The multi-armed bandit (MAB) is a classical online optimization model for the trade-off between exploration and exploitation. The traditional MAB is concerned with finding the arm that minimizes the mean cost. However, minimizing the mean does not take the risk of the problem into account. We now want to accommodate risk-averse decision makers. In this work, we introduce a coherent risk measure as the criterion to form a risk-averse MAB. In particular, we derive an index-based online sampling framework for the risk-averse MAB. We develop this framework in detail for three specific risk measures, i.e. the conditional value-at-risk, the mean-deviation and the shortfall risk measures. Under each risk measure, the convergence rate for the upper bound on the pseudo regret, defined as the difference between the expectation of the empirical risk based on the observation sequence and the true risk of the optimal arm, is established.
Download