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

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(λ) 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(λ) requires cal O( 1 /ε3) samples (up to a log factor) to attain a near-stationary solution with expected squared gradient norm less than ε > 0. Numerical experiments verify our analysis.