Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints (1311.2106v1)

Abstract: We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.

• The paper presents approximation algorithms that transform submodular cover and knapsack problems into scalable formulations with bounded guarantees.
• The paper employs surrogate functions and modular decompositions to achieve strong empirical performance in tasks like sensor placement and data selection.
• The paper establishes a polynomial transformation between cover and knapsack constraints, unifying these frameworks for broader practical applicability.

Overview of Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints

This paper presents significant advancements in the domain of submodular optimization, specifically focusing on two novel optimization constructs: minimizing a submodular function with a submodular lower bound constraint, termed as submodular cover, and maximizing a submodular function with an upper bound constraint, referred to as submodular knapsack. These constructs address complex issues in practical machine learning tasks, such as sensor placement and data subset selection, where one often aims to optimize a function measuring coverage or diversity while minimizing cooperative costs.

Submodular functions, known for their diminishing returns property, are vital in numerous machine learning applications due to their inherent properties that align with real-world operations. The paper elaborates on transforming traditional problem formulations into constrained optimizations, thereby achieving bounded approximation guarantees which are otherwise inapproximable when dealing with the minimization of differences between submodular functions.

Key Contributions and Theoretical Insights

1. Approximation Algorithms: The paper proposes algorithms that employ surrogate functions and decompositions based on modular approximations to provide approximation guarantees for both the submodular cover and submodular knapsack problems. Notably, these algorithms demonstrate good scalability and better empirical performance than worst-case theoretical bounds suggest.
2. Empirical and Theoretical Results: Strong empirical results accompanying the proposed algorithms substantiate their practical applicability in machine learning contexts. The paper also outlines theoretical bounds grounded in the curvature of submodular functions, offering deeper insights into the parametric dependencies affecting approximation guarantees.
3. Polynomial Transformation Between Problems: The paper rigorously establishes a connection between the submodular cost-submodular cover (SCSC) and submodular cost-submodular knapsack (SCSK) problems, showing they can be polynomially transformed into each other. Algorithms solving one problem can be adapted to provide guarantees for the other, thus unifying these problems' complexities.
4. Extensions and Practical Implementability: By abstracting the insights obtained from simpler special cases such as submodular set cover and submodular cost knapsack, the presented algorithms are extended to tackle the more intricate real-world problem instances that combine multiple submodular constraints.

Computational and Practical Implications

The paper addresses critical computational challenges associated with submodular optimization, improving the practical implementability of these algorithms. The derived bounds and algorithms imply significant advancements over existing methods, particularly for large-scale learning applications where previously available solutions were computationally prohibitive. The iterative and greedy nature of the proposed solutions make them suitable for real-time and large-data applications, which is essential in areas like sensor placement and data subset selection.

Future Prospects in AI

The research presented in this paper sets a foundation for further exploration into optimization problems constrained by submodularity. This domain will likely expand towards more generalized functions and hybrid constraints, catalyzing advancements in AI and its applications to complex data-driven decisions. The notion of extending these techniques to non-monotone submodular functions marks an essential step towards embracing real-world data unpredictability, fostering enhanced model adaptability and robustness.

While the empirical testing focused mainly on theoretical problem constructs, future work could extend towards integrating these algorithms into fully-fledged machine learning pipelines, thereby facilitating a seamless amalgamation of theory and practice in AI.

This paper is a testament to the profound influence of computational theoretics on practical machine learning components, identifying new trajectories for optimizing and extending submodular function applications in AI.