Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality
This work addresses decision-dependent distribution problems in machine learning, providing theoretical guarantees for algorithms in performative prediction, but it is incremental as it builds on prior work like Hájek and Le Cam.
The paper tackles stochastic approximation for decision-dependent problems, such as performative prediction, by proving that the average iterate's deviation from the solution is asymptotically normal with a covariance separating gradient noise and distributional shift, and shows the algorithm with averaging is locally minimax optimal.
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 Hájek and Le Cam, we show that the asymptotic performance of the algorithm with averaging is locally minimax optimal.