The field of algorithms with predictions aims to improve algorithm performance by integrating machine learning predictions into algorithm design. A central question in this area is how predictions can improve performance, and a key aspect of this analysis is the role of prediction accuracy. In this context, prediction accuracy is defined as a guaranteed probability that an instance drawn from the distribution belongs to the predicted set. As a performance measure that incorporates prediction accuracy, we focus on the distributionally-robust competitive ratio (DRCR), introduced by Sun et al.~(ICML 2024). The DRCR is defined as the expected ratio between the algorithm's cost and the optimal cost, where the expectation is taken over the worst-case instance distribution that satisfies the given prediction and accuracy requirement. A known structural property is that, for any fixed algorithm, the DRCR decreases linearly as prediction accuracy increases. Building on this result, we establish that the optimal DRCR value (i.e., the infimum over all algorithms) is a monotone and concave function of prediction accuracy. We further generalize the DRCR framework to a multiple-prediction setting and show that monotonicity and concavity are preserved in this setting. Finally, we apply our results to the ski rental problem, a benchmark problem in online optimization, to identify the conditions on prediction accuracies required for the optimal DRCR to attain a target value. Moreover, we provide a method for computing the critical accuracy, defined as the minimum accuracy required for the optimal DRCR to strictly improve upon the performance attainable without any accuracy guarantee.

翻译：带预测算法领域旨在通过将机器学习预测融入算法设计来提升算法性能。该领域的核心问题在于预测如何改善性能，而分析的关键环节在于预测精度的作用。在此背景下，预测精度被定义为从分布中抽取的实例属于预测集合的保证概率。作为纳入预测精度的性能度量，我们聚焦于由Sun等人（ICML 2024）提出的分布鲁棒竞争比。DRCR定义为算法成本与最优成本的期望比值，其中期望是在满足给定预测及精度要求的最坏情况实例分布上计算的。已知的结构性质表明：对于任意固定算法，DRCR随预测精度提升呈线性下降。基于此结果，我们证明了最优DRCR值（即所有算法的下确界）是预测精度的单调凹函数。我们进一步将DRCR框架推广至多预测场景，并证明单调性与凹性在该场景中得以保持。最后，我们将结果应用于在线优化的基准问题——滑雪租赁问题，以确定使最优DRCR达到目标值所需的预测精度条件。此外，我们提出了一种计算临界精度的方法，该临界精度定义为使最优DRCR严格优于无精度保证时可达到性能所需的最小精度。