State Dependent Performative Prediction with Stochastic Approximation

TL;DR

This paper introduces a state-dependent stochastic approximation framework for performative prediction, demonstrating convergence to stable solutions with finite-time guarantees and validating results through numerical experiments.

Contribution

It models performative prediction as a state-dependent stochastic approximation with biased gradients and provides a novel finite-time convergence analysis.

Findings

Expected squared distance decreases as O(1/k)

Algorithm converges to performative stable solution

Numerical experiments confirm theoretical results

Abstract

This paper studies the performative prediction problem which optimizes a stochastic loss function with data distribution that depends on the decision variable. We consider a setting where the agent(s) provides samples adapted to the learner's and agent's previous states. The said samples are used by the learner to optimize a loss function. This closed loop algorithm is studied as a state-dependent stochastic approximation (SA) algorithm, where we show that it finds a fixed point known as the performative stable solution. Our setting models the unforgetful nature and the reliance on past experiences of agent(s). Our contributions are three-fold. First, we demonstrate that the SA algorithm can be modeled with biased stochastic gradients driven by a controlled Markov chain (MC) whose transition probability is adapted to the learner's state. Second, we present a novel finite-time performance analysis of the state-dependent SA algorithm. We show that the expected squared distance to the performative stable solution decreases as ${\cal O}(1/k)$, where $k$ is the iteration number. Third, numerical experiments are conducted to verify our findings.