Submodular functions -- functions exhibiting diminishing returns -- are central to machine learning. When the objective is monotone and non-negative, the greedy algorithm achieves a tight $63\%$ approximation. But many practical objectives incorporate costs that make them negative on some inputs, and all existing multiplicative guarantees require non-negativity. Prior work handles negativity through additive bounds for the special class of decomposable functions and non-monotonicity through partial-monotonicity parameters, but these address each difficulty in isolation and neither extends the classical structural theory. We extend \emph{curvature} -- a parameter measuring how far a function deviates from linearity -- to all submodular functions, handling both non-monotonicity and negativity through a single classical concept. A greedy algorithm with pruning achieves a curvature-controlled multiplicative ratio for \emph{any} submodular function, including those taking negative values -- the first such guarantee beyond monotonicity and non-negativity. In the non-monotone regime $1 \le c_g < 2.2$, the bound strictly beats the best known uniform ratio of $0.401$ (for non-negative $f$), and it recovers the classical $(1-e^{-c_g})/c_g$ guarantee for monotone functions. A multilinear-extension variant extends the framework to general combinatorial constraints via multilinear relaxation. Experiments on cost-penalized experimental design, coverage, feature selection, and a curvature sweep on Multi-News passage selection support the theory.

翻译：次模函数——具有边际递减性质的函数——是机器学习中的核心概念。当目标函数为单调且非负时，贪婪算法可实现严格的$63\%$近似。然而，许多实际目标函数因引入成本而使得部分输入呈现负值，而现有的所有乘法保证均要求非负性。先前工作通过加法界处理可分解函数这一特殊类别的负值问题，并通过部分单调性参数处理非单调性问题，但这些方法仅孤立地解决单个困难，未能扩展经典结构理论。我们将\emph{曲率}——衡量函数偏离线性程度的参数——推广至所有次模函数，通过单一经典概念同时处理非单调性和负值问题。带剪枝的贪婪算法可为\emph{任意}次模函数（包括取负值的函数）实现曲率控制的乘法比率——这是首个超越单调性和非负性假设的此类保证。在非单调情形下（$1 \le c_g < 2.2$），该界限严格优于已知的最佳统一比率$0.401$（针对非负$f$），并恢复了单调函数的经典$(1-e^{-c_g})/c_g$保证。多线性扩展变体通过多线性松弛将框架推广至一般组合约束。在带成本惩罚的实验设计、覆盖问题、特征选择以及基于Multi-News文章选择的曲率扫描实验均支持该理论。