Optimal smoothings under non–inner-product norms (e.g., the infinity norm)

Develop optimal smoothing theory for sublinear convex functions when smoothness is measured with respect to norms that are not induced by an inner product, such as the infinity norm. Specifically, characterize the set of minimal and maximal optimal β-smooth approximations and the corresponding smoothability constants under these norms, so as to enable universal improvements over log-sum-exp smoothing for finite-maximum compositions g(x)=max_i G_i(x).

Background

The paper develops complete characterizations of optimal smoothings for sublinear functions and convex cones under Euclidean norms, yielding improved algorithmic performance over ad hoc smoothings such as log-sum-exp in certain settings.

However, the authors note that the improvement over log-sum-exp is not universal because log-sum-exp is also 1-smooth with respect to the infinity norm, and using norm-specific Lipschitz and smoothness constants can alter performance comparisons.

To achieve universal improvements, the authors highlight the need for new theory that extends optimal smoothing characterizations beyond inner-product-induced norms, particularly addressing the infinity norm.