second scenario is called function-independent structure, where the constructed network topology is independent of the choice of the target function in $ f\in\mathcal{F}_{d,n}$ with a given $\epsilon$. The principle behind these design choices (the network topology constructions and the choice of weights) is to achieve a tight upper bound as much as possible.

One might consider that we can transform an unquantized network within the error bound to a quantized one in a straightforward way by approximating every continuous-value weight with a combination of discrete weights with arbitrary accuracy. However, the complexity of such approximation (number of discrete weights needed) depends on the distribution of those continuous-value weights (e.g., their min and max), which may vary depending on the training data and network structure and a closed-form expression for the upper bounds is not possible. As such, a more elegant approach is needed. Below we will establish a constructive approach which allows us to bound the approximation analytically.

Function-independent Structure {#sec:independent}
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We start our analysis with function-independent structure, where the network topology is fixed for any $f\in \mathcal{F}_{d,n}$ and a given $\epsilon$. We first present the approximation of some basic functions by sub-networks in Section \[sqaure\]. We then present 