  An important theorem of Banaszczyk (Random Structures & Algorithms `98)
states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any
convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a
signed combination of these vectors which lands inside $K$. A major open
problem is to devise a constructive version of Banaszczyk's vector balancing
theorem, i.e. to find an efficient algorithm which constructs the signed
combination.
  We make progress towards this goal along several fronts. As our first
contribution, we show an equivalence between Banaszczyk's theorem and the
existence of $O(1)$-subgaussian distributions over signed combinations. For the
case of symmetric convex bodies, our equivalence implies the existence of a
universal signing algorithm (i.e. independent of the body), which simply
samples from the subgaussian sign distribution and checks to see if the
associated combination lands inside the body. For asymmetric convex bodies, we
provide a novel recentering procedure, which allows us to reduce to the case
where the body is symmetric.
  As our second main contribution, we show that the above framework can be
efficiently implemented when the vectors have length $O(1/\sqrt{\log n})$,
recovering Banaszczyk's results under this stronger assumption. More precisely,
we use random walk techniques to produce the required $O(1)$-subgaussian
signing distributions when the vectors have length $O(1/\sqrt{\log n})$, and
use a stochastic gradient ascent method to implement the recentering procedure
for asymmetric bodies.
