  A soft-max function has two main efficiency measures: (1) approximation -
which corresponds to how well it approximates the maximum function, (2)
smoothness - which shows how sensitive it is to changes of its input. Our goal
is to identify the optimal approximation-smoothness tradeoffs for different
measures of approximation and smoothness. This leads to novel soft-max
functions, each of which is optimal for a different application. The most
commonly used soft-max function, called exponential mechanism, has optimal
tradeoff between approximation measured in terms of expected additive
approximation and smoothness measured with respect to R\'enyi Divergence. We
introduce a soft-max function, called "piecewise linear soft-max", with optimal
tradeoff between approximation, measured in terms of worst-case additive
approximation and smoothness, measured with respect to $\ell_q$-norm. The
worst-case approximation guarantee of the piecewise linear mechanism enforces
sparsity in the output of our soft-max function, a property that is known to be
important in Machine Learning applications [Martins et al. '16, Laha et al.
'18] and is not satisfied by the exponential mechanism. Moreover, the
$\ell_q$-smoothness is suitable for applications in Mechanism Design and Game
Theory where the piecewise linear mechanism outperforms the exponential
mechanism. Finally, we investigate another soft-max function, called power
mechanism, with optimal tradeoff between expected \textit{multiplicative}
approximation and smoothness with respect to the R\'enyi Divergence, which
provides improved theoretical and practical results in differentially private
submodular optimization.
