  We consider high dimensional $M$-estimation in settings where the response
$Y$ is possibly missing at random and the covariates $\mathbf{X} \in
\mathbb{R}^p$ can be high dimensional compared to the sample size $n$. The
parameter of interest $\boldsymbol{\theta}_0 \in \mathbb{R}^d$ is defined as
the minimizer of the risk of a convex loss, under a fully non-parametric model,
and $\boldsymbol{\theta}_0$ itself is high dimensional which is a key
distinction from existing works. Standard high dimensional regression and
series estimation with possibly misspecified models and missing $Y$ are
included as special cases, as well as their counterparts in causal inference
using 'potential outcomes'.
  Assuming $\boldsymbol{\theta}_0$ is $s$-sparse ($s \ll n$), we propose an
$L_1$-regularized debiased and doubly robust (DDR) estimator of
$\boldsymbol{\theta}_0$ based on a high dimensional adaptation of the
traditional double robust (DR) estimator's construction. Under mild tail
assumptions and arbitrarily chosen (working) models for the propensity score
(PS) and the outcome regression (OR) estimators, satisfying only some
high-level conditions, we establish finite sample performance bounds for the
DDR estimator showing its (optimal) $L_2$ error rate to be $\sqrt{s (\log d)/
n}$ when both models are correct, and its consistency and DR properties when
only one of them is correct. Further, when both the models are correct, we
propose a desparsified version of our DDR estimator that satisfies an
asymptotic linear expansion and facilitates inference on low dimensional
components of $\boldsymbol{\theta}_0$. Finally, we discuss various of choices
of high dimensional parametric/semi-parametric working models for the PS and OR
estimators. All results are validated via detailed simulations.
