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An Elementary Proof of the Near Optimality of LogSumExp Smoothing

Published in ArXiv Preprint, 2025

The paper proves a sharp limitation on smoothing the max-of-coordinates function in \(d\) dimensions. Any convex surrogate with the desired smoothness must incur a worst-case error that grows like \(\log d\), so the standard LogSumExp smoothing is essentially optimal up to constants. This result is proved using elementary inequalities about smooth, convex functions. The paper also proves that in small dimensions (\(d = 2, 3\)) LogSumExp fails to be optimal.

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Teaching experience 1

Undergraduate course, University 1, Department, 2014

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Teaching experience 2

Workshop, University 1, Department, 2015

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