SlowGuess/NeurIPS_2023_OCR huggingface dataset:
By Step 4,  $\sqrt{\mathbb{E}[l(T,\delta,Z;\pi_n\nu_0,\pi_n\theta_0) - l(T,\delta,Z;\nu_0,\theta_0)]} = O(n^{-\frac{\beta}{2\beta + 2d + 2}})$ . Thus, we have  $d_{\mathsf{FN}}\left(\widehat{\psi}_n,\psi_0\right) = O(n^{-\frac{\beta}{2\beta + 2d + 2}}\log^2 n) = \widetilde{O}(n^{-\frac{\beta}{2\beta + 2d + 2}})$ .
B.4 Proofs of technical lemmas
Proof of lemma B.1. Since  $h_0(T) \in \mathcal{W}_M^\beta ([0,\tau])$  and  $m_0(Z) \in \mathcal{W}_M^\beta ([-1,1]^d)$ , we have that  $h_0(T) \leq M$ ,  $m_0(Z) \leq M$  and  $e^{m_0(Z)}\int_0^T h_0(s)ds \leq \tau e^{2M}$ . Let  $\mathcal{B} = [0,\tau e^{2M}]$ , we have that
$$
\begin{array}{l} \left| l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}) \right| \\ \leq \left| \log g _ {\theta_ {0}} \left(e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s\right) \right| + | h _ {0} (T) | + | m _ {0} (Z) | + \left| G _ {\theta_ {0}} \left(e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s\right) \right| \\ \leq 2 M + \sup  _ {x \in \mathcal {B}} | \log g _ {\theta_ {0}} (x) | + \sup  _ {x \in \mathcal {B}} | G _ {\theta_ {0}} (x) | \\ \end{array}
$$
By condition 5, we have that  $l(T,\delta,Z;h_0,m_0,\theta_0)$  is bounded for  $(T,\delta,Z) \in [0,\tau] \times \{0,1\} \times [-1,1]^d$ .
The proof of the boundness of  $l(T,\delta ,Z;\widehat{h},\widehat{m},\widehat{\theta})$  is similar.
Proof of lemma B.2. Since  $\nu_0(T,Z)\in \mathcal{W}_M^\beta ([0,\tau ]\times [-1,1]^d)$ , we have  $\nu_{0}(T,Z)\leq M$  and  $\int_0^T e^{\nu (s,Z)}ds\leq \tau e^M$ . Let  $\mathcal{B} = [0,\tau e^{M}]$ , we have that
$$
\begin{array}{l} \left| l (T, \delta , Z; \nu_ {0}, \theta_ {0}) \right| \\ \leq \left| \log G _ {\theta_ {0}} ^ {\prime} \left(\int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s\right) \right| + | \nu_ {0} (T, Z) | + \left| G _ {\theta_ {0}} \left(\int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s\right) \right| \\ \leq M + \sup  _ {x \in \mathcal {B}} \left| \log G _ {\theta_ {0}} ^ {\prime} (x) \right| + \sup  _ {x \in \mathcal {B}} \left| G _ {\theta_ {0}} (x) \right|. \\ \end{array}
$$
By condition 4.5, we have that  $l(T,\delta,Z;\nu_0,\theta_0)$  is bounded among  $(T,\delta,Z) \in [0,\tau] \times \{0,1\} \times [-1,1]^d$ . The proof of the boundness of  $l(T,\delta,Z;\widehat{\nu},\widehat{\theta})$  is similar.
