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Online Sampling and Decision Making with Low Entropy

Mohammad Taghi Hajiaghayi, Dariusz R. Kowalski, Piotr Krysta, Jan Olkowski

TL;DR

The paper tackles online selection with unknown, non-identical distributions by introducing a low-entropy distribution on arrival orders and a deterministic multi-threshold algorithm that achieves near-optimal competitive ratios for the free-order $k$-secretary problem. Its core contribution is a general framework that derandomizes threshold-based secretary algorithms via atomic/positive event decompositions, Chernoff-based concentration, and a single Reed-Solomon-based dimension reduction to enable polynomial-time construction. It proves tight entropy lower bounds and derives improved upper bounds, substantially expanding the range of feasible $k$ and tightening previously known gaps. The work has practical significance for randomness-efficient online decision making and offers a blueprint for applying similar derandomization and dimension-reduction techniques to broader online optimization problems such as prophet secretary and matroid secretary variants.

Abstract

Consider the problem: we are given $n$ boxes, labeled $\{1,2,\ldots, n\}$ by an adversary, each containing a single number chosen from an unknown distribution; these $n$ distributions are not necessarily identical. We are also given an integer $k \leq n$. We have to choose an order in which we will sequentially open these boxes, and each time we open the next box in this order, we learn the number in the box. Once we reject a number in a box, the box cannot be recalled. Our goal is to accept $k$ of these numbers, without necessarily opening all boxes, such that the accepted numbers are the $k$ largest numbers in the boxes (thus their sum is maximized). A natural approach to solve such problems is to use randomness to sample randomly ordered elements, however, as indicated in several sources, e.g., Turan et al. NIST'15, Bierhorst et al. Nature'18, pure randomness is hard to get in reality. We present an algorithm for this problem, which is provably and simultaneously near-optimal with respect to the achieved competitive ratio and the used amount of randomness. In particular, we construct a distribution on the orders with entropy $Θ(\log\log n)$ such that a deterministic multiple-threshold algorithm gives a competitive ratio $1-O(\sqrt{\log k/k})$, for $k < \log n/\log \log n$. Our competitive ratio is simultaneously optimal and uses optimal entropy $Θ(\log\log n)$, improving in three ways the previous best known algorithm, whose competitive ratio is $1 - O(1/k^{1/3}) - o(1)$.

Online Sampling and Decision Making with Low Entropy

TL;DR

The paper tackles online selection with unknown, non-identical distributions by introducing a low-entropy distribution on arrival orders and a deterministic multi-threshold algorithm that achieves near-optimal competitive ratios for the free-order -secretary problem. Its core contribution is a general framework that derandomizes threshold-based secretary algorithms via atomic/positive event decompositions, Chernoff-based concentration, and a single Reed-Solomon-based dimension reduction to enable polynomial-time construction. It proves tight entropy lower bounds and derives improved upper bounds, substantially expanding the range of feasible and tightening previously known gaps. The work has practical significance for randomness-efficient online decision making and offers a blueprint for applying similar derandomization and dimension-reduction techniques to broader online optimization problems such as prophet secretary and matroid secretary variants.

Abstract

Consider the problem: we are given boxes, labeled by an adversary, each containing a single number chosen from an unknown distribution; these distributions are not necessarily identical. We are also given an integer . We have to choose an order in which we will sequentially open these boxes, and each time we open the next box in this order, we learn the number in the box. Once we reject a number in a box, the box cannot be recalled. Our goal is to accept of these numbers, without necessarily opening all boxes, such that the accepted numbers are the largest numbers in the boxes (thus their sum is maximized). A natural approach to solve such problems is to use randomness to sample randomly ordered elements, however, as indicated in several sources, e.g., Turan et al. NIST'15, Bierhorst et al. Nature'18, pure randomness is hard to get in reality. We present an algorithm for this problem, which is provably and simultaneously near-optimal with respect to the achieved competitive ratio and the used amount of randomness. In particular, we construct a distribution on the orders with entropy such that a deterministic multiple-threshold algorithm gives a competitive ratio , for . Our competitive ratio is simultaneously optimal and uses optimal entropy , improving in three ways the previous best known algorithm, whose competitive ratio is .
Paper Structure (34 sections, 30 theorems, 127 equations, 1 table, 3 algorithms)

This paper contains 34 sections, 30 theorems, 127 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our results and techniques
  4. Main contribution: algorithmic results.
  5. Algorithms for free-order secretary problems.
  6. Lower bounds.
  7. Complementary results
  8. Further related work
  9. Secretary Problem.
  10. Prophet Inequality.
  11. Prophet Secretary.
  12. Probabilistic atomic and positive events for threshold algorithms
  13. Abstract derandomization of positive events via concentration bounds: Theorem \ref{['Thm:Derandomization_2_111']}
  14. Conditional probabilities and proof of Theorem \ref{['theorem:semi-random-conditional_2_111']}
  15. Missing details in proof of Theorem \ref{['Thm:Derandomization_2_111']}
  16. ...and 19 more sections

Key Result

Theorem 1

For any $k < \log{n}/\log \log n$, there exists a multi-set of $n$-element permutations $\mathcal{L}_{n}$ such that a deterministic multiple-threshold algorithm for the free order multiple-choice secretary achieves an expected competitive ratio when it uses the order chosen uniformly at random from $\mathcal{L}_{n}$. The set $\mathcal{L}_{n}$ is computable in time $O(\text{poly } (n))$ and the un

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1: Atomic events with respect to a bucketing
  • Definition 2: Positive events based on atomic family $\mathcal{A}_{k, \mathcal{B}}$
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1
  • ...and 22 more