Optimal Learning-Augmented Algorithm for Online Bidding

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.