Proof of lemma B.3. By definition, we have that
$$
\begin{array}{l} \left| l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {h}, \widehat {m}, \widehat {\theta}) \right| \\ \leq \left| \log g _ {\theta_ {0}} \left(e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s\right) - \log g _ {\widehat {\theta}} \left(e ^ {\widehat {m} (Z)} \int_ {0} ^ {T} e ^ {\widehat {h} (s)} d s\right) \right| + \left| h _ {0} (T) - \widehat {h} (T) \right| \\ + | m _ {0} (Z) - \widehat {m} (Z) | + \left| G _ {\theta_ {0}} \left(e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s\right) - G _ {\widehat {\theta}} \left(e ^ {\widehat {m} (Z)} \int_ {0} ^ {T} e ^ {\widehat {h} (s)} d s\right) \right|. \\ \end{array}
$$
Let  $\mathcal{B} = [0, \tau \max(e^{2M}, e^{M_h + M_m})]$ . By Taylor's expansion, we can further show that
$$
\begin{array}{l} \left| l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {h}, \widehat {m}, \widehat {\theta}) \right| \\ \leq \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| \\ + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left| e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s - e ^ {\hat {m} (Z)} \int_ {0} ^ {T} e ^ {\hat {h} (s)} d s \right| \\ + \left| h _ {0} (T) - \widehat {h} (T) \right| + \left| m _ {0} (Z) - \widehat {m} (Z) \right| + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| \\ + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left| e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s - e ^ {\widehat {m} (Z)} \int_ {0} ^ {T} e ^ {\widehat {h} (s)} d s \right|. \\ \end{array}
$$
Again, by Taylor's expansion, we have that
$$
\begin{array}{l} \left| e ^ {m _ {0} (Z)} \int_ {0} ^ {T} e ^ {h _ {0} (s)} d s - e ^ {\widehat {m} (Z)} \int_ {0} ^ {T} e ^ {\widehat {h} (s)} d s \right| \\ \leq \left| e ^ {m _ {0} (Z)} \int_ {0} ^ {T} \left(e ^ {h _ {0} (s)} - e ^ {\widehat {h} (s)}\right) d s \right| + \left| \left(e ^ {m _ {0} (Z)} - e ^ {\widehat {m} (Z)}\right) \int_ {0} ^ {T} e ^ {\widehat {h} (s)} d s \right| \\ \leq e ^ {M} \cdot \tau e ^ {\max (M, M _ {h})} \left\| h _ {0} - \widehat {h} \right\| _ {\infty} + \tau e ^ {M _ {h}} \cdot e ^ {\max (M, M _ {m})} \| m _ {0} - \widehat {m} \| _ {\infty}. \\ \end{array}
$$
$$
\begin{array}{l} \left| l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {h}, \widehat {m}, \widehat {\theta}) \right| \\ \leq \sup _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left[ e ^ {M} \cdot \tau e ^ {\max (M, M _ {h})} \| h _ {0} - \widehat {h} \| _ {\infty} + \tau e ^ {M _ {h}} \cdot e ^ {\max (M, M _ {m})} \| m _ {0} - \widehat {m} \| _ {\infty} \right] \\ + \sup _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| + \left| h _ {0} (T) - \widehat {h} (T) \right| + | m _ {0} (Z) - \widehat {m} (Z) | \\ + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| \\ + \sup _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left[ e ^ {M} \cdot \tau e ^ {\max (M, M _ {h})} \| h _ {0} - \widehat {h} \| _ {\infty} + \tau e ^ {M _ {h}} \cdot e ^ {\max (M, M _ {m})} \| m _ {0} - \widehat {m} \| _ {\infty} \right]. \\ \end{array}
$$
Taking supremum on both sides, we conclude that
$$
\| l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {h}, \widehat {m}, \widehat {\theta}) \| _ {\infty} \lesssim | \theta_ {0} - \widehat {\theta} | + \| h _ {0} - \widehat {h} \| _ {\infty} + \| m _ {0} - \widehat {m} \| _ {\infty}.
$$
The proof of the second inequality is similar.
