  In this paper, we study the contextual dynamic pricing problem where the
market value of a product is linear in its observed features plus some market
noise. Products are sold one at a time, and only a binary response indicating
success or failure of a sale is observed. Our model setting is similar to
Javanmard and Nazerzadeh [2019] except that we expand the demand curve to a
semiparametric model and need to learn dynamically both parametric and
nonparametric components. We propose a dynamic statistical learning and
decision-making policy that combines semiparametric estimation from a
generalized linear model with an unknown link and online decision-making to
minimize regret (maximize revenue). Under mild conditions, we show that for a
market noise c.d.f. $F(\cdot)$ with $m$-th order derivative ($m\geq 2$), our
policy achieves a regret upper bound of $\tilde{O}_{d}(T^{\frac{2m+1}{4m-1}})$,
where $T$ is time horizon and $\tilde{O}_{d}$ is the order that hides
logarithmic terms and the dimensionality of feature $d$. The upper bound is
further reduced to $\tilde{O}_{d}(\sqrt{T})$ if $F$ is super smooth whose
Fourier transform decays exponentially. In terms of dependence on the horizon
$T$, these upper bounds are close to $\Omega(\sqrt{T})$, the lower bound where
$F$ belongs to a parametric class. We further generalize these results to the
case with dynamically dependent product features under the strong mixing
condition.
