# Automatically Bounding the Taylor Remainder Series: Tighter Bounds and New Applications

arXiv: https://arxiv.org/abs/2212.11429 · HTML: https://mlsanity.com/paper/2212.11429/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 54 references that are in the map.
- First posted on arXiv 22 Dec 2022.

## Builds on (most cited first, 2 of 2)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)

## Built on by (most cited first, 5 of 5)

- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)
- [Scalable Optimization in the Modular Norm](https://mlsanity.com/paper/2405.14813/) (2024)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Universal Majorization-Minimization Algorithms](https://mlsanity.com/paper/2308.00190/) (2023)
- [No learning rates needed: Introducing SALSA -- Stable Armijo Line Search Adaptation](https://mlsanity.com/paper/2407.20650/) (2024)

## Abstract (arXiv)

We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function f: ℝ → ℝ, our algorithm takes as input a reference point x0, trust region [a, b], and integer k ≥ 1, and returns an interval I such that f(x) - sumi=0k-1 frac 1 i! f(i)(x0) (x - x0)i ∈ I (x - x0)k for all x ∈ [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.