Proof of lemma B.4. By definition, we have that
$$
\begin{array}{l} \left| l (T, \delta , Z; \nu_ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {\nu}, \widehat {\theta}) \right| \\ \leq \left| \log g _ {\theta_ {0}} \left(\int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s\right) - \log g _ {\widehat {\theta}} \left(\int_ {0} ^ {T} e ^ {\widehat {\nu} (s, Z)} d s\right) \right| + | \nu_ {0} (T, Z) - \widehat {\nu} (T, Z) | \\ + \left| G _ {\theta_ {0}} \left(\int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s\right) - G _ {\hat {\theta}} \left(\int_ {0} ^ {T} e ^ {\hat {\nu} (s, Z)} d s\right) \right|. \\ \end{array}
$$
Let  $\mathcal{B} = [0, \tau \max(e^M, e^{M_\nu})]$ . By Taylor's expansion, we can further show that
$$
\begin{array}{l} \left| l (T, \delta , Z; \nu_ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {\nu}, \widehat {\theta}) \right| \\ \leq \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left| \int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s - \int_ {0} ^ {T} e ^ {\widehat {\nu} (s, Z)} d s \right| \\ + \left| \nu_ {0} (T, Z) - \widehat {\nu} (T, Z) \right| + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| \\ + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \left| \int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s - \int_ {0} ^ {T} e ^ {\widehat {\nu} (s, Z)} d s \right|. \\ \end{array}
$$
Again, by Taylor's expansion,
$$
\left| \int_ {0} ^ {T} e ^ {\nu_ {0} (s, Z)} d s - \int_ {0} ^ {T} e ^ {\widehat {\nu} (s, Z)} d s \right| \leq \tau e ^ {\max  (M, M _ {\nu})} \| \nu_ {0} - \widehat {\nu} \| _ {\infty},
$$
$$
\begin{array}{l} \left| l (T, \delta , Z; \nu_ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {\nu}, \widehat {\theta}) \right| \\ \leq \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial \log g _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \tau e ^ {\max  (M, M _ {\nu})} \| \nu_ {0} - \widehat {\nu} \| _ {\infty} \\ + \left| \nu_ {0} (T, Z) - \widehat {\nu} (T, Z) \right| + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {\theta}}} \right| \cdot \left| \theta_ {0} - \widehat {\theta} \right| \\ + \sup  _ {\tilde {\theta} \in \Theta , \tilde {x} \in \mathcal {B}} \left| \frac {\partial G _ {\tilde {\theta}} (\tilde {x})}{\partial_ {\tilde {x}}} \right| \cdot \tau e ^ {\max  (M, M _ {\nu})} \| \nu_ {0} - \widehat {\nu} \| _ {\infty}. \\ \end{array}
$$
$$
\left\| l (T, \delta , Z; \nu_ {0}, \theta_ {0}) - l (T, \delta , Z; \widehat {\nu}, \widehat {\theta}) \right\| _ {\infty} \lesssim | \theta_ {0} - \widehat {\theta} | + \| \nu_ {0} - \widehat {\nu} \| _ {\infty},
$$
Proof of lemma B.5. According to [72, Theorem 1], there exist approximating functions  $\widehat{h}^*$  and  $\widehat{m}^*$  such that  $\| \widehat{h}^* - h_0 \|_{\infty} = O\left(n^{-\frac{\beta}{\beta + d}}\right)$  and  $\| \widehat{m}^* - m_0 \|_{\infty} = O\left(n^{-\frac{\beta}{\beta + d}}\right)$ . Let  $\pi_n h_0 = \widehat{h}^*$ ,
$\pi_n m_0 = \widehat{m}^*$ , and  $\pi_n \theta = \theta_0$ . We have that
$$
