Performative Prediction with Selective Labels

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom.

Jane: And I'm Jane, here with Lu, Meng and our in-house large language model, Lalam.

Tom: Today's paper: "Performative Prediction with Selective Labels".

Jane: The gist The work formalizes performative prediction with selective labels and shows that retraining only on observed data can misguide the retraining procedure and undermine convergence to a stable solution…

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So, we're looking at this paper today, "Performative Prediction with Selective Labels." Basically, it tackles a problem where how you train a model actually changes the data you see later on because of what the model predicts.

Jane: That sounds really abstract. Can you lay out what that means for someone just driving or cooking? What's the core idea here?

Tom: The core idea is this framework: every model we build induces its own distribution of features and labels, say D(theta). We define a risk function J(phi, theta) based on that. Now, the main result they show is that simply retraining repeatedly on new data—what they call repeated risk minimization or RRM—might get stuck in a solution that's stable only for the distribution it *created*, not necessarily for the real world you are trying to predict.

Lu: The paper formalizes this by defining a performatively optimal solution theta P O, which minimizes the risk on its induced distribution D(theta), and then they talk about performatively stable solutions, theta P S, which is robust to that induced distribution shift.

Meng: So if I train a model on my observed data and keep retraining, the paper suggests I might not be converging to the best possible solution for the real population because my training is only based on what I've seen so far. That sounds like a problem when we are trying to deploy something in production.

Jane: Exactly. The problem they focus on is "selective labels," where you don't see the label for every single person in the population, just the ones who were accepted or observed.

Tom: Right, and they point out that existing analyses usually assume we have access to all features and labels after deployment, but that's not always true. They call this "The Selective Performative Gap," showing that when observation is deterministic and based on a threshold, retraining with performativity can lead models arbitrarily far from the true stable solution theta P S when compared to RRM.

Lu: The naive fix they rule out is just using only the data from the accepted population, because that would result in a model that can produce "models arbitrarily far from the true performatively stable solution theta P S" when compared to RRM.

Meng: So, if we only train on our accepted data, we end up with a model that's way off where it should be? That makes sense because the acceptance criteria are based on the performance of models trained under different assumptions.

Jane: And to fix this gap, they introduce something called Robust Repeated Risk Minimization. This is an optimization approach inspired by partial identification techniques that uses a worst-case objective based on knowledge of a confidence interval on the probability of a positive label.

Tom: They formalize this in Definition four point two as the Worst-Case Loss, phi(x, theta), which defines it as the worst-case loss for sample x under distribution D(phi) <ref:2610.08272#pg1>. The Robust RRM procedure then uses that to update theta t+one = G(theta t), where G is defined by minimizing that worst-case objective.

Lu: Theorem four point six shows that one iteration performed both in G and G results in two models with a limited distance, which they link to the gap between gradients of the loss function on y equals zero and y equals one and the width of limits l and u over the induced distribution D(phi) <ref:2610.08272#pg1>.

Jane: That sounds like they are trying to keep those two iterates close together even when we only have partial information about the labels. What about their theoretical guarantees?

Tom: They prove that Robust RRM approximates the true performatively stable solution theta P S within a bounded distance. Theorem four point seven shows that for the robust iterates theta t+one equals G(theta t), there is a bound on the distance to the stable solution: theta t - theta P S squared is less than or equal to eta t theta zero - theta P S squared plus one minus eta t kappa, where kappa is some constant.

Meng: A bound that depends on how many steps we take and how far our starting point was from the stable solution. That's a concrete guarantee for deployment, I guess. But what about getting better information over time?

Lu: They also show in Theorem four point nine how to improve the knowledge of the label distribution by leveraging previously accepted samples and their feature sensitivity, leading to a bound on the gap between G and G that depends on the sensitivity constant Lx and coverage of observation <ref:2610.08272#pg1>.

Jane: So they aren't just static; they can use what they learn from accepted samples to refine how much information they need next. That sounds like a way to iteratively build better estimates of the label distribution as you go.

