Proposition: The supplied regrowth rule lacks this guarantee

Ranking coordinates by accumulated absolute minibatch gradients and initializing the chosen coordinates with independent centered Gaussians does not generally produce a signed first-order descent step.

In particular, random initialization can increase expected loss even at a coordinate with nonzero gradient.

Proof. Accumulated absolute gradients need not equal the absolute gradient of the averaged loss: gradients of opposite signs can cancel in the average but not in the score. Centered random initialization also has zero expected linear alignment with a fixed gradient.

For an exact example, take $L(w)=(w-v{)}^2/2$, $w=0$, $v\ne 0$, and initialize the selected coordinate by $sG$, with $G\sim N(0,1)$. The coordinate has nonzero gradient $-v$, but

Subsequent training may change that result, but is a separate map.