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Upper-Linearizability of Online Non-Monotone DR-Submodular Maximization over Down-Closed Convex Sets

Yiyang Lu, Haresh Jadav, Mohammad Pedramfar, Ranveer Singh, Vaneet Aggarwal

TL;DR

This work obtains static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback.

Abstract

We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is $1/e$-linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain $O(T^{1/2})$ static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.

Upper-Linearizability of Online Non-Monotone DR-Submodular Maximization over Down-Closed Convex Sets

TL;DR

This work obtains static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback.

Abstract

We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is -linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.
Paper Structure (23 sections, 18 theorems, 49 equations, 1 table, 10 algorithms)

This paper contains 23 sections, 18 theorems, 49 equations, 1 table, 10 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Notations
  5. Problem Setting: Adversarial Online Optimization
  6. Limited Feedback Setting
  7. Regret
  8. Upper Linearizability
  9. Non-monotone DR-submodular Functions over Down-Closed Convex Sets are Upper-Linearizable
  10. Regret Results Applied from Linearizable Optimization
  11. Static Regret
  12. Adaptive and Dynamic Regrets
  13. Semi-Bandit, Zeroth-order Full-Information, and Bandit Feedback
  14. Conclusion and Future Work
  15. Useful Lemmas
  16. ...and 8 more sections

Key Result

Lemma 1

Let $f$ be a non-monotone continuous DR-submodular function over a down-closed convex set. For any ${\mathbf{x}}, {\mathbf{y}} \in [0,1]^d$ and $z \in [0,1]$, let $h_z({\mathbf{x}})=\mathbf{1} - e^{-z{\mathbf{x}}}$, we have: where $\bar{x} = \max_j [{\mathbf{x}}]_j$. (here $[{\mathbf{x}}]_j$ is the $j$-th component of the vector ${\mathbf{x}}$).

Theorems & Definitions (28)

  • Remark 1: Optimal Approximation Ratio $\alpha$
  • Definition 1: Upper Linearizability
  • Lemma 1
  • Theorem 1: Main Theorem
  • proof
  • Remark 2: Expectation form of $\nabla F$
  • Lemma 2: Properties of BQND Estimator
  • proof
  • Proposition 1: Static Regret
  • proof
  • ...and 18 more