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We consider online convex optimization with a zero-order oracle feedback. Inparticular, the decision maker does not know the explicit representation of thetime-varying cost functions, or their gradients. At each time step, sheobserves the value of the corresponding cost function evaluated at her chosenaction (zero-order oracle). The objective is to minimize the regret, that is,the difference between the sum of the costs she accumulates and that of astatic optimal action had she known the sequence of cost functions a priori. Wepresent a novel algorithm to minimize regret in unconstrained action spaces.Our algorithm hinges on a classical idea of one-point estimation of thegradients of the cost functions based on their observed values. The algorithmis independent of problem parameters. Letting $T$ denote the number of queriesof the zero-order oracle and $n$ the problem dimension, the regret rateachieved is $O(n^{2/3}T^{2/3})$. Moreover, we adapt the presented algorithm tothe setting with two-point feedback and demonstrate that the adapted procedureachieves the theoretical lower bound on the regret of $(n^{1/2}T^{1/2})$.

Minimizing Regret of Bandit Online Optimization in Unconstrained Action Spaces