  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.

Optimization and Control