Runtime analysis, as a branch of the theory of AI, studies how the number of iterations algorithms take before finding a solution (its runtime) depends on the design of the algorithm and the problem structure. Drift analysis is a state-of-the-art tool for estimating the runtime of randomised algorithms, such as evolutionary and bandit algorithms. Drift refers roughly to the expected progress towards the optimum per iteration. This paper considers the problem of deriving concentration tail-bounds on the runtime/regret of algorithms. It provides a novel drift theorem that gives precise exponential tail-bounds given positive, weak, zero and even negative drift. Previously, such exponential tail bounds were missing in the case of weak, zero, or negative drift. Our drift theorem can be used to prove a strong concentration of the runtime/regret of algorithms in AI. For example, we prove that the regret of the \rwab bandit algorithm is highly concentrated, while previous analyses only considered the expected regret. This means that the algorithm obtains the optimum within a given time frame with high probability, i.e. a form of algorithm reliability. Moreover, our theorem implies that the time needed by the co-evolutionary algorithm RLS-PD to obtain a Nash equilibrium in a \bilinear max-min-benchmark problem is highly concentrated. However, we also prove that the algorithm forgets the Nash equilibrium, and the time until this occurs is highly concentrated. This highlights a weakness in the RLS-PD which should be addressed by future work.

翻译：运行时分析作为人工智能理论的一个分支，研究算法在找到解之前所需的迭代次数（即运行时）如何依赖于算法设计和问题结构。漂移分析是评估随机算法（如进化算法和赌博算法）运行时的最新工具。漂移大致指每轮迭代中算法向最优解期望进展的度量。本文考虑推导算法运行时/遗憾的浓度尾界问题。我们提出一个新颖的漂移定理，能够针对正漂移、弱漂移、零漂移甚至负漂移给出精确的指数尾界。此前在弱漂移、零漂移或负漂移情形下，此类指数尾界尚属缺失。我们的漂移定理可用于证明人工智能算法运行时/遗憾具有强浓度性质。例如，我们证明\rwab赌博算法的遗憾高度集中，而以往分析仅考虑期望遗憾。这意味着该算法在给定时间框架内以高概率获得最优解，即算法可靠性的体现。此外，该定理表明共进化算法RLS-PD在\bilinear max-min-benchmark问题中达到纳什均衡所需的时间高度集中。然而我们也证明该算法会遗忘纳什均衡，且遗忘发生的时间同样高度集中。这揭示了RLS-PD的缺陷，亟需未来工作加以改进。