Paper
An Elementary Proof of the Near Optimality of LogSumExp Smoothing
Authors
Thabo Samakhoana, Benjamin Grimmer
Abstract
We consider the design of smoothings of the (coordinate-wise) max function in in the infinity norm. The LogSumExp function provides a classical smoothing, differing from the max function in value by at most . We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least . Hence, LogSumExp is optimal up to constant factors. However, in small dimensions, we provide stronger, exactly optimal smoothings attaining our lower bound, showing that the entropy-based LogSumExp approach to smoothing is not exactly optimal.