Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $φ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.
We study the minimax rate of estimating a future value $μ_{t_n+h}$ of a curve $t\mapstoμ_t$ in the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\|\nabla_t^k v\|\le\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $γ_d(k+1)/(k+1+γ_d)$, $γ_d=\min(1/d,1/2)$ ($M$ the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable $W_2$-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are new. The bound interpolates a dimension-free extrapolation floor of order $\varepsilon h^{k+1}$ -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse $M^{-γ_d}$, recovering the static distribution-estimation rate as $k\to\infty$. We state the lower bound in a design-dependent form -- with a design-weighted effective sample size -- valid for arbitrary observation times, and obtain the closed-form exponent in the dense (equispaced) regime. The matching upper bound is established at $k=0$ (rate $M^{-1/(d+1)}$, $d\ge3$) and, in a translation submodel, for all $k$; for $k\ge1$ a covariant estimator attains the rate conditionally on two estimates (a comparison-geometry bias bound and an optimal-transport map-estimation rate), leaving the unconditional general-$k$ upper bound as an open problem. Numerical experiments on synthetic curved and flat families corroborate the predicted exponents.