Lipschitzian optimization without the Lipschitz constant
Journal of Optimization Theory and Applications
The Continuum-Armed Bandit Problem
SIAM Journal on Control and Optimization
The Nonstochastic Multiarmed Bandit Problem
SIAM Journal on Computing
Finite-time Analysis of the Multiarmed Bandit Problem
Machine Learning
Multi-armed bandits in metric spaces
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Optimal algorithms for global optimization in case of unknown Lipschitz constant
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
Improved rates for the stochastic continuum-armed bandit problem
COLT'07 Proceedings of the 20th annual conference on Learning theory
Regret Bounds and Minimax Policies under Partial Monitoring
The Journal of Machine Learning Research
The Journal of Machine Learning Research
Lipschitz bandits without the Lipschitz constant
ALT'11 Proceedings of the 22nd international conference on Algorithmic learning theory
Lipschitz bandits without the Lipschitz constant
ALT'11 Proceedings of the 22nd international conference on Algorithmic learning theory
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We consider the setting of stochastic bandit problems with a continuum of arms indexed by [0, 1]d. We first point out that the strategies considered so far in the literature only provided theoretical guarantees of the form: given some tuning parameters, the regret is small with respect to a class of environments that depends on these parameters. This is however not the right perspective, as it is the strategy that should adapt to the specific bandit environment at hand, and not the other way round. Put differently, an adaptation issue is raised. We solve it for the special case of environments whose mean-payoff functions are globally Lipschitz. More precisely, we show that the minimax optimal orders of magnitude Ld/(d+2) T(d+1)/(d+2) of the regret bound over T time instances against an environment whose mean-payoff function f is Lipschitz with constant L can be achieved without knowing L or T in advance. This is in contrast to all previously known strategies, which require to some extent the knowledge of L to achieve this performance guarantee.