonconvex settings reveal challenges like computational intractability and chaotic dynamics.

Performative stability is the central equilibrium concept in performative prediction and its extensions. It formalizes settings in which a deployed model, policy, or decision changes the very distribution or environment on which it is later evaluated. In its canonical form, one has a parameter space $\Theta$, a distribution map $D:\Theta\to\Delta(\mathcal Z)$, and loss $\ell(z;\theta)$. The deployed parameter $\theta$ induces data from $D(\theta)$, and a parameter is performatively stable if retraining on its own induced distribution returns the same parameter. In this sense, performative stability is a fixed point of a retraining or best-response map, and it differs from ordinary empirical risk minimization precisely because the data-generating process is endogenous to deployment (Perdomo et al., 2020).

1. Foundations and solution concepts

The original single-agent formulation distinguishes between the performative risk

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$

and the decoupled or cross-risk

$$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$

A parameter $\theta^\dagger$ is performatively stable when

$$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$

equivalently, a fixed point of

$G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$

The concept was introduced as the equilibrium notion appropriate for learning systems whose predictions reshape future outcomes (Perdomo et al., 2020).

Performative stability is distinct from performative optimality. A performative optimum minimizes the self-referential objective

$D:\Theta\to\Delta(\mathcal Z)$0

whereas a performatively stable point only requires self-consistency under retraining. The two notions need not coincide. The survey literature emphasizes that stability captures “stability under induced shifts”: once the environment has reacted to $D:\Theta\to\Delta(\mathcal Z)$1, the best counterfactual model for the resulting distribution is $D:\Theta\to\Delta(\mathcal Z)$2 itself (Kehrenberg et al., 10 Feb 2026).

The same fixed-point structure recurs in later generalizations. In multi-agent performative prediction, with joint profile $D:\Theta\to\Delta(\mathcal Z)$3 and joint-induced distribution $D:\Theta\to\Delta(\mathcal Z)$4, a profile $D:\Theta\to\Delta(\mathcal Z)$5 is performatively stable if for every agent $D:\Theta\to\Delta(\mathcal Z)$6,

$D:\Theta\to\Delta(\mathcal Z)$7

so no single agent can unilaterally improve its own loss on the distribution induced by the full profile (Piliouras et al., 2022). In reinforcement learning, a performatively stable policy is defined analogously as a policy that is optimal in the MDP induced by itself (Mandal et al., 2022). In control, a performatively stable control (PSC) solution is a fixed point of a distribution-shifted cost-minimizer mapping (Cai et al., 2024). These formulations differ in state space and objective, but preserve the same equilibrium template.

2. Existence, uniqueness, and contraction regimes

The classical positive theory is based on contraction. In the foundational performative-prediction model, if $D:\Theta\to\Delta(\mathcal Z)$8 is $D:\Theta\to\Delta(\mathcal Z)$9-jointly smooth and $\ell(z;\theta)$0-strongly convex in $\ell(z;\theta)$1, and if the distribution map is $\ell(z;\theta)$2-sensitive in Wasserstein-1,

$\ell(z;\theta)$3

then the retraining map satisfies

$\ell(z;\theta)$4

Hence, when $\ell(z;\theta)$5, $\ell(z;\theta)$6 is a contraction, the performatively stable point is unique, and repeated risk minimization converges linearly to it (Perdomo et al., 2020).

A related stateful generalization replaces the stateless map $\ell(z;\theta)$7 by a transition operator

$\ell(z;\theta)$8

so the population state itself carries memory. There, a stable pair $\ell(z;\theta)$9 satisfies both $\theta$0 and $\theta$1. If the transition map is $\theta$2-jointly sensitive and the loss is $\theta$3-strongly convex and $\theta$4-smooth, then the combined update is a contraction with coefficient

$\theta$5

and uniqueness plus linear convergence follow when $\theta$6 (Brown et al., 2020).

