My goal was to create a cover for my function class $\mathcal{H}$ in terms of the cover of $\mathcal{B}$ and $\mathcal{T}$
all ws here are weights. All functions are neural net parameterized functions hence follow lipschitz property. $L_B$ and $L_T$ are Lipschitz constant.
$$ \mathcal{B}:=\left\{\mathrm{B}_{\boldsymbol{w}} \mid \mathrm{B}: \mathbb{R}^{d_{1}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathrm{B}_{\boldsymbol{w}}\right) \leq L_{B} \&\|\boldsymbol{w}\| \leq W_{B}\right\} $$ $$ \mathcal{T}:=\left\{\mathbf{T}_{\boldsymbol{w}} \mid \mathbf{T}: \mathbb{R}^{d_{2}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathbf{T}_{\boldsymbol{w}}\right) \leq L_{T} \&\|\boldsymbol{w}\| \leq W_{T}\right\} $$ Now consider the following function class of all possible inner-products between the two, $$ \mathcal{H}:=\left\{h_{\boldsymbol{w}_{b}, \boldsymbol{w}_{t}} \mid \mathbb{R}^{d_{1}} \times \mathbb{R}^{d_{2}} \ni(s, p) \mapsto h(s, p):=\left\langle\mathrm{B}_{\boldsymbol{w}_{b}}(s), \mathbf{T}_{\boldsymbol{w}_{t}}(p)\right\rangle \in \mathbb{R}, \mathrm{B}_{\boldsymbol{w}_{b}} \in \mathcal{B} \& \mathbf{T}_{\boldsymbol{w}_{t}} \in \mathcal{T}\right\} $$
Then I find this theorem named as Pollard(1990)
Theorem 29.7. (POLLARD (1990)). Let $\mathcal{F}$ and $\mathcal{G}$ be classes of real functions on $\mathcal{R}^{d}$, bounded by $M_{1}$ and $M_{2}$, respectively. (That is, e.g., $|f(x)| \leq M_{1}$ for every $x \in \mathcal{R}^{d}$ and $f \in \mathcal{F}$.) For arbitrary fixed points $z_{1}^{n}=\left(z_{1}, \ldots, z_{n}\right)$ in $\mathcal{R}^{d}$ define the sets $\mathcal{F}\left(z_{1}^{n}\right)$ and $\mathcal{G}\left(z_{1}^{n}\right)$ in $\mathcal{R}^{n}$ as in Theorem 29.6. Introduce $$ \mathcal{J}\left(z_{1}^{n}\right)=\left\{\left(h\left(z_{1}\right), \ldots, h\left(z_{n}\right)\right) ; h \in \mathcal{J}\right\} $$ for the class of functions $$ \mathcal{J}=\{f g ; f \in \mathcal{F}, g \in \mathcal{G}\} $$ Then for every $\epsilon>0$ and $z_{1}^{n}$ $$ \mathcal{N}\left(\epsilon, \mathcal{J}\left(z_{1}^{n}\right)\right) \leq \mathcal{N}\left(\frac{\epsilon}{2 M_{2}}, \mathcal{F}\left(z_{1}^{n}\right)\right) \cdot \mathcal{N}\left(\frac{\epsilon}{2 M_{1}}, \mathcal{G}\left(z_{1}^{n}\right)\right) $$
It is kind of the theorem which I was looking for. But how to generalize it in terms of my $\mathcal{B}$, $\mathcal{T}$ and $\mathcal{H}$ so that I can use it. Clearly that theorem is set for q = 1. Can anyone help me with this? TIA
N.B :- The book link from which I have read about pollard theorem(1990),Page-500