As inthe classical problem, weights are fixed by an adversary and elements appear in random order. In contrast to previous variants of predictions, our algorithm only has access toamuch weakerpiece ofinformation: anadditive gapc.

Our main result is a matching upper and lower bound for the secretary ranking problem.

However,such predictions, although usually accurate, can be arbitrarily poor.

This strategy yields a probability1/eof selecting the best candidate.

The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take them into account. In particular, we study the following online selection problems: (i) the classical secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classical online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

We study the repeated optimal stopping problem, which generalizes the classical optimal stopping problem with an unknown distribution to a setting where the same problem is solved repeatedly over $T$ rounds. In this framework, we aim to design algorithms that guarantee a competitive ratio in each round while also achieving sublinear regret across all rounds. Our primary contribution is a general algorithmic framework that achieves these objectives simultaneously for a wide array of repeated optimal stopping problems. The core idea is to dynamically select an algorithm for each round, choosing between two candidates: (1) an empirically optimal algorithm derived from the history of observations, and (2) a sample-based algorithm with a proven competitive ratio guarantee. Based on this approach, we design an algorithm that performs no worse than the baseline sample-based algorithm in every round, while ensuring that the total regret is bounded by $\tilde{O}(\sqrt{T})$. We demonstrate the broad applicability of our framework to canonical problems, including the prophet inequality, the secretary problem, and their variants under adversarial, random, and i.i.d. input models. For example, for the repeated prophet inequality problem, our method achieves a $1/2$-competitive ratio from the second round on and an $\tilde{O}(\sqrt{T})$ regret. Furthermore, we establish a regret lower bound of $Ω(\sqrt{T})$ even in the i.i.d. model, confirming that our algorithm's performance is almost optimal with respect to the number of rounds.