\begin{array}{l} d _ {\mathrm {P F}} \left(\pi_ {n} \phi_ {0}, \phi_ {0}\right) \\ = \sqrt {\mathbb {E} _ {Z} \left[ \int | \sqrt {p (T , \delta \mid Z ; \pi_ {n} h _ {0} , \pi_ {n} m _ {0} , \pi_ {n} \theta_ {0})} - \sqrt {p (T , \delta \mid Z ; h _ {0} , m _ {0} , \theta_ {0})} | ^ {2} \mu (d T \times d \delta) \right]} \\ = \sqrt {\mathbb {E} _ {Z} \left[ \int [ e ^ {\frac {1}{2} l (T , \delta , Z ; \pi_ {n} h _ {0} , \pi_ {n} m _ {0} , \pi_ {n} \theta_ {0})} - e ^ {\frac {1}{2} l (T , \delta , Z ; h _ {0} , m _ {0} , \theta_ {0})} ] ^ {2} f _ {C | Z} (T) ^ {1 - \delta} S _ {C | Z} (T) ^ {\delta} \mu (d T \times d \delta) \right]} \\ \leq \left\| e ^ {\frac {1}{2} l (T, \delta , Z; \pi_ {n} h _ {0}, \pi_ {n} m _ {0}, \pi_ {n} \theta_ {0})} - e ^ {\frac {1}{2} l (T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0})} \right\| _ {\infty} \\ \times \sqrt {\mathbb {E} _ {Z} \left[ \int f _ {C | Z} (T) ^ {1 - \delta} S _ {C | Z} (T) ^ {\delta} \mu (d T \times d \delta) \right]}. \\ \end{array}
$$
By lemma B.1 and B.3, we have that
$$
\begin{array}{l} \left\| e ^ {\frac {1}{2} l \left(T, \delta , Z; \pi_ {n} h _ {0}, \pi_ {n} m _ {0}, \pi_ {n} \theta_ {0}\right)} - e ^ {\frac {1}{2} l \left(T, \delta , Z; h _ {0}, m _ {0}, \theta_ {0}\right)} \right\| _ {\infty} \\ \lesssim \| \pi_ {n} \theta_ {0} - \theta_ {0} \| + \| \pi_ {n} h _ {0} - h _ {0} \| _ {\infty} + \| \pi_ {n} m _ {0} - m _ {0} \| _ {\infty} \\ = O \left(n ^ {- \frac {\beta}{\beta + d}}\right). \\ \end{array}
$$
Since  $f_{C|Z}(T)^{1 - \delta} \leq f_{C|Z}(T) + 1$  and  $S_{C|Z}(T)^{\delta} \leq 1$ , we also have that
$$
\begin{array}{l} \sqrt {\mathbb {E} _ {Z} \left[ \int f _ {C | Z} (T) ^ {1 - \delta} S _ {C | Z} (T) ^ {\delta} \mu (d T \times d \delta) \right]} \leq \sqrt {\mathbb {E} _ {Z} \left[ \int (1 + f _ {C | Z} (T)) \mu (d T \times d \delta) \right]} \\ \leq \sqrt {2 + 2 \tau}. \\ \end{array}
$$
Thus, we obtain that  $d_{\mathsf{PF}}(\pi_n\phi_0,\phi_0) = O\left(n^{-\frac{\beta}{\beta + d}}\right)$ .
Proof of lemma B.6. According to [72, Theorem 1], there exists an approximating function  $\widehat{\nu}^*$  such that  $\| \widehat{\nu}^{*} - \nu_{0}\|_{\infty} = O\left(n^{-\frac{\beta}{\beta + d + 1}}\right)$ . Let  $\pi_n\nu_0 = \widehat{\nu}^*$  and  $\pi_n\theta_0 = \theta_0$ . We have that
$$
\begin{array}{l} d _ {\mathrm {F N}} \left(\pi_ {n} \psi_ {0}, \psi_ {0}\right) \\ = \sqrt {\mathbb {E} _ {Z} \left[ \int \left| \sqrt {p (T , \delta \mid Z ; \pi_ {n} \nu_ {0} , \pi_ {n} \theta_ {0})} - \sqrt {p (T , \delta \mid Z ; \nu_ {0} , \theta_ {0})} \right| ^ {2} \mu (d T \times d \delta) \right]} \\ = \sqrt {\mathbb {E} _ {Z} \left[ \int \left[ e ^ {\frac {1}{2} l (T , \delta , Z ; \pi_ {n} \nu_ {0} , \pi_ {n} \theta_ {0})} - e ^ {\frac {1}{2} l (T , \delta , Z ; \nu_ {0} , \theta_ {0})} \right] ^ {2} f _ {C | Z} (T) ^ {1 - \delta} S _ {C | Z} (T) ^ {\delta} \mu (d T \times d \delta) \right]} \\ \leq \left\| \frac {1}{2} e ^ {l (T, \delta , Z; \pi_ {n} \nu_ {0}, \pi_ {n} \theta_ {0})} - \frac {1}{2} e ^ {l (T, \delta , Z; \nu_ {0}, \theta_ {0})} \right\| _ {\infty} \sqrt {\mathbb {E} _ {Z} \left[ \int f _ {C | Z} (T) ^ {1 - \delta} S _ {C | Z} (T) ^ {\delta} \mu (d T \times d \delta) \right]}. \\ \end{array}
