Gaussian Two-Armed Bandit: Limiting Description

For a Gaussian two-armed bandit, which arises when batch data processing is analyzed, the minimax risk limiting behavior is investigated as the control horizon N grows infinitely. The minimax risk is searched for as the Bayesian one computed with respect to the worst-case prior distribution. We show that the highest requirements are imposed on the control in the domain of "close” distributions where mathematical expectations of incomes differ by a quantity of the order of N^−1/2. In the domain of "close” distributions, we obtain a recursive integro-difference equation for finding the Bayesian risk with respect to the worst-case prior distribution, in invariant form with control horizon one, and also a second-order partial differential equation in the limiting case. The results allow us to estimate the performance of batch processing. For example, the minimax risk corresponding to batch processing of data partitioned into 50 batches can be only 2% greater than its limiting value when the number of batches grows infinitely. In the case of a Bernoulli two-armed bandit, we show that optimal one-by-one data processing is not more efficient than batch processing as N grows infinitely.