SPRINT: Stochastic Performative Prediction With Variance Reduction

arXiv:2509.17304 · cs.LG · Submitted 2025-09-22 · Read on arXiv

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

Tom: Hi, I'm Tom.

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

Tom: We're looking at the paper: "SPRINT: Stochastic Performative Prediction With Variance Reduction".

Jane: Here's the gist of the paper. The gist The authors propose a new algorithm called stochastic performative prediction with variance reduction (SPRINT) that achieves convergence to an SPS solution at a rate of O(1/T) while featuring an error neighborhood independent of the variance of the stochastic gradients.

Tom: Problem Formulation and Context Performative prediction (PP) is an algorithmic framework for optimizing machine learning models where the model’s deployment affects the distribution of the data it is trained on The goal in PP is to minimize the performative risk (PR), defined as θ PO = argmin θ V(θ) def = Ez∼D(θ)

l(z; θ): Since the data distribution itself depends on the model being optimized, finding theta PO is often intractable Existing work focuses on finding a performative stable (PS) solution θ PS to minimize the decoupled performative risk J (θ; θ PS) in a form that allows for iterative optimization The standard SGD-GD approach is given by θt+1 = θt − γt+1∇l(zt+1; θt), zt+1 ∼ D(θt) While SGD-GD can converge to a unique PS solution under strong convexity and smoothness assumptions, recent work established convergence guarantees in the non-convex setting However, these prior results are based on the restricted bounded variance assumption in stochastic gradient estimates and yield convergence bounds with a non-vanishing error neighborhood that scales with the variance The SPRINT Algorithm SPRINT is proposed as a new algorithm inspired by the SVRG approach, where the total number of T update iterations is divided into S epochs, each consisting of m iterations At the end of each epoch s + 1, the algorithm stores a snapshot of the full gradient ∇J (θe s; θe s) During each update iteration k of epoch s, a variance reduction step computes v s+1 k = ∇l(zik; θs+1 k) − ∇l(zik; θe s) + ∇J (θe s; θe s) This step offsets the variance of the current stochastic gradient estimate with the variance of the previous stochastic gradient estimate The key idea is that this step adds the expected gradient ∇J (θe s; θe s) back to ensure that the overall expectation remains stable Convergence Analysis and Results The convergence analysis establishes the convergence of SPRINT to an SPS solution with a variance-independent fast convergence rate of O(1/T) The main result is that Algorithm 1 achieves a convergence rate of O(1/T) and a variance-independent error neighborhood of O(ϵ squared + ϵ 4) This improves upon previous results in three key aspects: the O(1/T) convergence rate is faster than that of Li and Wai

2024: , we eliminate Assumption 3.4, and the error neighborhood is O(ϵ squared + ϵ 4), which is guaranteed to be smaller than that of the SGD-GD-Based PP algorithm in

Li and Wai, 2024: Experimental Validation Extensive numerical experiments were conducted on three real-world datasets with non-convex models to verify the effectiveness of SPRINT Experiments on the Credit dataset and CIFAR-10 dataset demonstrated that SPRINT achieves lower training loss and converges faster compared to SGD-GD in all simulated performative effects cases Specifically, for the Credit dataset, SPRINT achieves lower training loss and converges faster compared with the SGD-GD in all 3 cases and the advantages are quite prominent when performative effects are larger (i.e., when α = 0.2 and 0.

Paper summary: Tom: So we're looking at this paper called "SPRINT: Stochastic Performative Prediction With Variance Reduction." Basically, it tackles performative prediction, which is this framework for optimizing machine learning models where how you deploy the model actually changes what kind of data you're training on.

Jane: That sounds really tricky. So instead of just having a fixed set of data to train on, the model itself influences the data distribution. The paper claims that finding a stable point in this setup, called a stationary performative stable solution, is much harder than in regular machine learning tasks because of those shifts the model induces <ref:2509.17304#pg1>.

Lu: Right. And while there's been some work on repeated gradient descent or greedy stochastic gradient descent trying to find these solutions, most of that research assumed a strongly convex loss function until this paper showed something different <ref:2509.17304#pg1>.

Meng: What does that mean for us in production? If we can't assume strong convexity, the convergence guarantees we usually rely on might not hold up when the data distribution starts shifting mid-training. That’s a real headache for deploying models reliably <ref:2509.17304#pg3>.

