Recent advances in machine learning have spurred significant interest in learning-augmented algorithms, particularly for online optimization. A growing body of work has studied online bidding in this framework, aiming to characterize the trade-off between robustness and consistency. While this trade-off is fully understood for deterministic algorithms, a gap between upper and lower bounds remains in the randomized setting. In this paper, we close this gap by presenting a Pareto-optimal randomized learning-augmented algorithm for this problem. Our approach introduces the notion of a bidding profile, a novel framework for representing the distribution over bids generated by an algorithm. We show that any bidding algorithm can be reduced, without loss of generality, to one driven by a bidding profile, and we characterize the optimal profile via a system of delayed differential equations. Finally, we demonstrate the broader applicability of our approach by extending it to the linear search problem, yielding a significant improvement over prior learning-augmented algorithms for linear search.

翻译：机器学习的最新进展极大地激发了人们对学习增强算法的兴趣，尤其是在在线优化领域。越来越多的研究在此框架下探索在线竞标问题，旨在刻画鲁棒性与一致性之间的权衡。尽管这一权衡在确定性算法中已被完全理解，但在随机设定下，上下界之间仍存在差距。本文通过提出一种帕累托最优的随机学习增强算法来解决该问题，从而填补了这一空白。我们的方法引入了竞标分布（bidding profile）的概念，这是一种用于表示算法生成的竞标分布的新框架。我们证明，任何竞标算法都可以在无一般性损失的情况下简化为由竞标分布驱动的算法，并通过延迟微分方程组刻画了最优分布。最后，我们通过将所提方法扩展到线性搜索问题，展示了其更广泛的适用性，相较于先前的学习增强算法取得了显著改进。