  We resolve an open question from (Christiano, 2014b) posed in COLT'14
regarding the optimal dependency of the regret achievable for online local
learning on the size of the label set. In this framework the algorithm is shown
a pair of items at each step, chosen from a set of $n$ items. The learner then
predicts a label for each item, from a label set of size $L$ and receives a
real valued payoff. This is a natural framework which captures many interesting
scenarios such as collaborative filtering, online gambling, and online max cut
among others. (Christiano, 2014a) designed an efficient online learning
algorithm for this problem achieving a regret of $O(\sqrt{nL^3T})$, where $T$
is the number of rounds. Information theoretically, one can achieve a regret of
$O(\sqrt{n \log L T})$. One of the main open questions left in this framework
concerns closing the above gap.
  In this work, we provide a complete answer to the question above via two main
results. We show, via a tighter analysis, that the semi-definite programming
based algorithm of (Christiano, 2014a), in fact achieves a regret of
$O(\sqrt{nLT})$. Second, we show a matching computational lower bound. Namely,
we show that a polynomial time algorithm for online local learning with lower
regret would imply a polynomial time algorithm for the planted clique problem
which is widely believed to be hard. We prove a similar hardness result under a
related conjecture concerning planted dense subgraphs that we put forth. Unlike
planted clique, the planted dense subgraph problem does not have any known
quasi-polynomial time algorithms.
  Computational lower bounds for online learning are relatively rare, and we
hope that the ideas developed in this work will lead to lower bounds for other
online learning scenarios as well.
