Continuum Armed Bandit Problem of Few Variables in High Dimensions

  • Hemant Tyagi
  • Bernd Gärtner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8447)


We consider the stochastic and adversarial settings of continuum armed bandits where the arms are indexed by [0,1] d . The reward functions r:[0,1] d  → ℝ are assumed to intrinsically depend on at most k coordinate variables implying \(r(x_1,\dots,x_d) = g(x_{i_1},\dots,x_{i_k})\) for distinct and unknown i 1,…,i k  ∈ {1,…,d} and some locally Hölder continuous g:[0,1] k  → ℝ with exponent α ∈ (0,1]. Firstly, assuming (i 1,…,i k ) to be fixed across time, we propose a simple modification of the CAB1 algorithm where we construct the discrete set of sampling points to obtain a bound of \(O(n^{\frac{\alpha+k}{2\alpha+k}} (\log n)^{\frac{\alpha}{2\alpha+k}} C(k,d))\) on the regret, with C(k,d) depending at most polynomially in k and sub-logarithmically in d. The construction is based on creating partitions of {1,…,d} into k disjoint subsets and is probabilistic, hence our result holds with high probability. Secondly we extend our results to also handle the more general case where (i 1,…,i k ) can change over time and derive regret bounds for the same.


Bandit problems continuum armed bandits functions of few variables online optimization 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Hemant Tyagi
    • 1
  • Bernd Gärtner
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH Zürich (ETHZ)ZürichSwitzerland

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