Best of both worlds: Stochastic & adversarial best-arm identification

Yasin Abbasi-Yadkori (1) , Peter Bartlett (2) , Victor Gabillon (3) , Alan Malek (4) , Michal Valko (5)

1 Adobe Research
2 LBNL - Lawrence Berkeley National Laboratory [Berkeley]
3 QUT - Queensland University of Technology [Brisbane]
4 MIT - Massachusetts Institute of Technology
5 SEQUEL - Sequential Learning

We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.

Mots clés: multi-armed bandits; adversarial and stochastic rewards; best-arm identification

Domaines: Machine Learning [stat.ML]

Yasin Abbasi-Yadkori, Peter Bartlett, Victor Gabillon, Alan Malek, Michal Valko. Best of both worlds: Stochastic & adversarial best-arm identification. Conference on Learning Theory, 2018, Stockholm, Sweden. ⟨hal-01808948v6⟩