Two-timescale Derivative Free Optimization for Performative Prediction with Markovian Data

TL;DR

This paper introduces a two-timescale derivative-free optimization algorithm for performative prediction with Markovian data, achieving near-stationary solutions efficiently in a non-convex setting.

Contribution

It proposes a novel two-timescale DFO($ extbackslash lambda$) algorithm that effectively handles Markovian data in performative prediction, with theoretical guarantees.

Findings

Requires ${ m O}(1/ ext{ extbackslash epsilon}^3)$ samples for near-stationary solutions

Balances DFO updates and bias reduction through two-timescale step sizes

Numerical experiments confirm theoretical analysis.

Abstract

This paper studies the performative prediction problem where a learner aims to minimize the expected loss with a decision-dependent data distribution. Such setting is motivated when outcomes can be affected by the prediction model, e.g., in strategic classification. We consider a state-dependent setting where the data distribution evolves according to an underlying controlled Markov chain. We focus on stochastic derivative free optimization (DFO) where the learner is given access to a loss function evaluation oracle with the above Markovian data. We propose a two-timescale DFO($\lambda$) algorithm that features (i) a sample accumulation mechanism that utilizes every observed sample to estimate the overall gradient of performative risk, and (ii) a two-timescale diminishing step size that balances the rates of DFO updates and bias reduction. Under a general non-convex optimization setting, we show that DFO($\lambda$) requires ${\cal O}( 1 /\epsilon^3)$ samples (up to a log factor) to attain a near-stationary solution with expected squared gradient norm less than $\epsilon > 0$. Numerical experiments verify our analysis.