Tom: They also address robustness to misspecification of those sensitivity constants, which is important because estimating Lx and Ltheta usually requires some domain knowledge or structure. They show that under correct logistic specification and maximum likelihood estimation conditions, an asymptotic one-sided confidence upper bound for Lx can be derived using the Delta method.

Meng: So even if we don't know the exact sensitivity constant L x, as long as we use a valid conservative upper bound, it still works? That’s reassuring because domain knowledge is always fuzzy in reality.

Lu: They suggest that any valid conservative upper bound is sufficient rather than identifying the exact model, and this approach remains valid even when Ltheta equals zero, as it requires estimating one alpha theta for each deployed theta.

Jane: It sounds like they've found a way to keep things stable even when we only have partial knowledge of the label distribution and imperfect estimates of those sensitivity constants. So, what's the practical validation?

Tom: They ran experiments in a lending application with fairness regularization to validate this worst-case approach. The results show that Robust RRM closely matches the performance of RRM with complete label access.

Meng: So it actually works when we compare it against the standard method where we have full labels? That's a big deal for deployment, because full labels are often an unrealistic assumption.

Jane: They also found that using learned limits l and u tightens the distance to theta P S, and they showed that underestimating Lx produced a smaller empirical distance to the stable solution.

Tom: The final result they show is that Robust RRM approximated the Oracle agent’s performance and had the smallest distance to the stable solution. It essentially matched what you'd get if you had perfect knowledge of everything.

Lu: In conclusion, they show this robust approach leverages a strongly convex upper-bound loss that does not depend on the true label, and it approximates the stable point up to a distance related to the acceptance rate, convexity of the loss function, and partial knowledge of the label distribution.

Jane: It sounds like they've built a method that is resilient to those kinds of real-world data limitations we encounter every day. What does this mean for how we deploy these kinds of predictive systems?

Tom: It means that even if your training data doesn't perfectly reflect the final deployment distribution, as long as you use this robust optimization procedure, you have a good chance of landing near the stable point.

Meng: I see it in terms of cost and maintenance. If we can rely on this method to stay within a bounded distance of stability without needing perfect label access, that simplifies our validation pipeline significantly.

Lu: And looking ahead, they show how to improve the knowledge of the label distribution by leveraging previously accepted samples and their feature sensitivity, which is a useful direction for future work in representation learning.

Jane: It's interesting how they tie it all together—from the initial problem of selective labels to a robust optimization that uses historical data to guide its learning. That’s a lot of mechanics packed into this paper on Performative Prediction with Selective Labels.

The paper's summary: Tom: So, to wrap up what we just discussed about selective labels and performative prediction, here's the gist: this work formalizes how a model's training process actually changes the data you see later on because of its predictions.

Jane: It’s like when you train a car and then only ever let it drive on perfectly smooth roads. Eventually, it learns to handle bumps, but if you suddenly put it on rough terrain, that initial training might not prepare it for that shift in the world.

Lu: Exactly. The paper introduces this concept of performative prediction where every model induces its own distribution D(theta), and then we have this risk function J(phi, theta) that measures how risky the model is under a specific scenario.

Meng: So, it’s not just about minimizing error on the data you feed it now. It’s about minimizing risk across all the scenarios your model might actually face later, even if those scenarios aren't perfectly represented in your current training set.

Tom: Right. And they show that simply retraining repeatedly on observed data can really mislead you, because the procedure itself can undermine convergence to a stable solution that works for the real deployment distribution.

Jane: It’s like if you only test your new software on ten specific kinds of users, and then you keep updating it based only on those ten users, but the actual customers are mostly from a completely different group.

Lu: They call this "The Selective Performative Gap," which shows that when labeling is deterministic and threshold-based, retraining with performativity can cause models trained on one distribution D(theta) to diverge arbitrarily from the stable solution theta P S when compared to retraining on a different distribution D˜(theta).

