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.
From: Benjamin Grimmer [view email]
[v1]
Thu, 11 Dec 2025 17:17:48 UTC (15 KB)
[v2]
Mon, 19 Jan 2026 15:57:45 UTC (19 KB)
[v3]
Mon, 13 Jul 2026 02:19:03 UTC (24 KB)
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