Cakewalk Sampling

Combinatorial optimization is a common theme in computer science which underlies a considerable variety of problems. In contrast to the continuous setting, combinatorial problems require special solution strategies, and it's hard to come by generic schemes like gradient methods for continuous domains. We follow a standard construction of a parametric sampling distribution that transforms the problem to the continuous domain, allowing us to optimize the expectation of a given objective using estimates of the gradient. In spite of the apparent generality, such constructions are known to suffer from highly variable gradient estimates, and thus require careful tuning that is done in a problem specific manner. We show that a simple trick of converting the objective values to their cumulative probabilities fixes the distribution of the objective, allowing us to derive an online optimization algorithm that can be applied in a generic fashion. As an experimental benchmark we use the task of finding cliques in undirected graphs, and we show that our method, even when blindly applied, consistently outperforms related methods. Notably, on the DIMACS clique benchmark, our method approaches the performance of the best clique finding algorithms without access to the graph structure, and only through objective function evaluations, thus providing significant evidence to the generality and effectivity of our method.