Refined Lower Bounds for Adversarial Bandits

Sébastien Gerchinovitz (1) , Tor Lattimore (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 University of Alberta

We provide new lower bounds on the regret that must be suffered by adversarial bandit algorithms. The new results show that recent upper bounds that either (a) hold with high-probability or (b) depend on the total loss of the best arm or (c) depend on the quadratic variation of the losses, are close to tight. Besides this we prove two impossibility results. First, the existence of a single arm that is optimal in every round cannot improve the regret in the worst case. Second, the regret cannot scale with the effective range of the losses. In contrast, both results are possible in the full-information setting.

Mots clés: bandits; lower bounds; online learning

Domaines: Statistiques [math.ST]; Machine Learning [stat.ML]; Apprentissage [cs.LG]

Sébastien Gerchinovitz, Tor Lattimore. Refined Lower Bounds for Adversarial Bandits. NIPS 2016, Dec 2016, Barcelona, Spain. pp.1198--1206. ⟨hal-01319572v2⟩