Adaptive Combinatorial Maximization: Beyond Approximate Greedy Policies
Shlomi Weitzman, Sivan Sabato
TL;DR
This work studies adaptive combinatorial maximization under Bayesian priors, introducing the maximal gain ratio $\beta_\pi(f,p)$ as a key policy-parameter that refines and sometimes surpasses traditional greedy-based guarantees. The authors establish comprehensive approximation guarantees for both utility maximization and minimum cost coverage that are parameterized by $\beta_\pi(f,p)$ and the adaptive submodularity ratio $\gamma^s_{n,k}(f,p)$, and show that $\beta_\pi(f,p) \le \alpha_\pi(f,p)$ with $\beta_\pi(f,p)$ potentially equal to 1 for non-greedy policies or arbitrarily small in some cases. They also extend guarantees to a modified prior $p'$, mitigating reliance on the smallest prior probability in Bayesian active learning. Overall, the paper unifies and strengthens adaptive guarantees across non-submodular utilities and non-greedy policies, and provides practical implications for active learning and adaptive experimentation.
Abstract
We study adaptive combinatorial maximization, which is a core challenge in machine learning, with applications in active learning as well as many other domains. We study the Bayesian setting, and consider the objectives of maximization under a cardinality constraint and minimum cost coverage. We provide new comprehensive approximation guarantees that subsume previous results, as well as considerably strengthen them. Our approximation guarantees simultaneously support the maximal gain ratio as well as near-submodular utility functions, and include both maximization under a cardinality constraint and a minimum cost coverage guarantee. In addition, we provided an approximation guarantee for a modified prior, which is crucial for obtaining active learning guarantees that do not depend on the smallest probability in the prior. Moreover, we discover a new parameter of adaptive selection policies, which we term the "maximal gain ratio". We show that this parameter is strictly less restrictive than the greedy approximation parameter that has been used in previous approximation guarantees, and show that it can be used to provide stronger approximation guarantees than previous results. In particular, we show that the maximal gain ratio is never larger than the greedy approximation factor of a policy, and that it can be considerably smaller. This provides a new insight into the properties that make a policy useful for adaptive combinatorial maximization.