$$
By lemma B.2 and B.4, we have that
$$
\begin{array}{l} \left\| e ^ {\frac {1}{2} l (T, \delta , Z; \pi_ {n} \nu_ {0}, \pi_ {n} \theta_ {0})} - e ^ {\frac {1}{2} l (T, \delta , Z; \nu_ {0}, \theta_ {0})} \right\| _ {\infty} \lesssim \| \pi_ {n} \theta_ {0} - \theta_ {0} \| + \| \pi_ {n} \nu_ {0} - \nu_ {0} \| _ {\infty} \\ = O \left(n ^ {- \frac {\beta}{\beta + d + 1}}\right). \\ \end{array}
$$
Thus, we obtain that  $d_{\mathsf{FN}}(\pi_n\psi_0,\psi_0) = O\left(n^{-\frac{\beta}{\beta + d + 1}}\right)$ .
Proof of lemma B.7. The left inequality is trivial according to the definition of covering number. We need to show that the correctness of the right inequality.
Suppose that we have \(\{B(g_i, \frac{\varepsilon}{2})\}, i = 1, \ldots, N\), where \(N = N(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|\)\), are the minimal number of \(\frac{\varepsilon}{2}\)-ball that covers \(\mathcal{F}\). Then there exists at least one \(f_i \in \mathcal{F}\) such that \(f_i \in B(g_i, \varepsilon)\). Consider the following \(\varepsilon\)-balls \(\{B(f_i, \varepsilon)\}, i = 1, \ldots, N\). For arbitrary \(f \in \mathcal{F} \cap B(g_i, \frac{\varepsilon}{2})\), we have that \(\| f - f_i \| \leq \| f - g_i \| + \| f_i - g_i \| \leq \varepsilon\). Thus \(\{B(f_i, \varepsilon)\}, i = 1, \ldots, N\) forms a \(\varepsilon\)-covering of \(\mathcal{F}\). By definition, we have that \(\widetilde{N}(\varepsilon, \mathcal{F}, \| \cdot \|) \leq N(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|)\).
Proof of lemma B.8. The proof of the first two inequalities follows exactly the same steps of lemma B.7. Here we just need to mention the rest of the statement that  $\widetilde{N}_{\mathbb{I}}(\varepsilon, \mathcal{F}, \| \cdot \|_{\infty}) = \widetilde{N}(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|_{\infty})$ . We first choose a set of  $\frac{\varepsilon}{2}$ -covering balls  $\{B(f_i, \frac{\varepsilon}{2})\}, i = 1, \ldots, N_1$ , where  $N_1 = \widetilde{N}(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|_{\infty})$ . Now we construct a set of brackets  $\{[l_i, u_i]\}$ ,  $i = 1, \ldots, N_1$ , where  $l_i = f_i - \frac{\varepsilon}{2}$  and  $u_i = f_i + \frac{\varepsilon}{2}$ . Noting that the bracket  $\{[l_i, u_i]\}$  is exactly the same as  $B(f_i, \frac{\varepsilon}{2})$ , the set  $\{[l_i, u_i]\}$ ,  $i = 1, \ldots, N_1$  covers  $\mathcal{F}$ , which leads to  $\widetilde{N}_{\mathbb{I}}(\varepsilon, \mathcal{F}, \| \cdot \|_{\infty}) \leq \widetilde{N}(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|_{\infty})$ . Likewise, we have that  $\widetilde{N}_{\mathbb{I}}(\varepsilon, \mathcal{F}, \| \cdot \|_{\infty}) \geq \widetilde{N}(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|_{\infty})$ . Consequently, we have that  $\widetilde{N}_{\mathbb{I}}(\varepsilon, \mathcal{F}, \| \cdot \|_{\infty}) = \widetilde{N}(\frac{\varepsilon}{2}, \mathcal{F}, \| \cdot \|_{\infty})$ .
