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A Unified Approach for Maximizing Continuous DR-submodular Functions

Mohammad Pedramfar, Christopher John Quinn, Vaneet Aggarwal

TL;DR

This work addresses maximizing continuous DR-submodular functions over convex feasible sets under varied oracle models (gradient or value, deterministic or stochastic). It introduces a unified Frank-Wolfe–style offline framework, featuring a Black-Box Gradient Estimator via smoothing and a shrunk constraint set to enable value queries inside the feasible region, achieving new or improved guarantees across $16$ settings and matching state-of-the-art in others, including projection-free variants. It extends offline results to online stochastic DR-submodular optimization with bandit and semi-bandit feedback, delivering the first regret bounds for bandit feedback in this domain. Collectively, the framework broadens the scope and practicality of DR-submodular maximization, offering a versatile toolkit for both offline and online problems with strong theoretical guarantees.

Abstract

This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in two cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining five cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions.

A Unified Approach for Maximizing Continuous DR-submodular Functions

TL;DR

This work addresses maximizing continuous DR-submodular functions over convex feasible sets under varied oracle models (gradient or value, deterministic or stochastic). It introduces a unified Frank-Wolfe–style offline framework, featuring a Black-Box Gradient Estimator via smoothing and a shrunk constraint set to enable value queries inside the feasible region, achieving new or improved guarantees across settings and matching state-of-the-art in others, including projection-free variants. It extends offline results to online stochastic DR-submodular optimization with bandit and semi-bandit feedback, delivering the first regret bounds for bandit feedback in this domain. Collectively, the framework broadens the scope and practicality of DR-submodular maximization, offering a versatile toolkit for both offline and online problems with strong theoretical guarantees.

Abstract

This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in two cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining five cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions.
Paper Structure (35 sections, 12 theorems, 118 equations, 5 figures, 3 tables, 3 algorithms)

This paper contains 35 sections, 12 theorems, 118 equations, 5 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work:
  3. Background and Notation
  4. Offline Algorithms and Guarantees
  5. Problem Setup
  6. Black Box Gradient Estimate
  7. Construction of Shrunk Constraint Set
  8. Generalized DR-Submodular Frank-Wolfe
  9. Approximation Guarantees for the Proposed Offline Algorithm
  10. Online DR-submodular optimization under bandit or semi-bandit feedback
  11. Problem Setup
  12. Algorithm for DR-submodular maximization with Bandit Feedback
  13. Regret Analysis for DR-submodular maximization with Bandit Feedback
  14. Conclusion
  15. Details of Related Works
  16. ...and 20 more sections

Key Result

Theorem 1

Suppose assumptions holds. Let $N \geq 4$, $B \geq 1$ and choose ${\mathbf{c}} \in {\mathcal{K}}$ and $r > 0$ according to Section SS:construction. If we have access to a gradient oracle, we choose $\delta = 0$, otherwise we choose $\delta \in (0, r/2)$. Then the following results hold for the outpu In all these cases, we have $C$ is a constant, $k = \operatorname{dim}({\mathcal{K}})$, $D = \oper

Figures (5)

  • Figure : Offline DR-submodular optimization results.
  • Figure : Online stochastic DR-submodular optimization.
  • Figure : Black Box Gradient Estimate (BBGE)
  • Figure : Generalized DR-Submodular Frank-Wolfe
  • Figure : DR-Submodular Explore-Then-Commit

Theorems & Definitions (29)

  • Remark 1
  • Definition 1: Smoothing Trick
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Lemma 1: Lemma 2.2 of mualem22_resol_approx_offlin_onlin_non
  • Lemma 2: Lemma 1 of durr19_nonmonotone
  • Lemma 3
  • ...and 19 more