  We investigate the problem of performing logistic regression on data
collected from privacy-sensitive sellers. Since the data is private, sellers
must be incentivized through payments to provide their data. Thus, the goal is
to design a mechanism that optimizes a weighted combination of test loss,
seller privacy, and payment, i.e., strikes a balance between multiple
objectives of interest. We solve the problem by combining ideas from game
theory, statistical learning theory, and differential privacy. The buyer's
objective function can be highly non-convex. However, we show that, under
certain conditions on the problem parameters, the problem can be convexified by
using a change of variables. We also provide asymptotic results characterizing
the buyer's test error and payments when the number of sellers becomes large.
Finally, we demonstrate our ideas by applying them to a real healthcare data
set.
