# Optimizing the Performative Risk under Weak Convexity Assumptions

arXiv: https://arxiv.org/abs/2209.00771 · HTML: https://mlsanity.com/paper/2209.00771/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 43 references that are in the map.
- First posted on arXiv 2 Sep 2022.

## Builds on (most cited first, 28 of 28)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [Strategic Classification](https://mlsanity.com/paper/1506.06980/) (2015)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence](https://mlsanity.com/paper/2002.10542/) (2020)
- [Differentially Private Empirical Risk Minimization Revisited: Faster and More General](https://mlsanity.com/paper/1802.05251/) (2018)
- [Outside the Echo Chamber: Optimizing the Performative Risk](https://mlsanity.com/paper/2102.08570/) (2021)
- [Stochastic Optimization for Performative Prediction](https://mlsanity.com/paper/2006.06887/) (2020)
- [Performative Prediction in a Stateful World](https://mlsanity.com/paper/2011.03885/) (2020)
- [How to Learn when Data Reacts to Your Model: Performative Gradient Descent](https://mlsanity.com/paper/2102.07698/) (2021)
- [Causal Strategic Linear Regression](https://mlsanity.com/paper/2002.10066/) (2020)
- [Stochastic optimization with decision-dependent distributions](https://mlsanity.com/paper/2011.11173/) (2020)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [A Local Convergence Theory for Mildly Over-Parameterized Two-Layer Neural Network](https://mlsanity.com/paper/2102.02410/) (2021)
- [Alternative Microfoundations for Strategic Classification](https://mlsanity.com/paper/2106.12705/) (2021)
- [Decisions, Counterfactual Explanations and Strategic Behavior](https://mlsanity.com/paper/2002.04333/) (2020)
- [From error bounds to the complexity of first-order descent methods for convex functions](https://mlsanity.com/paper/1510.08234/) (2015)
- [Regret Minimization with Performative Feedback](https://mlsanity.com/paper/2202.00628/) (2022)
- [State Dependent Performative Prediction with Stochastic Approximation](https://mlsanity.com/paper/2110.00800/) (2021)
- [Stateful Strategic Regression](https://mlsanity.com/paper/2106.03827/) (2021)
- [An Asynchronous Parallel Stochastic Coordinate Descent Algorithm](https://mlsanity.com/paper/1311.1873/) (2013)
- [Gradient methods for convex minimization: better rates under weaker conditions](https://mlsanity.com/paper/1303.4645/) (2013)
- [Asynchronous Stochastic Coordinate Descent: Parallelism and Convergence Properties](https://mlsanity.com/paper/1403.3862/) (2014)
- [Online Projected Gradient Descent for Stochastic Optimization with Decision-Dependent Distributions](https://mlsanity.com/paper/2107.09721/) (2021)
- [The restricted strong convexity revisited: Analysis of equivalence to error bound and quadratic growth](https://mlsanity.com/paper/1511.01635/) (2015)
- [Online Stochastic Optimization for Unknown Linear Systems: Data-Driven Synthesis and Controller Analysis](https://mlsanity.com/paper/2108.13040/) (2021)

## Built on by (most cited first, 5 of 5)

- [Addressing Polarization and Unfairness in Performative Prediction](https://mlsanity.com/paper/2406.16756/) (2024)
- [ProFL: Performative Robust Optimal Federated Learning](https://mlsanity.com/paper/2410.18075/) (2024)
- [When and How Human Curation Backfires: Preference Alignment under Multi-Model Self-Consuming Loop](https://mlsanity.com/paper/2605.29267/) (2026)
- [When Predictions Shape Reality: A Socio-Technical Synthesis of Performative Predictions in Machine Learning](https://mlsanity.com/paper/2601.04447/) (2026)
- [Self-Consuming Generative Models with Co-Evolving Human Preferences](https://mlsanity.com/paper/2610.09415/) (2026)

## Abstract (arXiv)

In performative prediction, a predictive model impacts the distribution that generates future data, a phenomenon that is being ignored in classical supervised learning. In this closed-loop setting, the natural measure of performance named performative risk (PR), captures the expected loss incurred by a predictive model after deployment. The core difficulty of using the performative risk as an optimization objective is that the data distribution itself depends on the model parameters. This dependence is governed by the environment and not under the control of the learner. As a consequence, even the choice of a convex loss function can result in a highly non-convex PR minimization problem. Prior work has identified a pair of general conditions on the loss and the mapping from model parameters to distributions that implies the convexity of the performative risk. In this paper, we relax these assumptions and focus on obtaining weaker notions of convexity, without sacrificing the amenability of the PR minimization problem for iterative optimization methods.