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Learning Submodular Sequencing from Samples

Jing Yuan, Shaojie Tang

TL;DR

This work tackles sequential submodular maximization when the utility functions are not directly accessible and only samples of sequences with observed utilities are available. It introduces a learning-from-samples framework under a two-stage uniform sampling distribution and develops a curvature-aware algorithm that reduces the problem to a maximum weight matching over item-position pairs. The main theoretical contribution is a bound showing that the produced sequence $\pi^{\diamond}$ achieves a near-optimal fraction of $F(\pi^*)$, with guarantees that depend on the curvature $c$ and a sampling-dependent factor $\alpha$, under polynomially many samples. Practically, this enables productive sequencing and ranking in settings like online retail, where the full utility function is difficult to estimate, demonstrating that limited data can yield strong sequence-level performance guarantees.

Abstract

This paper addresses the problem of sequential submodular maximization: selecting and ranking items in a sequence to optimize some composite submodular function. In contrast to most of the previous works, which assume access to the utility function, we assume that we are given only a set of samples. Each sample includes a random sequence of items and its associated utility. We present an algorithm that, given polynomially many samples drawn from a two-stage uniform distribution, achieves an approximation ratio dependent on the curvature of individual submodular functions. Our results apply in a wide variety of real-world scenarios, such as ranking products in online retail platforms, where complete knowledge of the utility function is often impossible to obtain. Our algorithm gives an empirically useful solution in such contexts, thus proving that limited data can be of great use in sequencing tasks. From a technical perspective, our results extend prior work on ``optimization from samples'' by generalizing from optimizing a set function to a sequence-dependent function.

Learning Submodular Sequencing from Samples

TL;DR

This work tackles sequential submodular maximization when the utility functions are not directly accessible and only samples of sequences with observed utilities are available. It introduces a learning-from-samples framework under a two-stage uniform sampling distribution and develops a curvature-aware algorithm that reduces the problem to a maximum weight matching over item-position pairs. The main theoretical contribution is a bound showing that the produced sequence achieves a near-optimal fraction of , with guarantees that depend on the curvature and a sampling-dependent factor , under polynomially many samples. Practically, this enables productive sequencing and ranking in settings like online retail, where the full utility function is difficult to estimate, demonstrating that limited data can yield strong sequence-level performance guarantees.

Abstract

This paper addresses the problem of sequential submodular maximization: selecting and ranking items in a sequence to optimize some composite submodular function. In contrast to most of the previous works, which assume access to the utility function, we assume that we are given only a set of samples. Each sample includes a random sequence of items and its associated utility. We present an algorithm that, given polynomially many samples drawn from a two-stage uniform distribution, achieves an approximation ratio dependent on the curvature of individual submodular functions. Our results apply in a wide variety of real-world scenarios, such as ranking products in online retail platforms, where complete knowledge of the utility function is often impossible to obtain. Our algorithm gives an empirically useful solution in such contexts, thus proving that limited data can be of great use in sequencing tasks. From a technical perspective, our results extend prior work on ``optimization from samples'' by generalizing from optimizing a set function to a sequence-dependent function.
Paper Structure (9 sections, 6 theorems, 21 equations, 1 algorithm)

This paper contains 9 sections, 6 theorems, 21 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Formulation
  4. Utility Function
  5. Problem Formulation
  6. Algorithm Design
  7. Performance Analysis
  8. Discussions
  9. Appendix

Key Result

Lemma 1

Assume $f_t$ is a monotone submodular function with curvature $c$ for all $t\in \{1, 2, \cdots, k\}$, for the case when $(1-c)^2 \geq \alpha\cdot \frac{1-c}{1+c-c^2}$, we have that, with a sufficiently large polynomial number of samples, where $\alpha=\frac{n-k}{n}\cdot \frac{n-k-1}{n-1} \cdot \ldots \cdot \frac{n-2k+1}{n-k+1}$.

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Lemma 5