# An Elementary Proof of the Near Optimality of LogSumExp Smoothing

arXiv: https://arxiv.org/abs/2512.10825 · HTML: https://mlsanity.com/paper/2512.10825/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 17 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [On Lower Complexity Bounds for Large-Scale Smooth Convex Optimization](https://mlsanity.com/paper/1307.5001/) (2013)
- [The Price of Adaptivity in Stochastic Convex Optimization](https://mlsanity.com/paper/2402.10898/) (2024)
- [Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions](https://mlsanity.com/paper/2010.11450/) (2020)
- [Radial Duality Part II: Applications and Algorithms](https://mlsanity.com/paper/2104.11185/) (2021)
- [Accelerated First-Order Methods for Hyperbolic Programming](https://mlsanity.com/paper/1512.07569/) (2015)
- [The Optimal Smoothings of Sublinear Functions and Convex Cones](https://mlsanity.com/paper/2508.06681/) (2025)

## Abstract (arXiv)

We consider the design of smoothings of the (coordinate-wise) max function in ℝd in the infinity norm. The LogSumExp function f(x)=ln(sumdiexp(xi)) provides a classical smoothing, differing from the max function in value by at most ln(d). We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least ~ 0.8145ln(d). Hence, LogSumExp is optimal up to small constant factors. However, we provide strictly stronger smoothings showing the entropy-based LogSumExp approach is not exactly optimal. In small dimensions, we propose exactly optimal smoothings, attaining our lower bound.