SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} L ^ {\mathrm {D R O}} (\tilde {\boldsymbol {W}}) + \check {\lambda} \left(\| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} - \bar {B}\right) = L ^ {\mathrm {D R O}} (\hat {\boldsymbol {W}}) + \lambda_ {B} ^ {*} \left(\langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle - \bar {B}\right) \\ \Longleftrightarrow L ^ {\mathrm {D R O}} (\tilde {\boldsymbol {W}}) + \check {\lambda} \left(\| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} - \bar {B}\right) - \lambda_ {B} ^ {*} \langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle = L ^ {\mathrm {D R O}} (\hat {\boldsymbol {W}}) + \lambda_ {B} ^ {*} \left(\langle \boldsymbol {Z} ^ {*}, \hat {\boldsymbol {W}} - \tilde {\boldsymbol {W}} \rangle - \bar {B}\right). \\ \end{array}
$$
The first-order optimality condition for  $\hat{W}$  says
$$
\nabla L ^ {\mathrm {D R O}} (\hat {\boldsymbol {W}}) + \lambda_ {B} ^ {*} \boldsymbol {Z} ^ {*} = \mathbf {0},
$$
$$
\check {\lambda} \left(\| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} - \bar {B}\right) + \lambda_ {B} ^ {*} (\bar {B} - \langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle) = L ^ {\mathrm {D R O}} (\hat {\boldsymbol {W}}) + \langle \nabla L ^ {\mathrm {D R O}} (\hat {\boldsymbol {W}}), \tilde {\boldsymbol {W}} - \hat {\boldsymbol {W}} \rangle - L ^ {\mathrm {D R O}} (\tilde {\boldsymbol {W}}).
$$
By definition,  $\| \tilde{\boldsymbol{W}} \|_{B,2,1} - \bar{B} = 0$  and  $\lambda_B^* > 0$ . Since  $L^{\mathrm{DRO}}(\cdot)$  is convex, the RHS of the above equation should be non-positive, or equivalently,
$$
\| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} \leqslant \langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle .
$$
$$
\langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle \leqslant \| \boldsymbol {Z} ^ {*} \| _ {B, 2, \infty} \| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} \leqslant \| \tilde {\boldsymbol {W}} \| _ {B, 2, 1}.
$$
Therefore, the equality holds for the above inequalities, which leads to
$$
\| \tilde {\boldsymbol {W}} \| _ {B, 2, 1} = \langle \boldsymbol {Z} ^ {*}, \tilde {\boldsymbol {W}} \rangle .
$$
Recall that  $\| Z_{\mathcal{S}_{\mathcal{C}}^{\circ}}^{*}\|_{B,2,\infty} < 1$ . In order for  $\| \tilde{W}\|_{B,2,1} = \langle Z^{*},\tilde{W}\rangle$  to hold, we must have
$$
\bar {W} _ {\mathcal {S} _ {c} ^ {*}} = 0.
$$
In that wise, all the optimal solutions  $\tilde{\boldsymbol{W}}$  have
$$
\tilde {\boldsymbol {W}} _ {i} = \boldsymbol {0} \quad \forall i \in \mathbf {C o} _ {r}.
$$
This implies that we have a unique solution that excludes all the non-neighbor nodes without information about the true skeleton. Until now, we have proven properties (a) and (b).
(iv) The set of correct neighbors is recovered.
Consider again the first-order optimality condition in Equation (17),
$$
\begin{array}{l} \hat {\boldsymbol {W}} _ {\mathcal {S} _ {r ^ {\cdot}}} - \boldsymbol {W} _ {\mathcal {S} _ {r ^ {\cdot}}} ^ {*} = (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}) ^ {- 1} (\mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} e ^ {\intercal} - \lambda_ {B} ^ {*} \boldsymbol {Z} _ {\mathcal {S} _ {r ^ {\cdot}}} ^ {*}) \\ \Rightarrow \| \hat {W} _ {\mathcal {S} _ {r}} - W _ {\mathcal {S} _ {r}} ^ {*} \| _ {B, 2, \infty} = \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}\right) ^ {- 1} \left(\mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} e ^ {\intercal} - \lambda_ {B} ^ {*} \boldsymbol {Z} _ {\mathcal {S} _ {r}} ^ {*}\right) \| _ {B, 2, \infty} \\ \leqslant \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}\right) ^ {- 1} \| _ {B, 1, \infty} \| \mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} \boldsymbol {e} ^ {\intercal} - \lambda_ {B} ^ {*} \boldsymbol {Z} _ {\mathcal {S} _ {r}} ^ {*} \| _ {2, \infty} \\ \leqslant \left\| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \left(\left\| \mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} e ^ {\intercal} \right\| _ {2, \infty} + \left\| \lambda_ {B} ^ {*} \boldsymbol {Z} _ {\mathcal {S} _ {r}} ^ {*} \right\| _ {2, \infty}\right) \\ \leqslant \rho_ {\max } \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}\right) ^ {- 1} \| _ {\infty , \infty} \left(\| \mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} \boldsymbol {e} ^ {\intercal} \| _ {2, \infty} + \lambda_ {B} ^ {*}\right) \\ \leqslant \rho_ {\max } \sqrt {| \mathcal {S} _ {r} |} \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q} ^ {*}}\right) ^ {- 1} \| _ {2, 2} \left(\| \mathbb {E} _ {\mathbb {Q} ^ {*}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} \boldsymbol {e} ^ {\intercal} \| _ {2, \infty} + \lambda_ {B} ^ {*}\right). \\ \end{array}
$$
According to Equation (13), with probability at least  $1 - 2|\mathcal{S}_r|^2 \exp \left(-\frac{m(\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$  and  $\varepsilon \leqslant \frac{\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}$ ,
$$
\| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \| _ {2, 2} \leqslant \sqrt {\frac {4}{\Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right)}}.
