Michel X. Goemans

Chiheon Kim, Afonso S. Bandeira, Michel X. Goemans

We study the problem of community detection in a random hypergraph model which we call the stochastic block model for $k$-uniform hypergraphs ($k$-SBM). We investigate the exact recovery problem in $k$-SBM and show that a sharp phase transition occurs around a threshold: below the threshold it is impossible to recover the communities with non-vanishing probability, yet above the threshold there is an estimator which recovers the communities almost asymptotically surely. We also consider a simple, efficient algorithm for the exact recovery problem which is based on a semidefinite relaxation technique.

Swati Gupta, Michel Goemans, Patrick Jaillet

In order to find Nash-equilibria for two-player zero-sum games where each player plays combinatorial objects like spanning trees, matchings etc, we consider two online learning algorithms: the online mirror descent (OMD) algorithm and the multiplicative weights update (MWU) algorithm. The OMD algorithm requires the computation of a certain Bregman projection, that has closed form solutions for simple convex sets like the Euclidean ball or the simplex. However, for general polyhedra one often needs to exploit the general machinery of convex optimization. We give a novel primal-style algorithm for computing Bregman projections on the base polytopes of polymatroids. Next, in the case of the MWU algorithm, although it scales logarithmically in the number of pure strategies or experts $N$ in terms of regret, the algorithm takes time polynomial in $N$; this especially becomes a problem when learning combinatorial objects. We give a general recipe to simulate the multiplicative weights update algorithm in time polynomial in their natural dimension. This is useful whenever there exists a polynomial time generalized counting oracle (even if approximate) over these objects. Finally, using the combinatorial structure of symmetric Nash-equilibria (SNE) when both players play bases of matroids, we show that these can be found with a single projection or convex minimization (without using online learning).