  Consider a stochastic interface $h(x,t)$, described by the $1+1$
Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\geq 0$. The interface is
initially flat, $h(x,t=0)=0$, and driven by a Neumann boundary condition
$\partial_x h(x=0,t)=A$ and by the noise. We study the short-time probability
distribution $\mathcal{P}\left(H,A,t\right)$ of the one-point height
$H=h(x=0,t)$. Using the optimal fluctuation method, we show that $-\ln
\mathcal{P}\left(H,A,t\right)$ scales as $t^{-1/2} s \left(H,A t^{1/2}\right)$.
For small and moderate $|A|$ this more general scaling reduces to the familiar
simple scaling $-\ln \mathcal{P}\left(H,A,t\right)\simeq t^{-1/2} s(H)$, where
$s$ is independent of $A$ and time and equal to one half of the corresponding
large-deviation function for the full-line problem. For large $|A|$ we uncover
two asymptotic regimes. At very short time the simple scaling is restored,
whereas at intermediate times the scaling remains more general and
$A$-dependent. The distribution tails, however, always exhibit the simple
scaling in the leading order.
