  We introduce the first direct policy search algorithm which provably
converges to the globally optimal $\textit{dynamic}$ filter for the classical
problem of predicting the outputs of a linear dynamical system, given noisy,
partial observations. Despite the ubiquity of partial observability in
practice, theoretical guarantees for direct policy search algorithms, one of
the backbones of modern reinforcement learning, have proven difficult to
achieve. This is primarily due to the degeneracies which arise when optimizing
over filters that maintain internal state.
  In this paper, we provide a new perspective on this challenging problem based
on the notion of $\textit{informativity}$, which intuitively requires that all
components of a filter's internal state are representative of the true state of
the underlying dynamical system. We show that informativity overcomes the
aforementioned degeneracy. Specifically, we propose a $\textit{regularizer}$
which explicitly enforces informativity, and establish that gradient descent on
this regularized objective - combined with a ``reconditioning step'' -
converges to the globally optimal cost a $\mathcal{O}(1/T)$ rate. Our analysis
relies on several new results which may be of independent interest, including a
new framework for analyzing non-convex gradient descent via convex
reformulation, and novel bounds on the solution to linear Lyapunov equations in
terms of (our quantitative measure of) informativity.