Meng: So, the naive fix they point out—just using only the data from the accepted population—it doesn't work. It actually results in a model that can be way off from where it should be compared to this robust repeated risk minimization procedure.

Tom: Because if you only look at what’s observed, you might end up with models arbitrarily far from that true performatively stable solution theta P S when measured against the robust RRM process.

Jane: To fix this, they bring in Robust Repeated Risk Minimization, which is a more careful optimization approach. Instead of just looking at the current data distribution, it uses a worst-case objective based on what we know about the confidence interval for positive labels.

Lu: They define this as Definition four point two, the Worst-Case Loss phi(x, theta), which accounts for both observed and unobserved samples using limits l and u over the induced distribution D(phi). This worst-case loss is strongly convex.

Meng: I like that part about the worst-case loss. It means we’re not assuming perfect knowledge of everything, which is a lot more realistic for production systems than assuming perfect data access all the time.

Tom: They prove that this Robust RRM procedure stays within a bounded distance of the true performatively stable solution theta P S. Theorem four point seven gives us a bound on that distance, depending on the number of iterations and how far we started from theta P S.

Jane: So, the main point for us listening is that even when you’re missing labels or only see a subset of data, this method keeps you anchored near where the model actually stabilizes in the real world.

Lu: They also show in Theorem four point nine how to get better about that information over time by leveraging previously accepted samples and their feature sensitivity, which tightens the gap between successive iterations G and G.

Meng: If we can use what we've already seen to refine our understanding of the label distribution, that makes the iterative training process much more efficient at scale.

Tom: And they show that even if our estimates for these sensitivity constants are slightly off—for instance, underestimating Lx—it actually leads to a smaller empirical distance to the stable solution.

Jane: It’s a lot of mechanics, but what this means for you is that we can build systems that don't get stuck in local optima just because the data we see isn't perfectly representative of what the model will actually encounter.

Lu: They essentially show how to use a strongly convex upper-bound loss that doesn’t depend on the true label, and it approximates the stable point up to a distance determined by things like your acceptance rate and how much you already know about the labels.

Meng: It sounds like a solid method for deployment because it has provable bounds on how far we might be from stability even when we have imperfect information.

Tom: The final result is that this Robust RRM approximates the Oracle agent’s performance, meaning it gets as good as you could get if you had perfect knowledge of everything.

Jane: So, if you’re building something where the training data won't perfectly match your real-world deployment data—like in finance or autonomous systems—this framework gives us a concrete way to stay close to the best possible outcome.

Lu: And they even show how to extend this idea to multi-class problems by defining a worst-case loss using multiple maximum operations over class distributions, which is useful for more complex scenarios.

Meng: I’m interested in the extension part. How much extra computational cost does that multi-class formulation add compared to binary classification? That’s where I need numbers to tell my team if it's worth the effort.

The paper's improvements: Tom: So, we’ve talked about how this framework works and what it means for stability under selective labels, and now we’re looking at the practical improvements they suggest for making this method even better.

Jane: Basically, the paper isn't just stopping there; they show ways to make those bounds tighter and more informative by using what you already have.

Lu: They introduce a way to improve the label distribution limits themselves by leveraging previously accepted samples and their feature sensitivity. This is formalized in Theorem four point nine, where the limits l and u are updated based on observations from earlier models.

Meng: So instead of guessing how many labels we need, we use the data we’ve already collected to refine our estimate of what’s missing. That makes sense for a system that has to learn on the fly.

Tom: It also shows robustness when you don't have perfect domain knowledge about those sensitivity constants, like Lx or Ltheta. They found that under certain conditions, even if we underestimate Lx, the empirical distance to the stable solution gets smaller.

Jane: That’s really important for real-world engineering because it means we don't need perfect mathematical definitions of how sensitive our model is to label changes before we can start retraining effectively.

Lu: They also tackle outcome performativity—that tricky situation where the model parameters directly affect the resulting label distribution. They use Definition four point seven to define dynamic limits that depend on things like the strong convexity of the loss function and how much you reject a prediction.

