Strong lottery ticket conjecture at virtually all sparsity levels

Let $g$ be a feed-forward network, let $K$ be a compact subset of the input space, and let $\epsilon>0$ and $\delta\in(0,1)$. For a randomly initialized network with equivalent input and output dimensions and widths

$$\min\{n_i:1\leq i\leq \ell-1\}\geq N,$$

write $f_{W\odot H}$ for the function obtained by applying a mask $H$ to its weights, and call $s$ a realizable sparsity when such a mask has sparsity $s$.

Strong lottery ticket conjecture at virtually all sparsity levels. There exists $N$ such that, with probability at least $\delta$, the network contains a mask $H$ with sparsity $s$ satisfying

$$\|f_{W\odot H}-g\|_{K,\infty}<\epsilon$$

for all realizable sparsities $s\in(0.5-\eta,0.5+\eta)$, where $\eta\to0.5$ as $N\to\infty$.

This conjecture strengthens the strong lottery ticket hypothesis by proposing subnetworks approximating any prescribed continuous target on a compact set across a widening range of sparsities, with the range approaching all sparsities between zero and one as the network width grows. The source reports empirical motivation from repeated applications of the double-scoring procedure, but gives no resolution of the conjecture.