Proof of lemma B.9. By lemma B.8, first we have that  $N_{\mathbb{I}}(\varepsilon ,\mathcal{F}_n,\| \cdot \|_\infty)\leq \widetilde{N}_{\mathbb{I}}(\varepsilon ,\mathcal{F}_n,\| \cdot \|_\infty)$ . By lemma B.3, there exists a constant  $c_{1} > 0$  such that for arbitrary  $\widehat{h}_1,\widehat{h}_2\in \mathcal{H}_n,\widehat{m}_1,\widehat{m}_2\in \mathcal{M}_n$  and  $\widehat{\theta}_1,\widehat{\theta}_2\in \Theta$ , we have that
$$
\left\| l (T, \delta , Z; \widehat {h} _ {1}, \widehat {m} _ {1}, \theta_ {1}) - l (T, \delta , Z; \widehat {h} _ {2}, \widehat {m} _ {2}, \theta_ {2}) \right\| _ {\infty} \leq c _ {1} \left[ \left| \widehat {\theta} _ {1} - \widehat {\theta} _ {2} \right| + \left\| \widehat {h} _ {1} - \widehat {h} _ {2} \right\| _ {\infty} + \left\| \widehat {m} _ {1} - \widehat {m} _ {2} \right\| _ {\infty} \right],
$$
which indicates that as long as  $|\widehat{\theta}_1 - \widehat{\theta}_2| \leq \frac{\varepsilon}{3c_1}$ ,  $\| \widehat{h}_1 - \widehat{h}_2 \|_{\infty} \leq \frac{\varepsilon}{3c_1}$  and  $\| \widehat{m}_1 - \widehat{m}_2 \|_{\infty} \leq \frac{\varepsilon}{3c_1}$ , we have that  $\| l(T, \delta, Z; \widehat{h}_1, \widehat{m}_1, \theta_1) - l(T, \delta, Z; \widehat{h}_2, \widehat{m}_2, \theta_2) \|_{\infty} \leq \varepsilon$ . Consequently, we have that
$$
\widetilde {N} _ {\mathbb {I}} (\varepsilon , \mathcal {F} _ {n}, \| \cdot \| _ {\infty}) \leq \widetilde {N} _ {\mathbb {I}} (\frac {\varepsilon}{3 c _ {1}}, \Theta , \| \cdot \| _ {\infty}) \times \widetilde {N} _ {\mathbb {I}} (\frac {\varepsilon}{3 c _ {1}}, \mathcal {H} _ {n}, \| \cdot \| _ {\infty}) \times \widetilde {N} _ {\mathbb {I}} (\frac {\varepsilon}{3 c _ {1}}, \mathcal {M} _ {n}, \| \cdot \| _ {\infty}).
$$
Since  $\Theta$  is a compact set on  $\mathbb{R}$ , by lemma B.8 and traditional volume argument, we have that  $\widetilde{N}_{\mathbb{I}}\left(\frac{\varepsilon}{3c_1},\Theta ,\| \cdot \|_{\infty}\right)\leq N_{\mathbb{I}}\left(\frac{\varepsilon}{6c_1},\Theta ,\| \cdot \|_{\infty}\right)\lesssim \frac{1}{\varepsilon}$ .
