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.

翻译：我们研究自适应组合最大化问题，这是机器学习中的核心挑战，在主动学习及众多其他领域具有广泛应用。我们聚焦贝叶斯设定，考虑基数约束下最大化与最小成本覆盖两种目标。本文提出了新的综合近似保证，不仅涵盖了先前研究成果，还显著强化了这些保证。我们的近似保证同时支持最大增益比与近子模效用函数，并涵盖基数约束下最大化与最小成本覆盖保证。此外，我们针对修正先验提供了近似保证，这对于获得不依赖于先验最小概率的主动学习保证至关重要。进一步地，我们发现了自适应选择策略的新参数——"最大增益比"。研究表明，该参数严格弱于先前近似保证中使用的贪婪近似参数，且能提供比先前结果更强的近似保证。具体而言，我们证明最大增益比永远不会超过策略的贪婪近似因子，且可能远小于后者。这为理解何种特性使策略适用于自适应组合最大化提供了全新视角。