  In this paper, we provide a novel construction of the linear-sized spectral
sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous
constructions required $\Omega(n^4)$ running time [BSS14, Zou12], our
sparsification routine can be implemented in almost-quadratic running time
$O(n^{2+\varepsilon})$.
  The fundamental conceptual novelty of our work is the leveraging of a strong
connection between sparsification and a regret minimization problem over
density matrices. This connection was known to provide an interpretation of the
randomized sparsifiers of Spielman and Srivastava [SS11] via the application of
matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we
explain how matrix MWU naturally arises as an instance of the
Follow-the-Regularized-Leader framework and generalize this approach to yield a
larger class of updates. This new class allows us to accelerate the
construction of linear-sized spectral sparsifiers, and give novel insights on
the motivation behind Batson, Spielman and Srivastava [BSS14].
