The cosine measure of a function at a point

What the math gives to ML

The paper introduces a function-dependent analogue of the cosine measure: instead of asking whether a direction set covers the unit sphere geometrically, it asks how much directional coverage is required to guarantee a descent direction for a particular smooth or nonsmooth function at a point. This is potentially useful for neural optimization near flat or nonsmooth regions, where the gradient is zero, unreliable, or unavailable, and ordinary gradient descent provides no escape direction. The most direct transfer is a blockwise directional-search optimizer that maintains a positively spanning perturbation set, estimates its directional descent coverage from minibatch loss evaluations, and invokes it only when gradient progress stalls.