We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function $f: \mathbb{R} \to \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 1$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$ for all $x \in [a, b]$. As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known atomic functions. At a high level, our algorithm has two steps. First, for a variety of commonly-used elementary functions (e.g., $\exp$, $\log$), we use recently-developed theory to derive sharp polynomial upper and lower bounds on the Taylor remainder series. We then recursively combine the bounds for the elementary functions using an interval arithmetic variant of Taylor-mode automatic differentiation. Our algorithm can make efficient use of machine learning hardware accelerators, and we provide an open source implementation in JAX. We then turn our attention to applications. Most notably, in a companion paper we use our new machinery to create the first universal majorization-minimization optimization algorithms: algorithms that iteratively minimize an arbitrary loss using a majorizer that is derived automatically, rather than by hand. We also show that our automatically-derived bounds can be used for verified global optimization and numerical integration, and to prove sharper versions of Jensen's inequality.

翻译：我们提出了一种自动界定泰勒余项级数的新算法。在标量函数 $f: \mathbb{R} \to \mathbb{R}$ 的特殊情形下，我们的算法以参考点 $x_0$、信任区间 $[a, b]$ 和整数 $k \ge 1$ 为输入，并返回一个区间 $I$，使得对所有 $x \in [a, b]$，有 $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$。与自动微分类似，函数 $f$ 以符号形式提供给算法，且必须由已知的原子函数组成。在高层次上，我们的算法包含两个步骤。首先，针对一系列常用初等函数（例如 $\exp$、$\log$），我们利用近期发展的理论推导出泰勒余项级数的尖锐多项式上界和下界。然后，我们使用泰勒模式自动微分的区间算术变体，递归地组合这些初等函数的界。我们的算法能高效利用机器学习硬件加速器，并提供了基于JAX的开源实现。随后，我们转向应用。最值得注意的是，在配套论文中，我们利用这一新工具创建了首个通用化最大-最小化优化算法：这类算法通过自动而非手动推导的优化算子，迭代地最小化任意损失函数。我们还展示了自动推导的界可应用于验证性全局优化、数值积分，以及证明更紧的詹森不等式形式。