An Elementary Proof of the Near Optimality of LogSumExp Smoothing

Thabo Samakhoana; Benjamin Grimmer

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arXiv · 2512.10825

An Elementary Proof of the Near Optimality of LogSumExp Smoothing

Thabo Samakhoana·Benjamin Grimmer

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Abstract

We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ 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 $\sim 0.8145\ln(d)$. 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.

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Thabo Samakhoana, Benjamin Grimmer. 2025-12-11. An Elementary Proof of the Near Optimality of LogSumExp Smoothing. https://arxiv.org/abs/2512.10825

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