Later work relaxes the object with respect to which sensitivity is measured. For neural networks, the key assumption is Lipschitz sensitivity of the induced distribution with respect to the model’s predictions rather than its parameters:

$\theta$7

Combined with a bounded density-ratio condition and strong convexity of the loss in its prediction argument, this yields contraction of repeated risk minimization in function space. The contraction factor is

$\theta$8

and if $\theta$9 there is linear convergence to a unique performatively stable classifier. For nonconvex function classes, an approximation version yields convergence to an $D(\theta)$0-neighborhood under squared-error loss (Mofakhami et al., 2023). This shifts the regularity burden from parameterization to predictive behavior.

A different line of work shows that contraction is sufficient but not necessary for stability-like guarantees. In online performative prediction, if an online algorithm achieves sublinear external regret on the adaptive loss sequence $D(\theta)$1 with $D(\theta)$2, then the uniform distribution over its iterates is $D(\theta)$3-performatively stable:

$D(\theta)$4

This reduction requires no convexity, no smoothness, and no continuity assumptions on $D(\theta)$5; it replaces deterministic fixed points by mixed equilibria (Farina et al., 27 Feb 2026).

3. Algorithms for reaching stable points

The basic computational procedures are repeated risk minimization (RRM) and repeated gradient descent (RGD). RRM iterates

$D(\theta)$6

whereas RGD performs one gradient step on the risk induced by the currently deployed model,

$D(\theta)$7

Under the standard contraction assumptions, both methods converge linearly to the same unique stable point (Perdomo et al., 2020).

For nonconvex losses, exact best-response maps are generally unavailable or multivalued, so the relevant target becomes a stationary performative stable (SPS) solution. In one formulation, $D(\theta)$8 is a $D(\theta)$9-SPS solution if

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$0

where

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$1

Greedy-deployment SGD,

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$2

converges in expectation to a biased SPS solution under smoothness, controlled gradient noise, and either Wasserstein-1 or total-variation sensitivity conditions. With constant-scale tuning, the asymptotic bias is

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$3

and a lazy deployment scheme reduces the dominant sensitivity-driven bias to $$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$4 (Li et al., 2024).

Variance reduction improves this picture. The SPRINT algorithm adapts the SVRG idea to performative prediction by combining a snapshot gradient

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$5

with inner-loop control variates

$$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$6

Under $$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$7-smoothness in $$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$8, $$PR(\theta)=\mathbb E_{Z\sim D(\theta)}[\ell(Z;\theta)]$$9-sensitivity of the distribution map, and pointwise Lipschitzness in $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$0, SPRINT attains an $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$1 rate to an SPS solution,

$$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$2

with

$$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$3

and this residual neighborhood is independent of stochastic-gradient variance (Xie et al., 22 Sep 2025).

The online-learning perspective yields a conceptually different algorithmic guarantee. Any no-regret method induces mixed performative stability through time-averaging. This includes standard online gradient descent with $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$4, whose iterate mixture is $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$5-stable for convex losses, and stronger $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$6 stability under strong convexity (Farina et al., 27 Feb 2026). A plausible implication is that time-averaging is not merely an analysis device but a structural way to stabilize systems in which deterministic retraining maps may be discontinuous.

4. Multi-agent, dynamical, and structured extensions

In multi-agent performative prediction, multiple decision makers simultaneously influence a common outcome. The decoupled loss of agent $$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$7 is

$$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$8

and performative stability becomes a Nash-like fixed point of unilateral retraining (Piliouras et al., 2022). In a location-scale regression model,

$$R(\theta_{\mathrm{induce}},\theta_{\mathrm{eval}}) =\mathbb E_{Z\sim D(\theta_{\mathrm{induce}})}[\ell(Z;\theta_{\mathrm{eval}})].$$9

small exponentiated-gradient learning rates satisfying thresholds $\theta^\dagger$0 and $\theta^\dagger$1 imply global convergence from any interior start. The stable profile is also the unique global minimizer of the total loss $\theta^\dagger$2, so in that regime stability implies optimality (Piliouras et al., 2022).