$$
According to Equation (15), with probability at least  $1 - |\mathcal{S}_r| \rho_r \exp \left( -\frac{m \mu^2}{2 \sigma^2} \right)$ ,  $\varepsilon \leqslant \frac{\mu}{\sigma}$  and  $\lambda_B^* > \frac{32 \mu \sqrt{\rho_r (1 - \alpha / 2)}}{\alpha}$ , we have
$$
\| \mathbb {E} _ {\mathbb {Q}} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) _ {\mathcal {S} _ {r}} \boldsymbol {e} ^ {\intercal} \| _ {2, \infty} \leqslant \frac {\lambda_ {B} ^ {*} \alpha}{8 (1 - \alpha / 2)}.
$$
On that account, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right) - |\mathcal{S}_r|\rho_r\exp \left(-\frac{m\mu^2}{2\sigma^2}\right)$  and  $\varepsilon \leqslant \min \left(\frac{\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}},\frac{\mu}{\sigma}\right)$  while requiring  $\lambda_B^* >\frac{32\mu\sqrt{\rho_r}(1 - \alpha / 2)}{\alpha}$ ,
$$
\| \hat {\boldsymbol {W}} _ {\mathcal {S} _ {r ^ {*}}} - \boldsymbol {W} _ {\mathcal {S} _ {r ^ {*}}} ^ {*} \| _ {B, 2, \infty} \leqslant \rho_ {\max } \sqrt {| \mathcal {S} _ {r} |} \sqrt {\frac {4}{\Lambda_ {\min } (\boldsymbol {H} _ {\mathcal {S} _ {r} , \mathcal {S} _ {r}})}} \lambda_ {B} ^ {*} \left(\frac {\alpha}{8 (1 - \alpha / 2)} + 1\right).
$$
By Assumption 2, if the condition  $\lambda_B^* < \frac{\beta}{2\left(\frac{\alpha}{8(1 - \alpha / 2)} + 1\right)\rho_{\max}\sqrt{|\mathcal{S}_r|}}\sqrt{\frac{\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{4}}$  is satisfied, the following inequality holds:
$$
\left\| \hat {W} _ {\mathcal {S} _ {r ^ {*}}} - W _ {\mathcal {S} _ {r ^ {*}}} ^ {*} \right\| _ {B, 2, \infty} <   \beta / 2.
$$
In this way, we are able to recover all the neighbor nodes with a threshold  $\beta /2$ . This proves (c).
(v) The true skeleton is recovered with high probability.
The above arguments tell us that with high probability and certain conditions for  $\varepsilon$  and  $\lambda_B^*$  satisfied, for each node  $r$ , we do not recover any non-neighbor and we do recover all the neighbor nodes. The correct  $\mathbf{N}\mathbf{e}_r$  and  $\mathbf{C}\mathbf{o}_r$  are thus identified. Now we are ready to prove (d).
