[Paper Review] Non-monotone DR-submodular Maximization: Approximation and Regret Guarantees

Christoph Dürr, Nguyễn Kim Thắng+2 more|arXiv (Cornell University)|May 23, 2019

Complexity and Algorithms in Graphs33 references4 citations

TL;DR

This paper presents the first approximation and regret guarantees for maximizing non-monotone DR-submodular functions over general and down-closed convex sets in offline and online settings. It introduces a Frank-Wolfe algorithm achieving a $(1 - \min_{\bm{x} \in \mathcal{K}} \|\bm{x}\|_\infty)/(3\sqrt{3})$ approximation ratio for general convex sets and a Stochastic Gradient Ascent algorithm with a $1/4$-approximation and $O(1/\sqrt{T})$ regret for down-closed sets.

ABSTRACT

Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we study the fundamental problem of maximizing non-monotone DR-submodular functions over down-closed and general convex sets in both offline and online settings. First, we show that for offline maximizing non-monotone DR-submodular functions over a general convex set, the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. Next, we show that the Stochastic Gradient Ascent algorithm achieves a 1/4-approximation ratio with the regret of $O(1/\sqrt{T})$ for the problem of maximizing non-monotone DR-submodular functions over down-closed convex sets. These are the first approximation guarantees in the corresponding settings. Finally we benchmark these algorithms on problems arising in machine learning domain with the real-world datasets.

Motivation & Objective

• Address the lack of theoretical performance guarantees for non-monotone DR-submodular maximization over constrained convex sets, a common setting in real-world applications.
• Provide the first approximation algorithm for non-monotone DR-submodular functions over general convex sets, with convergence rate dependent on the domain structure.
• Develop an online algorithm with provable $(1/4, O(1/\sqrt{T}))$-regret for non-monotone DR-submodular maximization over down-closed convex sets.
• Bridge the gap in theoretical understanding for non-monotone, constrained DR-submodular optimization in both offline and online learning frameworks.

Proposed method

• Adapt the Frank-Wolfe algorithm for offline maximization over general convex sets, leveraging the DR-submodular structure to derive an approximation ratio dependent on the minimum infinity norm of points in the domain.
• Design a Stochastic Gradient Ascent algorithm for the online setting, where at each time step $t$, the algorithm selects $\bm{x}^t \in \mathcal{K}$ before observing the function $F^t$, and updates based on gradient feedback.
• Prove that the Stochastic Gradient Ascent algorithm achieves a $1/4$-approximation ratio with $O(1/\sqrt{T})$ regret, using properties of DR-submodularity and convexity.
• Introduce a specialized projection operator for the down-closed polytope $\mathcal{K} = \{ \bm{x} \geq 0 \mid \sum_i x_i \leq 1 \}$, enabling efficient computation via sorting and iterative averaging.
• Establish theoretical bounds on convergence and approximation by analyzing the curvature and submodular structure of the function $F$ over the feasible set $\mathcal{K}$.
• Implement and benchmark both algorithms on real-world datasets (Advogato, Facebook), comparing online performance against offline Frank-Wolfe baselines.

Experimental results

Research questions

• RQ1 What approximation ratio can be achieved for non-monotone DR-submodular maximization over general convex sets using the Frank-Wolfe algorithm?
• RQ2 Can a stochastic online algorithm achieve constant approximation and sublinear regret for non-monotone DR-submodular functions over down-closed convex sets?
• RQ3 How does the approximation ratio of the Frank-Wolfe algorithm depend on the geometry of the convex set, particularly $\min_{\bm{x} \in \mathcal{K}} \|\bm{x}\|_\infty$?
• RQ4 What is the convergence rate of the Frank-Wolfe algorithm for non-monotone DR-submodular functions over general convex sets?
• RQ5 How do the proposed online and offline algorithms compare empirically on real-world machine learning problems with non-monotone DR-submodular objectives?

Key findings

• The Frank-Wolfe algorithm achieves an approximation ratio of $\left(1 - \min_{\bm{x} \in \mathcal{K}} \|\bm{x}\|_\infty\right)/(3\sqrt{3})$ for non-monotone DR-submodular maximization over general convex sets, with a convergence rate of $O(n / \ln^2 T)$.
• When the convex set $\mathcal{K}$ contains the origin, the algorithm guarantees a $1/(3\sqrt{3}) \approx 0.192$ approximation after $T = O(e^{\sqrt{n/\epsilon}})$ iterations for any $\epsilon > 0$.
• The Stochastic Gradient Ascent algorithm achieves a $1/4$-approximation ratio with $O(1/\sqrt{T})$ regret for non-monotone DR-submodular maximization over down-closed convex sets.
• Empirical evaluation on the Advogato and Facebook datasets confirms the theoretical guarantees, showing the online algorithm's performance ratio relative to the offline baseline improves over time.
• The projection operator for the down-closed polytope is efficiently implementable in $O(n \log n)$ time via sorting and iterative averaging, enabling practical deployment.
• The performance ratio of the online algorithm relative to the offline optimal solution increases over time, consistent with the $O(1/\sqrt{T})$ regret bound.