Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach

Amir Ali Farzin, Yuen-Man Pun, Philipp Braun, Tyler Summers, Iman Shames

TL;DR

The paper tackles offline and online minimax optimization for non-smooth submodular-concave costs by introducing a zeroth-order extragradient method that uses the Lovász extension for subgradients with respect to the minimiser and Gaussian smoothing for gradient estimation with respect to the maximiser, relying only on function evaluations. It proves convergence to an $\epsilon$-saddle point in the offline setting with $O(m^2\epsilon^{-2})$ queries and obtains an online duality gap bound of $O(\sqrt{N\bar{P}_N})$ in the online setting, along with guidance on hyperparameters and complexity. The approach is validated through numerical experiments on adversarial image segmentation, showing robustness and real-time performance without pretraining, and it outperforms baseline U-Nets in online and adversarial scenarios. This work provides a practical, theoretically grounded framework for zeroth-order minimax optimization of combinatorial, non-smooth objectives with applications to robust perception tasks.

Abstract

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lovász extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an $ε$-saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves $O(\sqrt{N\bar{P}_N})$ online duality gap, where $N$ is the number of iterations and $\bar{P}_N$ is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.

Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach

TL;DR

The paper tackles offline and online minimax optimization for non-smooth submodular-concave costs by introducing a zeroth-order extragradient method that uses the Lovász extension for subgradients with respect to the minimiser and Gaussian smoothing for gradient estimation with respect to the maximiser, relying only on function evaluations. It proves convergence to an -saddle point in the offline setting with queries and obtains an online duality gap bound of in the online setting, along with guidance on hyperparameters and complexity. The approach is validated through numerical experiments on adversarial image segmentation, showing robustness and real-time performance without pretraining, and it outperforms baseline U-Nets in online and adversarial scenarios. This work provides a practical, theoretically grounded framework for zeroth-order minimax optimization of combinatorial, non-smooth objectives with applications to robust perception tasks.

Abstract

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lovász extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an -saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves online duality gap, where is the number of iterations and is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.
Paper Structure (23 sections, 7 theorems, 104 equations, 12 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 7 theorems, 104 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Min-Max Problems and Saddle Points
  4. Submodular Functions and the Lovász Extension
  5. Relations between \ref{['mainp']} and \ref{['mainp2']}
  6. Gaussian smoothing
  7. Main Results
  8. Zeroth-order Framework
  9. Offline Settings
  10. Online Settings
  11. Numerical Example
  12. Offline Adversarial Image Segmentation
  13. Online Adversarial Image Segmentation
  14. Conclusions and Future Work
  15. Complementary Materials on Submodular Functions
  16. ...and 8 more sections

Key Result

Lemma 2.7

Under Assumption assum:main, let $f^L:[0,1]^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be the Lovász extension of $f$ with respect to its first variable. Then, $f^L$ is a convex-concave function. Moreover, $\max_{y\in\mathcal{Y}}f^L(x,y) = \max_{y\in\mathcal{Y}}E_\tau[f(S_\tau,y)]$, where $S_\tau

Figures (12)

  • Figure 1: Offline Image Segmentation
  • Figure 2: Online image segmentation under adversarial attack.
  • Figure 3: Offline Image Segmentation
  • Figure 4: Lovász extension history over iterations for online image segmentation.
  • Figure 5: Online image segmentation under adversarial attack using Algorithm \ref{['alg:ZOEG']} at different times.
  • ...and 7 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Submodular Functions hazan2012online, Sec 2.1
  • Definition 2.4: Lovász Extension hazan2012online, Def 5
  • Remark 2.5: hazan2012online, Def. 5
  • Lemma 2.7
  • Definition 2.8: Maximal Chain Associated with $x$ hazan2012online, Def. 6
  • Definition 2.9: Lovász Extension Subgradient hazan2012online, Prop. 7
  • Proposition 2.10
  • Proposition 2.11
  • ...and 18 more