Title: Gaussian One-Armed Bandit and Optimization of Batch Data Processing,
Abstract: We consider the minimax setup for Gaussian one-armed bandit problem, i.e. the
two-armed bandit problem with Gaussian distributions of incomes and known
distribution corresponding to the first arm. This setup naturally arises when
the optimization of batch data processing is considered and there are two
alternative processing methods available with a priori known efficiency of the
first method. One should estimate the efficiency of the second method and
provide predominant usage of the most efficient of both them. According to the
main theorem of the theory of games minimax strategy and minimax risk are
searched for as Bayesian ones corresponding to the worst-case prior
distribution. As a result, we obtain the recursive integro-difference equation
and the second order partial differential equation in the limiting case as the
number of batches goes to infinity. This makes it possible to determine minimax
risk and minimax strategy by numerical methods. If the number of batches is
large enough we show that batch data processing almost does not influence the
control performance, i.e. the value of the minimax risk. Moreover, in case of
Bernoulli incomes and large number of batches, batch data processing provides
almost the same minimax risk as the optimal one-by-one data processing.