证明对最大值函数的平滑逼近中，LogSumExp几乎最优。
An Elementary Proof of the Near Optimality of LogSumExp Smoothing
- 通过简洁构造证明所有上界平滑必须至少差0.8145ln(d)
- LogSumExp与最优值仅差常数因子，接近理论极限
- 在小维度下给出精确最优平滑，突破熵基方法局限
我们研究了在R^d中对坐标方向最大值函数在无穷范数下的平滑设计。经典LogSumExp函数f(x)=ln(∑_i^d exp(x_i))提供了平滑近似，其与最大值函数的差异不超过ln(d)。本文通过一个初等构造给出了下界证明，表明所有上界平滑至少需相差∼0.8145ln(d)。因此，LogSumExp在常数因子意义下是近似最优的。然而，我们还构造出严格更优的平滑方案，说明基于熵的LogSumExp方法并非完全最优。在低维情形下，我们提出了能精确达到该下界的最优平滑。
原文摘要 · Abstract (English)
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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