Competitive strategies to use "warm start" algorithms with predictions

TL;DR

This work studies how predictions can accelerate warm-start algorithms when solving sequences of related problems. It develops three complementary avenues: (i) offline guarantees against a set of $k$ fixed predictions, achieving a constant-factor approximation to the best $k$-point predictor; (ii) leveraging coarse information via learnable $k$-wise partitions to remove the linear $k$-factor in runtime; and (iii) online guarantees against moving trajectories, presenting a deterministic algorithm that is $O(k^4\ln^2 k)$-competitive with any set of $k$ trajectories while keeping total runtime within a constant factor of the work done by the subroutine. The paper also connects online ball search to the $k$-server problem, showing reductions that yield $O(k^2)$-type guarantees in some regimes, and provides a unified framework for learning and deploying multiple predictions in warm-start contexts. Collectively, these results establish that warm starts with multiple, learnable predictions can yield substantial improvements over single-prediction baselines in both offline and online settings, with broad potential for practical speedups in related optimization and search tasks.

Abstract

We consider the problem of learning and using predictions for warm start algorithms with predictions. In this setting, an algorithm is given an instance of a problem, and a prediction of the solution. The runtime of the algorithm is bounded by the distance from the predicted solution to the true solution of the instance. Previous work has shown that when instances are drawn iid from some distribution, it is possible to learn an approximately optimal fixed prediction (Dinitz et al, NeurIPS 2021), and in the adversarial online case, it is possible to compete with the best fixed prediction in hindsight (Khodak et al, NeurIPS 2022). In this work we give competitive guarantees against stronger benchmarks that consider a set of $k$ predictions $\mathbf{P}$. That is, the "optimal offline cost" to solve an instance with respect to $\mathbf{P}$ is the distance from the true solution to the closest member of $\mathbf{P}$. This is analogous to the $k$-medians objective function. In the distributional setting, we show a simple strategy that incurs cost that is at most an $O(k)$ factor worse than the optimal offline cost. We then show a way to leverage learnable coarse information, in the form of partitions of the instance space into groups of "similar" instances, that allows us to potentially avoid this $O(k)$ factor. Finally, we consider an online version of the problem, where we compete against offline strategies that are allowed to maintain a moving set of $k$ predictions or "trajectories," and are charged for how much the predictions move. We give an algorithm that does at most $O(k^4 \ln^2 k)$ times as much work as any offline strategy of $k$ trajectories. This algorithm is deterministic (robust to an adaptive adversary), and oblivious to the setting of $k$. Thus the guarantee holds for all $k$ simultaneously.

Theorems & Definitions (53)

• Definition 2.1: Warm start algorithm
• Lemma 3.1: Using $k$ predictions
• proof
• Definition 3.3: Clustering cost of a set
• Definition 3.4: Clustering cost of a distribution
• Lemma 3.4: Learning (k) fixed points
• proof
• Theorem 3.5: Competing against $k$ fixed points offline
• proof
• Remark 3.6