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We address the scaling behaviour of contour-shape-dependent ultra-violetsingularities of the light-like cusped Wilson loops in Yang-Mills and ${\cal N}= 4$ super-Yang-Mills theories in the higher orders of the perturbativeexpansion. We give the simple arguments to support the idea that identifying ofa special type of non-local infinitesimal shape variations of the light-likeWilson polygons with the Fr\'echet differentials results in the combinedgeometric and renormalization-group evolution equation, which is applicablebeyond the leading order exponentiated Wilson loops.

On Geometric Scaling of Light-Like Wilson Polygons: Higher Orders in $\alpha_s$