For  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{3c_1},\mathcal{H}_n,\| \cdot \|_{\infty})$  , by lemma B.8, we have that  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{3c_1},\mathcal{H}_n,\| \cdot \|_{\infty}) = \widetilde{N} (\frac{\varepsilon}{3c_1},\mathcal{H}_n,\| \cdot \|_{\infty}).$  By [12, Lemma 2], there exists a constant  $c_{2} > 0$  such that  $\| \widehat{h}_1 - \widehat{h}_2\|_{\infty}\leq c_2\| \widehat{h}_1 - \widehat{h}_2\| _2^{s_h}$  , which leads to  $\widetilde{N} (\frac{\varepsilon}{3c_1},\mathcal{H}_n,\| \cdot \|_{\infty})\leq \widetilde{N} (\frac{\varepsilon^{1 / s_h}}{(3c_1c_2)^{1 / s_h}},\mathcal{H}_n,\| \cdot \| _2)$  . By lemma B.7 we further have that  $\widetilde{N} (\frac{\varepsilon^{1 / s_h}}{(3c_1c_2)^{1 / s_h}},\mathcal{H}_n,\| \cdot \| _2)\leq N(\frac{\varepsilon^{1 / s_h}}{2(3c_1c_2)^{1 / s_h}},\mathcal{H}_n,\| \cdot \| _2)$  . Let  $c_{h} = \frac{1}{2(3c_{1}c_{2})^{1 / s_{h}}}$  . We have that  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{3c_1},\mathcal{H}_n,\| \cdot \|_{\infty})\leq N(c_h\varepsilon^{1 / s_h},\mathcal{H}_n,\| \cdot \| _2)$
Similarly, there exists a constant  $c_{m} > 0$  such that  $\widetilde{N}_{\mathbb{I}}\left(\frac{\varepsilon}{3c_1},\mathcal{M}_n,\| \cdot \|_{\infty}\right)\leq N(c_m\varepsilon^{1 / s_m},\mathcal{M}_n,\| \cdot \| _2)$ . Thus, finally we can obtain that
$$
N _ {[ ]} (\varepsilon , \mathcal {F} _ {n}, \| \cdot \| _ {\infty}) \lesssim \frac {1}{\varepsilon} N (c _ {h} \varepsilon^ {1 / s _ {h}}, \mathcal {H} _ {n}, \| \cdot \| _ {2}) \times N (c _ {m} \varepsilon^ {1 / s _ {m}}, \mathcal {M} _ {n}, \| \cdot \| _ {2}).
$$
Proof of lemma B.10. By lemma B.8, first we have  $N_{\mathbb{I}}(\varepsilon ,\mathcal{G}_n,\| \cdot \|_{\infty})\leq \widetilde{N}_{\mathbb{I}}(\varepsilon ,\mathcal{G}_n,\| \cdot \|_{\infty})$  . By lemma B.4, there exists a constant  $c_{3} > 0$  such that for arbitrary  $\widehat{\nu}_1,\widehat{\nu}_2\in \mathcal{V}_n$  and  $\widehat{\theta}_1,\widehat{\theta}_2\in \Theta$  , we have that
$$
\left\| l (T, \delta , Z; \widehat {\nu} _ {1}, \widehat {\theta} _ {1}) - l (T, \delta , Z; \widehat {\nu} _ {2}, \widehat {\theta} _ {2}) \right\| _ {\infty} \leq c _ {3} [ | \widehat {\theta} _ {1} - \widehat {\theta} _ {2} | + \| \widehat {\nu} _ {1} - \widehat {\nu} _ {2} \| _ {\infty} ],
$$
which indicates that as long as  $|\widehat{\theta}_1 - \widehat{\theta}_2| \leq \frac{\varepsilon}{2c_3}$  and  $\| \widehat{\nu}_1 - \widehat{\nu}_2 \|_{\infty} \leq \frac{\varepsilon}{2c_3}$ , we have that  $\| l(T, \delta, Z; \widehat{\nu}_1, \widehat{\theta}_1) - l(T, \delta, Z; \widehat{\nu}_2, \widehat{\theta}_2) \|_{\infty} \leq \varepsilon$ . Thus, we have:
$$
\widetilde {N} _ {\mathbb {I}} (\varepsilon , \mathcal {G} _ {n}, \| \cdot \| _ {\infty}) \leq \widetilde {N} _ {\mathbb {I}} (\frac {\varepsilon}{2 c _ {3}}, \Theta , \| \cdot \| _ {\infty}) \times \widetilde {N} _ {\mathbb {I}} (\frac {\varepsilon}{2 c _ {3}}, \mathcal {V} _ {n}, \| \cdot \| _ {\infty}).
