  Many practical learning systems aggregate data across many users, while
learning theory traditionally considers a single learner who trusts all of
their observations. A case in point is the foundational learning problem of
prediction with expert advice. To date, there has been no theoretical study of
the general collaborative version of prediction with expert advice, in which
many users face a similar problem and would like to share their experiences in
order to learn faster. A key issue in this collaborative framework is
robustness: generally algorithms that aggregate data are vulnerable to
manipulation by even a small number of dishonest users.
  We exhibit the first robust collaborative algorithm for prediction with
expert advice. When all users are honest and have similar tastes our algorithm
matches the performance of pooling data and using a traditional algorithm. But
our algorithm also guarantees that adding users never significantly degrades
performance, even if the additional users behave adversarially. We achieve
strong guarantees even when the overwhelming majority of users behave
adversarially. As a special case, our algorithm is extremely robust to
variation amongst the users.
