  We study an off-policy contextual pricing problem where the seller has access
to samples of prices that customers were previously offered, whether they
purchased at that price, and auxiliary features describing the customer and/or
item being sold. This is in contrast to the well-studied setting in which
samples of the customer's valuation (willingness to pay) are observed. In our
setting, the observed data is influenced by the previous pricing policy, and we
do not know how customers would have responded to alternative prices. We
introduce suitable loss functions for this setting that can be directly
optimized to find an effective pricing policy with expected revenue guarantees,
without the need for estimation of an intermediate demand function. We focus on
convex loss functions. This is particularly relevant when linear pricing
policies are desired for interpretability reasons, resulting in a tractable
convex revenue optimization problem. We propose generalized hinge and quantile
pricing loss functions that price at a multiplicative factor of the conditional
expected valuation or a particular quantile of the prices that sold, despite
the valuation data not being observed. We prove expected revenue bounds for
these pricing policies respectively when the valuation distribution is
log-concave, and we provide generalization bounds for the finite sample case.
Finally, we conduct simulations on both synthetic and real-world data to
demonstrate that this approach is competitive with, and in some settings
outperforms, state-of-the-art methods in contextual pricing.
