  This paper studies a risk minimization problem with decision dependent data
distribution. The problem pertains to the performative prediction setting where
a trained model can affect the outcome that the model estimates. Such
dependency creates a feedback loop that influences the stability of
optimization algorithms such as stochastic gradient descent (SGD). We present
the first study on performative prediction with smooth but possibly non-convex
loss. We analyze a greedy deployment scheme with SGD (SGD-GD). Note that in the
literature, SGD-GD is often studied with strongly convex loss. We first propose
the definition of stationary performative stable (SPS) solutions through
relaxing the popular performative stable condition. We then prove that SGD-GD
converges to a biased SPS solution in expectation. We consider two conditions
of sensitivity on the distribution shifts: (i) the sensitivity is characterized
by Wasserstein-1 distance and the loss is Lipschitz w.r.t.~data samples, or
(ii) the sensitivity is characterized by $\chi^2$-divergence and the loss is
bounded. In both conditions, the bias levels are proportional to stochastic
gradient's variance and sensitivity level. Our analysis is extended to a lazy
deployment scheme where models are deployed once per several SGD updates, and
we show that it converges to a bias-free SPS solution. Numerical experiments
corroborate our theories.
