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.

对于在分析批处理数据时出现的高斯双臂匪徒，随着控制范围N的无限增长，研究了minimax风险限制行为。在针对最坏情况的先验分布进行贝叶斯计算时搜索最小最大风险。我们表明，对“封闭”分布域中的控制施加了最高要求，在该域中，收入的数学期望相差数量级为N ^-1/2。在“接近”分布的域中，我们获得了一个递归积分差方程，用于找到相对于最坏情况先验分布的贝叶斯风险（具有控制范围为1的不变形式），以及一个二阶偏微分方程。结果使我们能够估计批处理的性能，例如，当批数无限增长时，与分为50个批的数据的批处理相对应的minimax风险仅比其极限值大2％。对于伯努利（Bernoulli）双臂土匪，我们证明随着N无限增长，最佳的一对一数据处理没有比批处理更有效。