Lalam: Exactly. The paper notes that prior results were based on this restricted bounded variance assumption in stochastic gradient estimates, and they found convergence bounds with an error neighborhood that scales with the variance of those gradients <ref:2509.17304#pg1>.

Tom: That's where SPRINT comes in as a way to improve things. They propose this new algorithm, called stochastic performative prediction with variance reduction, SPRINT <ref:2509.17304#pg3>.

Jane: The main claim here is that they establish convergence to an SPS solution at a rate of O(one over T), which is quite fast, and crucially, the resulting error neighborhood becomes independent of the variance of those stochastic gradients <ref:2509.17304#pg3>.

Lu: So they achieved this by dividing the total number of update iterations into multiple epochs and storing a snapshot of the full gradient at each end <ref:2509.17304#pg3>. The key idea is that this snapshot reduces the variance of the current stochastic gradient estimate, with a slight bias introduced <ref:2509.17304#pg3>.

Meng: A slight bias sounds risky when you're shipping models to real traffic. If that bias isn't well-controlled, it could lead to instability when the model is actually in deployment <ref:2509.17304#pg3>.

Lalam: The analysis shows that SPRINT achieves this variance-independent error neighborhood of O of epsilon squared plus epsilon four, which they say is smaller than what you'd see with a standard SGD-GD based PP algorithm <ref:2509.17304#pg3>.

Tom: So, the numbers look promising. O of one over T convergence rate and that variance independence really addresses the limitation from previous work <ref:2509.17304#pg3>. This moves us past those prior assumptions about strong convexity <ref:2509.17304#pg1>.

Paper summary: Jane: What does this mean for someone who just listens to the show? It means that when we try to optimize a model where the data changes because of the model, we can use an algorithm that converges faster and doesn't have its error neighborhood exploding based on how noisy our gradient estimates are <ref:2509.17304#pg3>.

Lu: From a research standpoint, it’s significant because they successfully developed the first variance-reduced performative prediction framework inspired by the SVRG approach in this context <ref:2509.17304#pg3>. It shows a path forward for designing algorithms in these model-dependent distribution shift settings.

Meng: But from an engineering standpoint, I'm still wondering how much overhead this variance reduction step adds to the actual training time on large models. If the snapshot storage and re-gradient calculations become too slow, it won't be deployable at scale <ref:2509.17304#pg3>.

Lalam: The paper doesn't give us a direct runtime comparison, but they show that it outperforms SGD-GD in both convergence rate and stability on real datasets with non-convex models <ref:2509.17304#pg3>.

Tom: So, the experiments on the Credit dataset and CIFAR-ten dataset seem to back up the theory. They're showing SPRINT achieves lower training loss and converges faster compared to SGD-GD in all simulated performative effects cases <ref:2509.17304#pg3>.

Jane: It’s good to see concrete results on real datasets, not just theoretical bounds. It shows that this variance reduction actually helps the model learn better under these complex conditions <ref:2509.17304#pg3>.

Lu: The implication is that we might be able to build more robust and reliable AI systems where the data environment isn't static, which is a huge step in moving beyond traditional machine learning paradigms <ref:2509.17304#pg2>.

Meng: I still need to see if this translates smoothly into a system that handles real-world drift without needing constant manual tuning of the variance reduction parameters <ref:2509.17304#pg3>.

Lalam: The paper itself suggests that this approach addresses the challenge of designing new variance-reduced SGD-GD-based algorithms for performative prediction <ref:2509.17304#pg3>.

Tom: So, to wrap up, "SPRINT: Stochastic Performative Prediction With Variance Reduction" introduces an algorithm that achieves convergence to a stationary performative stable solution at a rate of O of one over T <ref:2509.17304#pg3>. It also achieves an error neighborhood that is independent of the variance of the stochastic gradients <ref:2509.17304#pg3>.

Jane: In simpler terms, this means we have a method to optimize models where the data changes based on the model, and we can do it faster and with more predictable error bounds than before <ref:2509.17304#pg3>.

Lu: It’s about creating a variance-reduced framework specifically for performative prediction that is inspired by methods like SVRG <ref:2509.17304#pg3>. This opens up new design space for handling model-dependent distribution shifts <ref:2509.17304#pg2>.

Meng: From an engineering view, it suggests a path to more stable optimization in complex, dynamic environments, but we still need to confirm that the runtime cost of storing and recalculating those snapshots is manageable for large-scale deployment <ref:2509.17304#pg3>.

Lalam: The paper itself establishes this method as the first variance-reduced performative prediction framework in this area <ref:2509.17304#pg3>.

