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Lower Bound on the Greedy Approximation Ratio for Adaptive Submodular Cover

Blake Harris, Viswanath Nagarajan

TL;DR

The paper examines min-cost ASC under adaptive-submodular utilities and proves a fundamental limitation: the greedy policy cannot guarantee a $(1+\ln Q)$-style approximation in general ASC, instead exhibiting a lower bound of $1.3\times(1+\ln Q)$. It constructs a concrete $4$-item instance with $Q=1$ showing a gap between greedy and optimal policies, and discusses extensions that push the gap to at least $1.3$. This finding contradicts earlier claims of a $(1+\ln Q)^2$-style bound in the adaptive setting and refines our understanding of greedy performance in adaptive-submodular cover. The work also delineates the problem's framework, adaptive-greedy mechanics, and the capacity to generalize the hard instances, highlighting implications for active learning and stochastic optimization under adaptive submodularity.

Abstract

We show that the greedy algorithm for adaptive-submodular cover has approximation ratio at least 1.3*(1+ln Q). Moreover, the instance demonstrating this gap has Q=1. So, it invalidates a prior result in the paper ``Adaptive Submodularity: A New Approach to Active Learning and Stochastic Optimization'' by Golovin-Krause, that claimed a (1+ln Q)^2 approximation ratio for the same algorithm.

Lower Bound on the Greedy Approximation Ratio for Adaptive Submodular Cover

TL;DR

The paper examines min-cost ASC under adaptive-submodular utilities and proves a fundamental limitation: the greedy policy cannot guarantee a -style approximation in general ASC, instead exhibiting a lower bound of . It constructs a concrete -item instance with showing a gap between greedy and optimal policies, and discusses extensions that push the gap to at least . This finding contradicts earlier claims of a -style bound in the adaptive setting and refines our understanding of greedy performance in adaptive-submodular cover. The work also delineates the problem's framework, adaptive-greedy mechanics, and the capacity to generalize the hard instances, highlighting implications for active learning and stochastic optimization under adaptive submodularity.

Abstract

We show that the greedy algorithm for adaptive-submodular cover has approximation ratio at least 1.3*(1+ln Q). Moreover, the instance demonstrating this gap has Q=1. So, it invalidates a prior result in the paper ``Adaptive Submodularity: A New Approach to Active Learning and Stochastic Optimization'' by Golovin-Krause, that claimed a (1+ln Q)^2 approximation ratio for the same algorithm.
Paper Structure (14 sections, 1 theorem, 10 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 14 sections, 1 theorem, 10 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Random items.
  4. Utility function.
  5. Min-cost adaptive-submodular cover (${\sf ASC}$).
  6. Remark:
  7. Adaptive Greedy Policy
  8. Prior Results on the Greedy Policy
  9. Our Result
  10. Hard Instance for Greedy
  11. Adaptive submodularity.
  12. The optimal policy.
  13. The greedy policy.
  14. Other hard instances.

Key Result

Theorem 1.1

There are instances of adaptive submodular cover with integer-valued function $f$ and $Q=1$ where the greedy policy has expected cost at least $\rho > 1$ times the optimum.

Figures (1)

  • Figure 1: The r.v.s $\{X_i\}_{i=1}^4$ and items $\{a, b , c, d\}$.

Theorems & Definitions (5)

  • Definition 1.1: Monotonicity
  • Definition 1.2: Coverable
  • Definition 1.3: Marginal benefit
  • Definition 1.4: Adaptive submodularity
  • Theorem 1.1