be reduced by first doing interpolation at a coarser scale and then fill the details in the intervals to make the error go down to $\epsilon$. By assigning a “cached” function to every interval depending on specific function and proper scaling, the number of weights is reduced to $\mathcal{O}\left(\left(\log^{-1}\left(1/\epsilon\right)\right)1/\epsilon\right)$ when there is no constraint on weight values [@yarotsky2017error].

The key difficulty in applying this approach to quantized ReLU networks is that the required linear interpolation at $\frac{i}{T}$ exactly where $i = 1,2,\cdots,T$ is not feasible because of the constraint on weight selection. To this end, we transform $f(x)$ to $\widetilde{f}(x)$ such that the approximation error is bounded; the Lipschitz constant is preserved; $\widetilde{f}\left(\frac{i}{T}\right)$ are reachable for the network under the constraints of weight selection without increasing the requirement on weight precision. Then we can apply the interpolation and cached function method on $\widetilde{f}(x)$ and finally approximate $f(x)$ with a quantized ReLU network. Formally, we get the following proposition and the proof can be found in Appendix \[PP4\].

\[ftp\] For any $f\in \mathcal{F}_{1,1}$, $t\in \mathbb{Z^+}$, and $T\in \mathbb{Z^+}$, there exists a function $\widetilde{f}(x)$ such that (1) $\widetilde{f}(x)$ is a continuous function with Lipschitz constant 1; (2) $\widetilde{f}(\frac{i}{T})$ = $\left\lceil T f\left(\frac{i}{T}\right) 