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Online Uniform Sampling: Randomized Learning-Augmented Approximation Algorithms with Application to Digital Health

Xueqing Liu, Kyra Gan, Esmaeil Keyvanshokooh, Susan Murphy

TL;DR

This work presents the first randomized algorithm designed for the novel problem of online uniform sampling and subsequently extends it to incorporate learning augmentation and provides worst-case approximation guarantees for both algorithms.

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 $τ^* \in [b,T]$, which is revealed to the algorithm online. At each decision time $i \in [τ^*]$, 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 $τ^*$. 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.

Online Uniform Sampling: Randomized Learning-Augmented Approximation Algorithms with Application to Digital Health

TL;DR

This work presents the first randomized algorithm designed for the novel problem of online uniform sampling and subsequently extends it to incorporate learning augmentation and provides worst-case approximation guarantees for both algorithms.

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 and a time horizon , and an adversary then chooses a value , which is revealed to the algorithm online. At each decision time , the algorithm must determine a sampling probability that maximizes the budget spent throughout the horizon, respecting budget constraint , while achieving as uniform a distribution as possible over . 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.
Paper Structure (25 sections, 4 theorems, 44 equations, 8 figures, 8 algorithms)

This paper contains 25 sections, 4 theorems, 44 equations, 8 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Framework
  4. OUS as An Online Optimization Problem
  5. Offline Clairvoyant and Competitive Ratio
  6. With Learning Augmentation
  7. Randomized Algorithm
  8. Learning-Augmented Algorithm
  9. Experiments
  10. Synthetic Experiments
  11. Real-World Experiments on HeartSteps
  12. Extension to Multiple Risk Levels
  13. The Penalty Term for Uniformity
  14. Proof for Algorithm 1
  15. Proof of Lemma 3.1: Budget constraint
  16. ...and 10 more sections

Key Result

Lemma 3.1

Let $p_i^{A1}$ be the probability returned by Algorithm alg:rand at risk time $i \in [\tau^*]$. This solution always satisfies the budget constraint in expectation, $\mathbb{E}\left[\sum_{i=1}^{\tau^*} p_{i}^{A1}\right] \leq b$, where the expectation is taken over the randomness of the algorithm.

Figures (8)

  • Figure 1: Average competitive ratio under non-learning augmented setting with $b=3$. The scenarios correspond to $T\leq b e$, $be<T\leq be^2$, and $T>be^2$, respectively.
  • Figure 2: Average competitive ratio under learning augmented setting with $b=3$. The scenarios correspond to $U\leq b e$, $be<U\leq be^2$, and $U>be^2$, respectively.
  • Figure 3: Average competitive ratio across user days under various prediction interval widths on HeartSteps V1 dataset. The shaded area indicates the $\pm 1.96$ standard error bounds across user days.
  • Figure 4: Average competitive ratio under learning-augmented setting with $b=3$.
  • Figure 5: Average competitive ratio under non-learning augmented setting with $b=3$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Definition 2.2: $\gamma$-competitive
  • Remark 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 5.1: Design choice of $b$ and $T$ in absence of prediction confidence intervals
  • ...and 5 more