Regret Minimization with Noisy Observations

In a typical optimization problem, the task is to pick one of a number of options with the lowest cost or the highest value. In practice, these cost/value quantities are often not accurately known: they can come from physical measurements with imperfect tools, estimations of machine learning algorithms, or observations from differentially private mechanisms. In many of these situations, the cost/value quantities are noisy, but with quantifiable noise distributions. To take these noise distributions into account, one approach is to assume a prior distribution for the values, use it to build a posterior, and then apply standard stochastic optimization to pick a solution. However, in many practical applications, such prior distributions may not be available. In this paper, we study such scenarios using a regret minimization model.