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

Hierarchies of Relaxations for Online Prediction Problems with Evolving Constraints