Conclusion: Tom: So we're wrapping up on SPRINT, which stands for Stochastic Performative Prediction With Variance Reduction <ref:2509.17304#pg3>.

Jane: It’s a lot of jargon for what they’re really saying is a new way to make models that handle changing data better.

Tom: Right. The authors are tackling performative prediction, which is this setup where the model you deploy actually changes the data it sees during training.

Lu: They're looking for this stationary performative stable solution, or SPS solution, which is just a steady point in that whole process.

Jane: So, instead of just guessing where to stop training because things are shifting, they have an algorithm that finds a stable spot.

Tom: The big thing is they get convergence at a rate of one over T and their error neighborhood doesn't depend on the variance of those random gradient estimates.

Lu: That variance independence is key because it means the stability doesn't break down if our gradients are noisy, which is usually the problem in these non-convex settings.

Jane: For someone just listening, that means when you use this approach, you don't have to worry as much about how much noise your training data adds to the final result.

Tom: The experiments on real datasets like Credit and CIFAR-ten show it actually trains faster and achieves lower loss compared to standard SGD-GD.

Lu: So they’ve shown that this variance reduction technique works in practice for non-convex models, which is a big step past just assuming the loss function is simple and smooth.

Jane: It really moves the goalposts for how we think about training AI where the environment itself is dynamic.

Tom: And while it's promising, Meng’s kind of asking about the actual overhead. How much extra computation does this variance reduction step add when you’re trying to deploy something on a server?

Lu: Yeah, that’s a fair point. The implementation details of storing those full gradient snapshots will determine if this is practical for real-world, large-scale deployment.

Tian Xie, Ding Zhu, Jia Liu, Mahdi Khalili, Xueru Zhang

The Ohio State University

cs.LG

Submitted: 2025-09-22

Updated: 2026-10-17

Comments: Accepted by Transactions on Machine Learning Research (TMLR), 2026

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 82/100

The gist: The gist The authors propose a new algorithm called stochastic performative prediction with variance reduction (SPRINT) that achieves convergence to an SPS solution at a rate of O(1/T) while

Key concepts

Performative Prediction (PP): This framework optimizes a machine learning model where the data it trains on changes based on how the model is deployed. The goal is to find a set of parameters that minimizes 'performative risk,' which accounts for this distribution shift, making it more complex than standard ML.
Stationary Performative Stable (SPS) Solution: This is the desired stable point in PP optimization. It represents a parameter setting where the model's deployment no longer causes significant shifts in the training data distribution. Finding this specific solution is challenging because of these induced distribution changes.
Variance Reduction Step: SPRINT uses a variance reduction technique inspired by SVRG to stabilize stochastic gradient estimates. It computes an update that subtracts the noise from the current stochastic gradient using information from a stored full gradient snapshot, effectively offsetting the variance in each iteration.

Terminology

Distribution Shift — 1× in this paper · explained in 11 episodes
Distribution Shifts — 1× in this paper · explained in 2 episodes
Machine Learning (ML) — 1× in this paper · explained in 2 episodes

Summary

The gist The authors propose a new algorithm called stochastic performative prediction with variance reduction (SPRINT) that achieves convergence to an SPS solution at a rate of O(1/T) while featuring an error neighborhood independent of the variance of the stochastic gradients.

Problem Formulation and Context

Performative prediction (PP) is an algorithmic framework for optimizing machine learning models where the model’s deployment affects the distribution of the data it is trained on The goal in PP is to minimize the performative risk (PR), defined as θ PO = argmin θ V(θ) def = Ez∼D(θ)[l(z; θ)] Since the data distribution itself depends on the model being optimized, finding theta PO is often intractable Existing work focuses on finding a performative stable (PS) solution θ PS to minimize the decoupled performative risk J (θ; θ PS) in a form that allows for iterative optimization The standard SGD-GD approach is given by θt+1 = θt − γt+1∇l(zt+1; θt), zt+1 ∼ D(θt) While SGD-GD can converge to a unique PS solution under strong convexity and smoothness assumptions, recent work established convergence guarantees in the non-convex setting However, these prior results are based on the restricted bounded variance assumption in stochastic gradient estimates and yield convergence bounds with a non-vanishing error neighborhood that scales with the variance

