💻 computer science

Partially Performative Prediction

This paper introduces the framework of partially performative prediction, which generalizes traditional performative prediction by accounting for both endogenous distribution shifts caused by model deployment and exogenous shifts driven by external factors, while extending concepts of stability and optimality to analyze adaptive learning heuristics in such dynamic environments.

Original authors: Jaewook Lee, Tijana Zrnic

Published 2026-06-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jaewook Lee, Tijana Zrnic

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Core Problem: The Self-Fulfilling Prophecy

Imagine you are a weather forecaster. In the old days, your job was simple: look at the clouds, predict if it will rain, and be right. The weather didn’t care what you said; it just happened. This is what computer scientists call "classical prediction."

But now, imagine you are a stock market analyst. If you predict that a certain stock will go up, people might buy it, causing the price to actually go up. Your prediction changed the reality you were trying to predict. This is called "Performative Prediction." The data shifts because of your actions.

The Problem with Current Models:
Most existing research assumes one of two extremes:

Purely Performative: The only reason the data changes is because of your prediction (like the stock market example).
Purely Exogenous: The data changes for reasons completely outside your control (like the weather changing due to a hurricane), and your predictions don’t affect it at all.

The Reality:
In the real world, it’s almost always a mix of both.

Example: A credit scoring algorithm predicts who is likely to default.
- Performative part: If the bank denies loans to certain people, those people might change their behavior (or stop applying), changing the data.
- Exogenous part: The economy crashes, or a new law passes. These things change the data regardless of what the algorithm says.

This paper introduces "Partially Performative Prediction," a framework that handles this messy, real-world mix.

The Analogy: Driving a Car in a Moving Crowd

Think of training an AI model like driving a car.

The Goal: You want to stay in the center of the lane (minimize error/loss).
The Performative Shift (You): If you steer left, the other cars around you might move to avoid you. Your action changes the traffic pattern.
The Exogenous Shift (The Road): The road itself might be curving, or there might be wind pushing the car. These are forces you didn’t create.

Classical Performative Prediction assumes the road is straight and flat, and the only thing moving is the traffic reacting to you.
Classical Online Learning assumes the traffic is frozen, but the road is constantly shifting under your tires.

This Paper’s Approach: It acknowledges that both are happening. You are steering (affecting traffic), and the road is curving (external forces).

Key Concepts Explained

1. The "Moving Target"

In the old models, the goal was to find a "stable point"—a perfect model that, once deployed, wouldn’t cause the data to shift anymore. It was like finding a parking spot.

In this new "Partial" world, there is no permanent parking spot. The "perfect model" is a moving target.

Because the external world (the exogenous shift) is changing, the "best" model changes every day.
Because your model affects the world (the performative shift), your attempt to catch the target moves the target slightly.

2. Regret: How Far Behind Are You?

Since you can’t park, you can’t measure success by "did I stop?" Instead, you measure "Regret."

Regret is the difference between how well you did and how well you could have done if you had magically known the perfect model for every single day.
The paper defines two types of regret:
- Stability Regret: How far are you from the model that would be stable right now?
- Optimality Regret: How far are you from the model that would give the best results right now?

3. Path Length: How Fast is the World Changing?

The paper introduces a key metric called "Path Length."

Imagine the "perfect model" is a runner. The Path Length is how far that runner moves over time.
If the external world is calm (slow exogenous shift), the runner moves slowly. It’s easy to track.
If the external world is chaotic (fast exogenous shift), the runner sprints and zig-zags. It’s hard to track.
The paper proves that your error (regret) depends heavily on this Path Length. The more the world changes on its own, the harder it is for your algorithm to keep up.

What Did They Find? (The Results)

The authors tested several common ways to update AI models (like "Retraining" or "Gradient Descent") in this mixed environment. Here is what they discovered:

1. The Trade-off of "Alpha" ( $\alpha_t$ )
They use a variable called $\alpha$ to represent how much the external world matters vs. how much your model matters.

If $\alpha$ is low (Mostly Performative): The world is mostly reacting to you. The paper shows that standard retraining works well here. The "Path Length" is small because the world is stable unless you touch it.
If $\alpha$ is high (Mostly Exogenous): The world is changing on its own. Your model has less power to shape the data. The paper shows that retraining still works, but your error will grow if the external world changes too fast. You are essentially chasing a runaway train.

2. Retraining is Still Good (With Caveats)
They analyzed "Repeated Risk Minimization" (RRM)—which is just fancy talk for "retrain the model on the latest data."

They proved that if you retrain often enough, you can keep your regret low.
However, the "cost" of this regret is tied to how fast the external environment is drifting. If the external drift is unpredictable, your error will be higher.

3. Optimality is Harder
Finding the "Stable" point (where the model stops causing shifts) is easier than finding the "Optimal" point (the absolute best performance).

To find the Optimal point, you need to anticipate how your model will change the future.
The paper uses "Zeroth-order optimization" (essentially poking the system to see how it reacts) to find these optimal points.
They showed that even with this complex method, you can track the moving optimal target, provided the target doesn’t move too erratically.

