# Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality

arXiv: https://arxiv.org/abs/2207.04173 · HTML: https://mlsanity.com/paper/2207.04173/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 9 Jul 2022.

## Builds on (most cited first, 18 of 18)

- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [Strategic Classification](https://mlsanity.com/paper/1506.06980/) (2015)
- [Strategic Classification from Revealed Preferences](https://mlsanity.com/paper/1710.07887/) (2017)
- [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)
- [Strategic Classification is Causal Modeling in Disguise](https://mlsanity.com/paper/1910.10362/) (2019)
- [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)
- [Stochastic optimization with decision-dependent distributions](https://mlsanity.com/paper/2011.11173/) (2020)
- [Multiplayer Performative Prediction: Learning in Decision-Dependent Games](https://mlsanity.com/paper/2201.03398/) (2022)
- [Regret Minimization with Performative Feedback](https://mlsanity.com/paper/2202.00628/) (2022)
- [Asymptotic Optimality in Stochastic Optimization](https://mlsanity.com/paper/1612.05612/) (2016)
- [Multi-agent Performative Prediction: From Global Stability and Optimality to Chaos](https://mlsanity.com/paper/2201.10483/) (2022)
- [Decision-Dependent Risk Minimization in Geometrically Decaying Dynamic Environments](https://mlsanity.com/paper/2204.08281/) (2022)
- [Online Projected Gradient Descent for Stochastic Optimization with Decision-Dependent Distributions](https://mlsanity.com/paper/2107.09721/) (2021)
- [Gaming Helps! Learning from Strategic Interactions in Natural Dynamics](https://mlsanity.com/paper/2002.07024/) (2020)
- [Stochastic Optimization under Distributional Drift](https://mlsanity.com/paper/2108.07356/) (2021)
- [Stochastic Saddle Point Problems with Decision-Dependent Distributions](https://mlsanity.com/paper/2201.02313/) (2022)

## Built on by (most cited first, 12 of 12)

- [Performative Prediction: Past and Future](https://mlsanity.com/paper/2310.16608/) (2023)
- [Asymptotic normality and optimality in nonsmooth stochastic approximation](https://mlsanity.com/paper/2301.06632/) (2023)
- [Statistical Inference under Performativity](https://mlsanity.com/paper/2505.18493/) (2025)
- [Statistical Inference for Online Algorithms](https://mlsanity.com/paper/2505.17300/) (2025)
- [Contextual Strongly Convex Simulation Optimization: Optimize then Predict with Inexact Solutions](https://mlsanity.com/paper/2512.06270/) (2025)
- [When and How Human Curation Backfires: Preference Alignment under Multi-Model Self-Consuming Loop](https://mlsanity.com/paper/2605.29267/) (2026)
- [The radius of statistical efficiency](https://mlsanity.com/paper/2405.09676/) (2024)
- [Non-Asymptotic Convergence of Stochastic Iterative Algorithms: A Lyapunov Framework](https://mlsanity.com/paper/2605.31309/) (2026)
- [Towards Performatively Stable Equilibria in Decision-Dependent Games for Arbitrary Data Distribution Maps](https://mlsanity.com/paper/2509.02619/) (2025)
- [When Predictions Shape Reality: A Socio-Technical Synthesis of Performative Predictions in Machine Learning](https://mlsanity.com/paper/2601.04447/) (2026)
- [Performative Learning Theory](https://mlsanity.com/paper/2602.04402/) (2026)
- [Adaptive Conditional Forest Sampling for Spectral Risk Optimisation under Decision-Dependent Uncertainty](https://mlsanity.com/paper/2603.12507/) (2026)

## Abstract (arXiv)

We analyze a stochastic approximation algorithm for decision-dependent problems, wherein the data distribution used by the algorithm evolves along the iterate sequence. The primary examples of such problems appear in performative prediction and its multiplayer extensions. We show that under mild assumptions, the deviation between the average iterate of the algorithm and the solution is asymptotically normal, with a covariance that clearly decouples the effects of the gradient noise and the distributional shift. Moreover, building on the work of H'ajek and Le Cam, we show that the asymptotic performance of the algorithm with averaging is locally minimax optimal.