Guarantees for Greedy Maximization of Non-submodular Functions with Applications
This provides theoretical guarantees for a widely used algorithm in optimization problems, addressing a gap for non-submodular functions, which is incremental but important for applications like experimental design.
The paper tackles the problem of maximizing non-submodular set functions using the Greedy algorithm, proving a tight approximation guarantee of 1/α(1 - e^{-γα}) based on curvature and submodularity ratio, and applies this to real-world objectives like Bayesian A-optimality.
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 $α$ and the submodularity ratio $γ$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $\frac{1}α(1- e^{-γα})$ 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.