Covariance-adapting algorithm for semi-bandits with application to sparse outcomes

Pierre Perrault (1, 2, 3) , Vianney Perchet (4) , Michal Valko (5)

1 ENS Paris Saclay - Ecole Normale Supérieure Paris-Saclay
2 SEQUEL - Sequential Learning
3 Scool - Scool
4 ENSAE - Ecole Nationale de la Statistique et de l'Analyse Economique
5 DeepMind [Paris]

We investigate stochastic combinatorial semi-bandits, where the entire joint distribution of outcomes impacts the complexity of the problem instance (unlike in the standard bandits). Typical distributions considered depend on specific parameter values, whose prior knowledge is required in theory but quite difficult to estimate in practice; an example is the commonly assumed sub-Gaussian family. We alleviate this issue by instead considering a new general family of sub-exponential distributions, which contains bounded and Gaussian ones. We prove a new lower bound on the regret on this family, that is parameterized by the unknown covariance matrix, a tighter quantity than the sub-Gaussian matrix. We then construct an algorithm that uses covariance estimates, and provide a tight asymptotic analysis of the regret. Finally, we apply and extend our results to the family of sparse outcomes, which has applications in many recommender systems.

Mots clés: combinatorial stochastic semi-bandits; covariance; confidence ellipsoid; sparsity

Domaines: Mathématiques [math]; Statistiques [math.ST]; Informatique [cs]; Apprentissage [cs.LG]

Pierre Perrault, Vianney Perchet, Michal Valko. Covariance-adapting algorithm for semi-bandits with application to sparse outcomes. Conference on Learning Theory, 2020, Graz, Austria. ⟨hal-02876102⟩