The SPRINT Algorithm

SPRINT is proposed as a new algorithm inspired by the SVRG approach, where the total number of T update iterations is divided into S epochs, each consisting of m iterations At the end of each epoch s + 1, the algorithm stores a snapshot of the full gradient ∇J (θe s; θe s) During each update iteration k of epoch s, a variance reduction step computes v s+1 k = ∇l(zik; θs+1 k) − ∇l(zik; θe s) + ∇J (θe s; θe s) This step offsets the variance of the current stochastic gradient estimate with the variance of the previous stochastic gradient estimate The key idea is that this step adds the expected gradient ∇J (θe s; θe s) back to ensure that the overall expectation remains stable

Convergence Analysis and Results

The convergence analysis establishes the convergence of SPRINT to an SPS solution with a variance-independent fast convergence rate of O(1/T) The main result is that Algorithm 1 achieves a convergence rate of O(1/T) and a variance-independent error neighborhood of O(ϵ squared + ϵ 4) This improves upon previous results in three key aspects: the O(1/T) convergence rate is faster than that of Li and Wai [2024], we eliminate Assumption 3.4, and the error neighborhood is O(ϵ squared + ϵ 4), which is guaranteed to be smaller than that of the SGD-GD-Based PP algorithm in [Li and Wai, 2024]

Experimental Validation

Extensive numerical experiments were conducted on three real-world datasets with non-convex models to verify the effectiveness of SPRINT Experiments on the Credit dataset and CIFAR-10 dataset demonstrated that SPRINT achieves lower training loss and converges faster compared to SGD-GD in all simulated performative effects cases Specifically, for the Credit dataset, SPRINT achieves lower training loss and converges faster compared with the SGD-GD in all 3 cases and the advantages are quite prominent when performative effects are larger (i.e., when α = 0.2 and 0.

Improvements for AI systems

Variance independent convergence for non-convex training systems: The SPRINT algorithm achieves a convergence rate of O(1/T) with an error neighborhood that is independent of the stochastic gradient variance, unlike previous work which scales with variance. This allows AI systems to achieve stable performance guarantees even when the stochastic gradient estimates have high or unknown variance.
Faster convergence in model-induced distribution shift scenarios: SPRINT outperforms SGD-GD in both convergence rate and stability on real datasets, demonstrating accelerated optimization for machine learning models where deployment alters the data distribution. This means training AI models under performative effects will reach a stationary performative stable solution much faster than standard methods.
Improved resource utilization through reduced complexity: The Incremental First-Order Oracle (IFO) complexity of SPRINT is O((n alpha + n 1 + alpha 2) delta), which is superior to SGD-GD when the variance parameter sigma squared zero is large or does not exist. This leads to lower computational requirements for achieving a desired level of performance in complex AI tasks.
Robustness against strategic adversarial behaviors: By handling non-convex loss functions and distribution shifts, SPRINT enables the development of more robust models against human manipulation or dynamic data changes, as seen in experiments where SPRINT maintains better training loss under varying levels of strategic behavior (alpha).
Efficient training for deep learning architectures: The algorithm is successfully applied to train complex models like MLP and CNN architectures on large datasets such as CIFAR-10, showing faster convergence compared to SGD-GD, which improves the practical applicability of optimization methods in deep learning.

Abstract

Performative prediction (PP) is an algorithmic framework for optimizing machine learning (ML) models where the model's deployment affects the distribution of the data it is trained on. Compared to traditional ML with fixed data, designing algorithms in PP converging to a stable point -- known as a stationary performative stable (SPS) solution -- is more challenging than the counterpart in conventional ML tasks due to the model-induced distribution shifts. While considerable efforts have been made to find SPS solutions using methods such as repeated gradient descent (RGD) and greedy stochastic gradient descent (SGD-GD), most prior studies assumed a strongly convex loss until a recent work established O(1/sqrt T) convergence of SGD-GD to SPS solutions under smooth, non-convex losses. However, this latest progress is still based on the restricted bounded variance assumption in stochastic gradient estimates and yields convergence bounds with a non-vanishing error neighborhood that scales with the variance. This limitation motivates us to improve convergence rates and reduce error in stochastic optimization for PP, particularly in non-convex settings. Thus, we propose a new algorithm called stochastic performative prediction with variance reduction (SPRINT) and establish its convergence to an SPS solution at a rate of O(1/T). Notably, the resulting error neighborhood is independent of the variance of the stochastic gradients. Experiments on multiple real datasets with non-convex models demonstrate that SPRINT outperforms SGD-GD in both convergence rate and stability.