This paper points out the limitation of existing online learning algorithms that assume that all mistakes can be recovered, and proposes a new online learning problem that considers the case where some mistakes are irreversible 'fatal' mistakes. We define the reward of each round as the 'catastrophe avoidance probability', and aim to maximize the product of the catastrophe avoidance probabilities (total catastrophe avoidance probability) within a limited number of mentor queries. We allow knowledge transfer between similar inputs, and prove that in general, the mentor query rate is linear or the catastrophe occurrence probability is close to 1. However, in a standard online model, we present an algorithm in which the mentor query rate and regret converge to 0 as the time horizon increases in an environment where the mentor policy class is learnable. Although we focus on the product of rewards, we also present a bound on general additive regret. In essence, we show that if a policy class is learnable in the absence of fatal risk, it is learnable even in the presence of fatal risk if it can receive help from a mentor.