Performative control extends the same idea to policy-dependent dynamical systems. With disturbance-action policy parameter $\theta^\dagger$3, cost

$\theta^\dagger$4

and policy-dependent disturbance laws $\theta^\dagger$5, a PSC solution is a fixed point

$\theta^\dagger$6

Existence and uniqueness follow if a weighted sensitivity-propagation condition holds:

$\theta^\dagger$7

For almost-sure strongly stable dynamics this reduces to a small-sensitivity sum condition, whereas for almost-sure unstable dynamics it requires temporally backward decay of the sensitivities. A repeated stochastic gradient descent scheme then converges to the PSC solution with a non-asymptotic bound whose second term is $\theta^\dagger$8 (Cai et al., 2024).

Performative reinforcement learning replaces decision-dependent supervised data with policy-dependent rewards and transition kernels. In one formulation, a performatively stable policy $\theta^\dagger$9 satisfies

$$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$0

meaning that once the environment freezes at $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$1, no alternative policy performs better there. Regularized repeated optimization of the occupancy-measure program converges under Lipschitz dependence of rewards and transitions on the occupancy measure and sufficiently large regularization $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$2; projected-gradient and finite-sample versions also converge (Mandal et al., 2022). Subsequent work in performative RL separates stability from optimality and develops a performative policy gradient method for the latter, which underscores that the two solution concepts remain distinct in sequential decision problems (Basu et al., 23 Dec 2025).

Several later frameworks preserve the fixed-point idea while changing the ambient optimization problem. In decision-dependent chance-constrained optimization, a performative solution is a self-feasible optimizer $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$3, existence follows from Kakutani’s fixed-point theorem, and a stochastic scenario iteration with

$$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$4

samples converges almost surely to the unique fixed point under a deterministic contraction condition $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$5 (Zhu et al., 31 Mar 2026). In federated bilevel performative prediction, the federated bilevel performatively stable (FBPS) point is the fixed point of a decoupled upper-level response map, and a contraction condition of the form

$$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$6

implies uniqueness and linear convergence of federated repeated-risk minimization, while a stochastic federated hypergradient method converges at rate $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$7 in mean square under diminishing step sizes and sufficiently small sensitivities (Qian et al., 18 Jun 2026).

5. Instability, phase transitions, and computational hardness

Performative stability is not universally benign. The multi-agent theory exhibits a phase transition from convergence to instability and then to formal chaos. In the two-dimensional location-scale setting with common learning rate $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$8 and total influence $$\theta^\dagger\in\arg\min_{\theta'}R(\theta^\dagger,\theta') \quad\Longleftrightarrow\quad \theta^\dagger=\arg\min_{\theta'}\mathbb E_{Z\sim D(\theta^\dagger)}[\ell(Z;\theta')],$$9 sufficiently large, the exponentiated-gradient dynamics become Li–Yorke chaotic. Under symmetric initialization, the system reduces to the one-dimensional map

$G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$0

and existence of a period-3 orbit implies Li–Yorke chaos and periodic orbits of all periods. Simulations show monotone convergence for settings such as $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$1 or $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$2, but aperiodic sensitive oscillations for $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$3 (Piliouras et al., 2022).

A separate line of work establishes a sharp complexity transition at the classical contraction threshold

$G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$4

When $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$5, repeated risk minimization is a contraction and converges linearly to the unique stable point. At $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$6, even simple one-dimensional quadratic examples can cycle. For any sufficiently small $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$7, computing an $G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$8-performatively stable point is PPAD-complete even when

$G(\theta)=\arg\min_{\theta'}R(\theta,\theta').$9

and this hardness persists for quadratic loss with linear distribution shifts,

$D:\Theta\to\Delta(\mathcal Z)$00

The hardness also extends from hypercubes to general well-bounded convex domains. In strategic classification, computing a strategic local optimum is PLS-hard (Anagnostides et al., 28 Jan 2026).

