Guarantees for Greedy Maximization of Non-submodular Functions with Applications (1703.02100v4)

Abstract: We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of the (generalized) curvature $\alpha$ and the submodularity ratio $\gamma$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $\frac{1}{\alpha}(1- e^{-\gamma\alpha})$ for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, including the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for both synthetic and real-world applications.

• The paper derives a tight approximation guarantee for Greedy maximization by combining generalized curvature and submodularity ratios.
• It validates theoretical results with extensive experiments in applications like experimental design and sparse modeling.
• The study provides parameter boundaries to assess submodularity, guiding practical improvements in algorithm performance.

Insights into Greedy Maximization of Non-submodular Functions

The research paper, "Guarantees for Greedy Maximization of Non-submodular Functions with Applications," by Bian et al., addresses a critical gap in understanding the performance of the Greedy algorithm when applied to non-submodular, nondecreasing set functions subjected to cardinality constraints. While the empirical effectiveness of the Greedy algorithm in such settings is acknowledged, the theoretical underpinnings have been less explored, which this paper aims to rectify.

The fundamental contribution of the paper is the derivation of theoretical guarantees for the Greedy algorithm, characterized by a novel synthesis of two parameters: the generalized curvature $\alpha$ and the submodularity ratio $\gamma$. The authors demonstrate that the Greedy algorithm can provide a tight approximation guarantee of $\frac{1}{\alpha}(1-e^{-\gamma\alpha})$, significantly enhancing the understanding of its application to broader classes of functions beyond the traditional submodular context.

Theoretical Contributions

1. Approximation Guarantees: The paper establishes that the Greedy algorithm can achieve an approximation ratio of $\frac{1}{\alpha}(1-e^{-\gamma\alpha})$ for maximizing non-submodular, monotone set functions. This is a critical extension of the classical $(1-1/e)$ guarantee associated with submodular functions, offering a more comprehensive theoretical framework for analyzing Greedy's performance across different types of functions.
2. Empirical Validation: The authors substantiate their theoretical findings through extensive experiments with real-world applications, such as Bayesian A-optimality in experimental design and determinantal functions. They demonstrate that the Greedy algorithm frequently performs near-optimal and is computationally efficient, making it a viable alternative to more complex optimization methods like semidefinite programming (SDP).
3. Parameter Boundaries: The paper provides bounds on the submodularity ratio and curvature for several practical objectives, including those related to Gaussian processes and linear programs with combinatorial constraints. These parameter bounds are crucial as they measure how close a function is to being submodular, informing the approximate performance of Greedy in practice.
4. Extensions and Generalizations: By considering the generalized notions of curvature and submodularity ratio, this paper extends the applicability of the Greedy algorithm to a wider collection of functions, paving the way for novel applications in sparse modeling and broader experimental design problems.

Implications and Future Work

The implications of this research are twofold: firstly, it provides a stronger theoretical foundation for utilizing Greedy in non-traditional settings, potentially extending to more complex combinatorial problems often encountered in machine learning and data science. Secondly, it suggests avenues for further research in tuning the (generalized) curvature and submodularity parameters to optimize the algorithm's performance in specific scenarios.

The research prompts further exploration into non-submodular function classes and their inherent properties, which affect optimization performance. Future work could focus on developing adaptive or heuristic algorithms leveraging these insights to automatically adjust to the curvature and submodularity characteristics of the function landscape, thereby optimizing Greedy's efficacy further.

In conclusion, this paper transforms the empirical effectiveness of the Greedy algorithm in non-submodular scenarios into a robust theoretical framework, equipping researchers with the tools to predict and adjust its performance across diverse applications. The balance between theoretical rigor and empirical validation sets a foundation for advancing optimization algorithms tailored for complex, non-submodular environments.