Meng: Outcome performativity is a big headache at scale because it means your training data is fundamentally changing every time you update your model, which makes convergence much harder to guarantee.

Tom: But by defining those dynamic limits, they can calculate a one-step gap that depends on the rejection rate and the dependence of the loss gradient on y. It gives us a more realistic picture of how much drift to expect in just one step.

Jane: So, if you're dealing with systems where model predictions directly influence the data stream itself, this method allows you to calculate a much tighter bound on your error rather than relying on a fixed assumption.

Lu: They also mention extending this framework to multi-class problems by using Definition five point zero. This involves defining a worst-case loss that uses multiple maximum operations over class distributions, which helps bound the risk when you're only observing a subset of possible outcomes.

Meng: Multi-class adds complexity, but bounding the risk using those multiple maximums seems like it gives us the necessary mathematical guarantee we need for production systems.

Tom: The overall message is that this robust approach doesn’t just work; it can be improved by being smarter about how much information you use and how much uncertainty you account for.

Jane: It means for anyone building a predictive system, the future involves not just training on data, but intelligently using your historical performance to guide the training process itself.

Lu: And this ties into their exploration of better representations; they show how leveraging previously accepted samples can lead to better feature sensitivity estimates, which is a direction for improving the underlying model structure.

Meng: If we can use those learned sensitivities to dynamically adjust our bounds, that reduces the guesswork in deploying these complex models.

Tom: So, yeah, this isn't just a static learning algorithm; it’s an iterative optimization procedure designed to stay close to stability even when you’re operating under real-world label constraints.

Jane: And next time we talk about this, we can look at how these improvements apply specifically to those complex multi-class scenarios they mentioned.

Conclusion: Tom: So we’ve covered how they tackle those selective labels using worst-case objectives and robust retraining to keep models stable under real deployment conditions.

Jane: It boils down to this: they show that you don't need perfect data access to get near a good solution if you use the right optimization procedure.

Lu: The core idea of Performative Prediction with Selective Labels is formalizing how a model’s training process induces its own distribution, D(theta), and then defining a risk function J(phi, theta) based on that.

Meng: So, it’s about making sure the training doesn't just optimize for the data we see right now but also stays safe when it hits the real world.

Tom: Exactly. And they prove that Robust RRM approximates the true stable solution theta P S within a bounded distance, which is a solid number to work with.

Jane: It changes things for us because it gives us a concrete way to validate models where we can’t be one hundred percent sure about every single label in the whole population.

Lu: And they even show how leveraging previous accepted samples helps tighten the gap between iterations G and G, which is key for improving knowledge over time.

Meng: If you can get better estimates from past deployments, that cuts down on the iteration count we need to keep running models at scale.

Tom: It’s about practical stability under uncertainty in deployment settings.

Jane: So, if you’re looking at a system where labels are sparse or only observed for a subset of users, this paper gives you the mathematical framework to stay anchored near the best possible outcome.

Lu: They also extended it to multi-class problems using worst-case loss definitions with multiple maximum operations over class distributions.

Meng: That extension is interesting because it means we can apply this same robust thinking to more complex classification tasks where there are many potential outcomes.

Tom: Yeah, so the paper is about building a resilient training loop that accounts for real-world data gaps and prediction uncertainty.

Jane: It’s a lot of mechanics, but it shows that even when the input distribution isn't perfect, we can still find a stable point within reach.

Lu: The main implication is that this method provides a way to get reliable performance bounds without needing full access to the true label distribution.

Meng: For production, that means fewer surprises when we roll out new models because we have a better guarantee on how far they might drift from what’s actually stable.

Tom: So, taking the paper Performative Prediction with Selective Labels, the bottom line is that you can achieve stability under selective labels by using robust optimization instead of just naive repeated risk minimization.

Jane: And for anyone building predictive systems in the real world, this means we can stop worrying so much about having perfect data and start focusing on a procedure that handles the gaps intelligently.