Thanks for the suggestion! The reason average pooling is faster than max pooling is that the former uses only addition, which is quite a bit faster than comparison in MPC. The smooth max requires both exponentiation and division, both of which are significantly more expensive than doing the comparison in ReLU or max pooling.
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[ 1.7 ms ] story [ 23.7 ms ] threadWhy not use: https://en.m.wikipedia.org/wiki/Smooth_maximum