  We study online prediction where regret of the algorithm is measured against
a benchmark defined via evolving constraints. This framework captures online
prediction on graphs, as well as other prediction problems with combinatorial
structure. A key aspect here is that finding the optimal benchmark predictor
(even in hindsight, given all the data) might be computationally hard due to
the combinatorial nature of the constraints. Despite this, we provide
polynomial-time \emph{prediction} algorithms that achieve low regret against
combinatorial benchmark sets. We do so by building improper learning algorithms
based on two ideas that work together. The first is to alleviate part of the
computational burden through random playout, and the second is to employ
Lasserre semidefinite hierarchies to approximate the resulting integer program.
Interestingly, for our prediction algorithms, we only need to compute the
values of the semidefinite programs and not the rounded solutions. However, the
integrality gap for Lasserre hierarchy \emph{does} enter the generic regret
bound in terms of Rademacher complexity of the benchmark set. This establishes
a trade-off between the computation time and the regret bound of the algorithm.
