A nonmonotone GRASP

A greedy randomized adaptive search procedure (GRASP) is an itera- tive multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the con- struction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solu- tion. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut prob- lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP)

First-order regret bounds for combinatorial semi-bandits

We consider the problem of online combinatorial optimization under semi-bandit feedback, where a learner has to repeatedly pick actions from a combinatorial decision set in order to minimize the total losses associated with its decisions. After making each decision, the learner observes the losses associated with its action, but not other losses. For this problem, there are several learning algorithms that guarantee that the learner's expected regret grows as $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this paper, we propose an algorithm that improves this scaling to $\widetilde{O}(\sqrt{{L_T^*}})$, where $L_T^*$ is the total loss of the best action. Our algorithm is among the first to achieve such guarantees in a partial-feedback scheme, and the first one to do so in a combinatorial setting.Comment: To appear at COLT 201

Cakewalk Sampling

We study the task of finding good local optima in combinatorial optimization problems. Although combinatorial optimization is NP-hard in general, locally optimal solutions are frequently used in practice. Local search methods however typically converge to a limited set of optima that depend on their initialization. Sampling methods on the other hand can access any valid solution, and thus can be used either directly or alongside methods of the former type as a way for finding good local optima. Since the effectiveness of this strategy depends on the sampling distribution, we derive a robust learning algorithm that adapts sampling distributions towards good local optima of arbitrary objective functions. As a first use case, we empirically study the efficiency in which sampling methods can recover locally maximal cliques in undirected graphs. Not only do we show how our adaptive sampler outperforms related methods, we also show how it can even approach the performance of established clique algorithms. As a second use case, we consider how greedy algorithms can be combined with our adaptive sampler, and we demonstrate how this leads to superior performance in k-medoid clustering. Together, these findings suggest that our adaptive sampler can provide an effective strategy to combinatorial optimization problems that arise in practice.Comment: Accepted as a conference paper by AAAI-2020 (oral presentation