  In this work, we examine the security of InstaHide, a scheme recently
proposed by [Huang, Song, Li and Arora, ICML'20] for preserving the security of
private datasets in the context of distributed learning. To generate a
synthetic training example to be shared among the distributed learners,
InstaHide takes a convex combination of private feature vectors and randomly
flips the sign of each entry of the resulting vector with probability 1/2. A
salient question is whether this scheme is secure in any provable sense,
perhaps under a plausible hardness assumption and assuming the distributions
generating the public and private data satisfy certain properties.
  We show that the answer to this appears to be quite subtle and closely
related to the average-case complexity of a new multi-task, missing-data
version of the classic problem of phase retrieval. Motivated by this
connection, we design a provable algorithm that can recover private vectors
using only the public vectors and synthetic vectors generated by InstaHide,
under the assumption that the private and public vectors are isotropic
Gaussian.