Summary in One Sentence

This paper provides the mathematical rules for how AI models should learn when they are influencing the world they are predicting, while the world is also changing independently of them—proving that as long as the external world doesn’t change too chaotically, standard retraining methods can still keep the AI on track.

Technical Summary: Partially Performative Prediction

Problem Formulation
The paper addresses the limitations of classical performative prediction, which assumes that distribution shifts are purely endogenous—induced solely by the deployment of the predictive model $\theta$ . In realistic settings, however, data distributions shift due to both endogenous feedback (the model’s impact on the population) and exogenous factors (external drifts such as seasonal changes, policy shifts, or market forces). The authors introduce partially performative prediction, a framework that models the data distribution at time $t$ as a mixture of these two sources:
$D_t(\theta) = (1 - \alpha_t)D(\theta) + \alpha_t P_t$
where $D(\theta)$ is the performative response map, $P_t$ is an exogenous, time-varying distribution, and $\alpha_t \in [0,1]$ controls the relative strength of the exogenous shift. This setting is fundamentally non-stationary, meaning the target stable and optimal points drift over time. Consequently, the standard objectives of converging to a fixed performatively stable or optimal point are replaced by minimizing dynamic regret relative to time-varying sequences of performatively stable points ( $\theta^{PS}_t$ ) and performatively optimal points ( $\theta^{PO}_t$ ).

Methodology and Key Contributions
The authors extend the concepts of performative stability and optimality to this non-stationary setting by defining performative stability regret and performative optimality regret. These metrics compare the learner’s cumulative loss against the cumulative loss of the sequence of ideal stable or optimal models. The analysis relies on standard assumptions: the loss function $\ell(z, \theta)$ is strongly convex and smooth in $\theta$ , and the distribution map $D$ is $\epsilon$ -sensitive (Lipschitz with respect to the Wasserstein-1 distance).

The paper provides theoretical guarantees for several learning algorithms:

Repeated Risk Minimization (RRM): The authors analyze RRM, where the model is retrained on data generated by the previous model. They prove that under a relaxed sensitivity condition ( $\epsilon < \frac{\mu}{(1-\alpha_{\min})\beta_z}$ ), the stability regret is bounded by:
$\text{Reg}^{PS}_T(\text{ARRM}) = O\left(1 + \sum_{t=1}^{T-1} \Delta^{PS}_t\right)$
where $\Delta^{PS}_t = \|\theta^{PS}_t - \theta^{PS}_{t+1}\|$ is the path length of the stable points. This result highlights a tradeoff: when shifts are mostly performative ( $\alpha_t \approx 0$ ), the path length vanishes and regret is constant; when shifts are mostly exogenous ( $\alpha_t \approx 1$ ), regret depends on the non-stationarity of $P_t$ .
Gradient-Based Methods: The authors extend these bounds to Repeated Gradient Descent (RGD) and stochastic variants (Greedy and Lazy SGD). For RGD, they derive similar path-length-dependent regret bounds without requiring the stable point to lie in the interior of the parameter space, weakening previous convergence conditions. For stochastic settings, they show that Lazy SGD (which takes multiple offline steps between deployments) achieves sublinear regret under certain decay rates of the exogenous component, whereas Greedy SGD exhibits higher regret dependence on the path length.
Performative Optimality via Zeroth-Order Methods: Since computing gradients of the performative risk is often infeasible, the authors analyze zeroth-order algorithms that estimate gradients via perturbation. They propose two-deployment and single-deployment zeroth-order gradient descent methods. They prove that these algorithms achieve dynamic optimality regret bounds scaling with the path length of the optimal points ( $\sum \Delta^{PO}_t$ ). Specifically, for convex performative risk, the regret scales as $O(d\sqrt{T} \cdot (1 + \sum \Delta^{PO}_t))$ . The analysis shows that while exogenous variation makes the target harder to track (increasing path length), a larger endogenous component can make the performative risk non-convex, complicating optimization.

Results and Significance
The theoretical results establish a clear tradeoff between endogenous and exogenous shifts. A large exogenous component ( $\alpha_t$ ) can increase the path length (making the target move faster), but it also weakens the performative feedback, potentially preserving the convexity of the risk function. Conversely, strong performativity ensures the target is stationary but may induce non-convexity.

The paper validates these findings through experiments on credit scoring data and synthetic Gaussian distributions. The experiments demonstrate that:

When the exogenous component $P_t$ varies randomly, faster decay of $\alpha_t$ (suppressing the exogenous term) leads to lower stability regret, consistent with the path-length dominance in the bounds.
When $P_t$ is stationary, keeping $\alpha_t$ larger (closer to 1) yields lower regret because the path length is zero, and the algorithm benefits from the stability of the non-performative component.
Empirical regret curves align with the theoretical upper bounds, confirming the qualitative correctness of the path-length dependence.

The significance of this work lies in generalizing performative prediction to realistic, non-stationary environments. By introducing dynamic regret notions and analyzing standard learning heuristics under partial performativity, the paper provides a unified framework for understanding how algorithms adapt when distribution shifts are driven by both the learner’s actions and external forces.