Online Uniform Sampling: Randomized Learning-Augmented Approximation Algorithms with Application to Digital Health (2402.01995v6)

Abstract: Motivated by applications in digital health, this work studies the novel problem of online uniform sampling (OUS), where the goal is to distribute a sampling budget uniformly across unknown decision times. In the OUS problem, the algorithm is given a budget $b$ and a time horizon $T$, and an adversary then chooses a value $\tau^* \in [b,T]$, which is revealed to the algorithm online. At each decision time $i \in [\tau^*]$, the algorithm must determine a sampling probability that maximizes the budget spent throughout the horizon, respecting budget constraint $b$, while achieving as uniform a distribution as possible over $\tau^*$. We present the first randomized algorithm designed for this problem and subsequently extend it to incorporate learning augmentation. We provide worst-case approximation guarantees for both algorithms, and illustrate the utility of the algorithms through both synthetic experiments and a real-world case study involving the HeartSteps mobile application. Our numerical results show strong empirical average performance of our proposed randomized algorithms against previously proposed heuristic solutions.