  Observations from dynamical systems often exhibit irregularities, such as
censoring, where values are recorded only if they fall within a certain range.
Censoring is ubiquitous in practice, due to saturating sensors,
limit-of-detection effects, and image-frame effects. In light of recent
developments on learning linear dynamical systems (LDSs), and on censored
statistics with independent data, we revisit the decades-old problem of
learning an LDS, from censored observations (Lee and Maddala (1985); Zeger and
Brookmeyer (1986)). Here, the learner observes the state $x_t \in \mathbb{R}^d$
if and only if $x_t$ belongs to some set $S_t \subseteq \mathbb{R}^d$. We
develop the first computationally and statistically efficient algorithm for
learning the system, assuming only oracle access to the sets $S_t$. Our
algorithm, Stochastic Online Newton with Switching Gradients, is a novel
second-order method that builds on the Online Newton Step (ONS) of Hazan et al.
(2007). Our Switching-Gradient scheme does not always use (stochastic)
gradients of the function we want to optimize, which we call "censor-aware"
function. Instead, in each iteration, it performs a simple test to decide
whether to use the censor-aware, or another "censor-oblivious" function, for
getting a stochastic gradient.
  In our analysis, we consider a "generic" Online Newton method, which uses
arbitrary vectors instead of gradients, and we prove an error-bound for it.
This can be used to appropriately design these vectors, leading to our
Switching-Gradient scheme. This framework significantly deviates from the
recent long line of works on censored statistics (e.g., Daskalakis et al.
(2018); Kontonis et al. (2019); Daskalakis et al. (2019)), which apply
Stochastic Gradient Descent (SGD), and their analysis reduces to establishing
conditions for off-the-shelf SGD-bounds.