$$
Table 3: Descriptive statistics of benchmark datasets
METABRIC	RotGBSG	FLCHAIN	SUPPORT	MIMIC-III	KKBOX
Size	1904	2232	6524	8873	35953	2646746
Censoring rate	0.423	0.432	0.699	0.320	0.901	0.280
Features	9	7	8	14	26	15
Since  $\Theta$  is a compact set on  $\mathbb{R}$ , by lemma B.8 and traditional volume argument, we have that  $\widetilde{N}_{\mathbb{I}}\left(\frac{\varepsilon}{2c_3},\Theta ,\| \cdot \|_{\infty}\right)\leq N_{\mathbb{I}}\left(\frac{\varepsilon}{4c_3},\Theta ,\| \cdot \|_{\infty}\right)\lesssim \frac{1}{\varepsilon}$ .
For  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{2c_3},\mathcal{V}_n,\| \cdot \|_{\infty})$  , by lemma B.8, we have that  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{2c_3},\mathcal{V}_n,\| \cdot \|_{\infty}) = \widetilde{N} (\frac{\varepsilon}{2c_3},\mathcal{V}_n,\| \cdot \|_{\infty})$  . By [12, Lemma 2], there exists a constant  $c_{4} > 0$  such that  $\| \widehat{\nu}_1 - \widehat{\nu}_2\|_{\infty}\leq c_4\| \widehat{\nu}_1 - \widehat{\nu}_2\| _2^{s_h}$  which leads to  $\widetilde{N} (\frac{\varepsilon}{2c_3},\mathcal{V}_n,\| \cdot \|_{\infty})\leq \widetilde{N} (\frac{\varepsilon^{1 / s_\nu}}{(2c_3c_4)^{1 / s_\nu}},\mathcal{V}_n,\| \cdot \| _2)$  . By lemma B.7 we further have  $\widetilde{N} (\frac{\varepsilon^{1 / s_\nu}}{(2c_3c_4)^{1 / s_\nu}},\mathcal{V}_n,\| \cdot \| _2)\leq N(\frac{\varepsilon^{1 / s_\nu}}{2(2c_3c_4)^{1 / s_\nu}},\mathcal{V}_n,\| \cdot \| _2)$  . Let  $c_{\nu} = \frac{1}{2(2c_{3}c_{4})^{1 / s_{\nu}}}$  , we have that  $\widetilde{N}_{\mathbb{I}}(\frac{\varepsilon}{2c_3},\mathcal{V}_n,\| \cdot \|_{\infty})\leq$ $N(c_{\nu}\varepsilon^{1 / s_{\nu}},\mathcal{V}_n,\| \cdot \| _2)$
Thus, finally we can obtain that
$$
N _ {\mathbb {I}} \left(\varepsilon , \mathcal {G} _ {n}, \| \cdot \| _ {\infty}\right) \lesssim \frac {1}{\varepsilon} N \left(c _ {\nu} \varepsilon^ {1 / s _ {\nu}}, \mathcal {V} _ {n}, \| \cdot \| _ {2}\right).
$$
C.1 Dataset summary
We report summaries of descriptive statistics of the 6 benchmark datasets used in section 5.2 in table 3.
C.2 Details of synthetic experiments
Since the true model is assumed to be of PF form, we generate event time according to the following transformed regression model [18]:
$$
\log H (\tilde {T}) = - m (Z) + \epsilon , \tag {15}
$$