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From Linear to Linearizable Optimization: A Novel Framework with Applications to Stationary and Non-stationary DR-submodular Optimization

Mohammad Pedramfar, Vaneet Aggarwal

TL;DR

The notion of upper-linearizable/quadratizable functions is introduced, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different convex sets, offering a unified approach to tackling concave and DR-submodular optimization problems.

Abstract

This paper introduces the notion of upper-linearizable/quadratizable functions, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different convex sets. A general meta-algorithm is devised to convert algorithms for linear/quadratic maximization into ones that optimize upper-linearizable/quadratizable functions, offering a unified approach to tackling concave and DR-submodular optimization problems. The paper extends these results to multiple feedback settings, facilitating conversions between semi-bandit/first-order feedback and bandit/zeroth-order feedback, as well as between first/zeroth-order feedback and semi-bandit/bandit feedback. Leveraging this framework, new algorithms are derived using existing results as base algorithms for convex optimization, improving upon state-of-the-art results in various cases. Dynamic and adaptive regret guarantees are obtained for DR-submodular maximization, marking the first algorithms to achieve such guarantees in these settings. Notably, the paper achieves these advancements with fewer assumptions compared to existing state-of-the-art results, underscoring its broad applicability and theoretical contributions to non-convex optimization.

From Linear to Linearizable Optimization: A Novel Framework with Applications to Stationary and Non-stationary DR-submodular Optimization

TL;DR

The notion of upper-linearizable/quadratizable functions is introduced, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different convex sets, offering a unified approach to tackling concave and DR-submodular optimization problems.

Abstract

This paper introduces the notion of upper-linearizable/quadratizable functions, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different convex sets. A general meta-algorithm is devised to convert algorithms for linear/quadratic maximization into ones that optimize upper-linearizable/quadratizable functions, offering a unified approach to tackling concave and DR-submodular optimization problems. The paper extends these results to multiple feedback settings, facilitating conversions between semi-bandit/first-order feedback and bandit/zeroth-order feedback, as well as between first/zeroth-order feedback and semi-bandit/bandit feedback. Leveraging this framework, new algorithms are derived using existing results as base algorithms for convex optimization, improving upon state-of-the-art results in various cases. Dynamic and adaptive regret guarantees are obtained for DR-submodular maximization, marking the first algorithms to achieve such guarantees in these settings. Notably, the paper achieves these advancements with fewer assumptions compared to existing state-of-the-art results, underscoring its broad applicability and theoretical contributions to non-convex optimization.
Paper Structure (27 sections, 23 theorems, 78 equations, 1 figure, 3 tables)

This paper contains 27 sections, 23 theorems, 78 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Setup and Definitions
  3. Formulation of Upper-Quadratizable Functions and Regret Relation to that of Quadratic Functions
  4. Up-concave function optimization is upper-quadratizable function optimization
  5. Monotone up-concave optimization over general convex sets
  6. Monotone up-concave optimization over convex sets containing the origin
  7. Non-monotone up-concave optimization over general convex sets
  8. Meta algorithms for other feedback cases
  9. Applications
  10. Conclusions
  11. Acknowledgement
  12. Related works
  13. DR-submodular maximization
  14. Up-concave maximization
  15. Projection-free optimization
  16. ...and 12 more sections

Key Result

Theorem 1

Let ${\mathcal{A}}$ be algorithm for online optimization with semi-bandit feedback. Also let ${\mathbf{F}}$ be a function class over ${\mathcal{K}}$ that is quadratizable with $\mu \geq 0$ and maps ${\mathfrak{g}} : {\mathbf{F}} \times {\mathcal{K}} \to {\mathbb{R}}^d$ and $h : {\mathcal{K}} \to {\m

Figures (1)

  • Figure 1: Summary of applications (See Section \ref{['sec:applications']})

Theorems & Definitions (37)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Corollary 1
  • Lemma 2
  • Theorem 3
  • Corollary 2
  • Lemma 3
  • Theorem 4
  • Corollary 3
  • ...and 27 more