These negative results clarify a common misunderstanding. Stability is often introduced through contraction-based sufficient conditions, but those conditions do not characterize the full landscape. Outside the weak-performativity regime, deterministic stable points may fail to exist, may be hard to find, or may be dynamically unstable. The mixed-equilibrium reduction via no-regret learning provides one route around this barrier by replacing deterministic fixed points with distributions over iterates (Farina et al., 27 Feb 2026).

6. Fairness, nonstationarity, and statistical inference

Performative stability can conflict with social desiderata. In group-structured performative prediction with demographic groups $D:\Theta\to\Delta(\mathcal Z)$01, the literature defines loss disparity

$D:\Theta\to\Delta(\mathcal Z)$02

and participation disparity

$D:\Theta\to\Delta(\mathcal Z)$03

Examples show that repeated risk minimization can converge to a performatively stable point whose induced distribution is highly polarized, including cases where one group vanishes and group-wise loss disparity becomes large (Jin et al., 2024).

Standard static-data fairness interventions may also break stability. Group-loss-variance regularization can render the per-round objective non-convex, and repeated DRO with $D:\Theta\to\Delta(\mathcal Z)$04 constraints can generate oscillation with no fixed point. To address this, Fair-RRM replaces the standard per-round objective with one of three modified objectives: a group-level penalty, a sample-level penalty, or loss-guided re-weighting. Under the same strong convexity and joint-smoothness assumptions as standard performative prediction, but with modified smoothness parameter $D:\Theta\to\Delta(\mathcal Z)$05, uniqueness and convergence are recovered when

$D:\Theta\to\Delta(\mathcal Z)$06

In the retention setting, larger fairness weight $D:\Theta\to\Delta(\mathcal Z)$07 yields smaller group-loss disparity at the Fair-PS point (Jin et al., 2024).

Performative stability has also been extended to environments with both endogenous and exogenous drift. In partially performative prediction, the round-$D:\Theta\to\Delta(\mathcal Z)$08 data law is

$D:\Theta\to\Delta(\mathcal Z)$09

and the relevant comparator is the time-varying stable point $D:\Theta\to\Delta(\mathcal Z)$10. The corresponding online criterion is stability regret,

$D:\Theta\to\Delta(\mathcal Z)$11

Under strong convexity, smoothness, sensitivity, and a path-length measure

$D:\Theta\to\Delta(\mathcal Z)$12

repeated risk minimization satisfies

$D:\Theta\to\Delta(\mathcal Z)$13

while repeated gradient descent achieves

$D:\Theta\to\Delta(\mathcal Z)$14

This recasts stability as a tracking problem rather than convergence to a single fixed point (Lee et al., 5 Jun 2026).

Finally, recent work develops an inferential theory for performative stability itself. In a unified single- and multi-player setting, repeated risk minimization produces estimators $D:\Theta\to\Delta(\mathcal Z)$15 satisfying

$D:\Theta\to\Delta(\mathcal Z)$16

with a recursive closed-form covariance formula involving the Jacobian of the solution map and the gradient covariance under the induced distribution. The same work shows that the estimator attains the semiparametric efficiency bound under the stated regularity conditions (Zhang et al., 3 Feb 2026). This shifts performative stability from a purely optimization-theoretic target to an estimable statistical object with uncertainty quantification.

Performative stability therefore occupies a broad middle ground between optimization, equilibrium analysis, and dynamical systems. In weakly performative regimes it is a contractive fixed point reachable by straightforward retraining; in richer settings it becomes a mixed equilibrium, a moving target, or a stability notion for control, reinforcement learning, bilevel optimization, and chance-constrained decision-making. The modern literature shows both its utility and its limits: stability can imply exact convergence and, in some regimes, optimality, but it can also coexist with polarization, undergo transitions to chaos, or become computationally intractable once feedback strength crosses the contraction threshold (Perdomo et al., 2020).