Putting everything together and taking the union bound for all nodes  $r \in [n]$ , with probability at least  $1 - \mathcal{O}(n\exp (-\frac{Cm\mu^2}{\sigma^2\rho_{\max}^4\rho_{[n]}^3} + 2\log \rho_{[n]}))$ ,  $\varepsilon \leqslant \frac{C\mu}{\sigma\rho_{\max}\rho_{[n]}^{3 / 2}}$  and  $\frac{32\mu\rho_{\max}}{\alpha} < \lambda_B^* < \frac{\beta}{2(\frac{\alpha}{8(1 - \alpha / 2)} + 1)\rho_{\max}\sqrt{\rho_{[n]}}}\sqrt{\frac{\Lambda}{4}}$ , where  $C$  only depends on  $\alpha, \Lambda$ , we have
$$
\hat {\mathcal {G}} _ {\mathrm {s k e l}} = \mathcal {G} _ {\mathrm {s k e l}}.
$$
Setting  $\varepsilon = \frac{\varepsilon_0}{m}$  and making the dependence on the sample size more explicit. We draw the conclusion that, if the number of samples satisfies
$$
m = \mathcal {O} (\frac {C (\varepsilon_ {0} + \log (n / \delta) + \log \rho_ {[ n ]}) \sigma^ {2} \rho_ {\max} ^ {4} \rho_ {[ n ]} ^ {3}}{\min (\mu^ {2} , 1)}),
$$
where  $C$  only depends on  $\alpha, \Lambda$ , and if  $\lambda_B^*$  satisfies
$$
\frac {3 2 \mu \rho_ {\mathrm {m a x}}}{\alpha} <   \lambda_ {B} ^ {*} <   \frac {\beta}{(\alpha / (4 - 2 \alpha) + 2) \rho_ {\mathrm {m a x}} \sqrt {\rho_ {[ n ]}}} \sqrt {\frac {\Lambda}{4}},
$$
then with probability at least  $1 - \delta$  for  $\delta \in (0, 1]$ :
Moreover, if we assume that the target graph has a bounded degree of  $d$ , the sample complexity becomes logarithmic in  $n$ :
$$
m = \mathcal {O} (\frac {C (\varepsilon_ {0} + \log (n / \delta) + \log n + \log \rho_ {\mathrm {m a x}}) \sigma^ {2} \rho_ {\mathrm {m a x}} ^ {7} d ^ {3}}{\min (\mu^ {2} , 1)}).
$$
where  $C$  depends on  $\alpha, \Lambda$  while independent of  $n$ , and if the Lagrange multiplier satisfies the same condition as in Theorem 9, then for any  $\delta \in (0,1]$ ,  $r \in [n]$ , with probability at least  $1 - \delta$ , the properties (a)-(d) in Theorem 9 hold.
$$
\ell_ {\boldsymbol {W}} (\boldsymbol {X}) := \frac {1}{2} \| \mathcal {E} (X _ {r}) - \boldsymbol {W} ^ {\intercal} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) \| _ {2} ^ {2}.
$$
According to Theorem 7 in Lam [2019], the worst-case risk with a KL divergence ambiguity set can be bounded as follows:
$$
\begin{array}{l} \sup  _ {\mathbb {Q} \in \mathcal {A} _ {\varepsilon} ^ {D} (\tilde {\mathbb {P}} _ {m})} \mathbb {E} _ {\mathbb {Q}} \ell_ {\boldsymbol {W}} (\boldsymbol {X}) \leqslant \mathbb {E} _ {\tilde {\mathbb {P}} _ {m}} \ell_ {\boldsymbol {W}} (\boldsymbol {X}) + \sqrt {\varepsilon} \sqrt {\frac {1}{m} \sum_ {i \in [ m ]} (\ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \bar {\ell_ {\boldsymbol {W}} ^ {-}}) ^ {2}} + C \varepsilon \frac {\sum_ {i \in [ m ]} | \ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \bar {\ell_ {\boldsymbol {W}} ^ {-}} | ^ {3}}{\sum_ {i \in [ m ]} (\ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \bar {\ell_ {\boldsymbol {W}} ^ {-}}) ^ {2}} \\ \leqslant \mathbb {E} _ {\tilde {\mathbb {P}} _ {m}} \ell_ {\boldsymbol {W}} (\boldsymbol {X}) + \sqrt {\varepsilon} \max _ {i \in [ m ]} | \ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \ell_ {\boldsymbol {W}} ^ {-} | + C \varepsilon \max _ {i \in [ m ]} | \ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \ell_ {\boldsymbol {W}} ^ {-} |, \\ \end{array}
$$
where  $\ell_{\mathbf{W}}^{-} = \frac{1}{m}\sum_{i\in [m]}\ell_{\mathbf{W}}(\pmb{x}^{(i)})$  and  $C > 0$  is constant independent of  $n$ .
$$
\begin{array}{l} \max  _ {i \in [ m ]} | \ell_ {\boldsymbol {W}} (\boldsymbol {x} ^ {(i)}) - \bar {\ell_ {\boldsymbol {W}}} | \leqslant \max  _ {\boldsymbol {W}, \boldsymbol {W} ^ {\prime}, \boldsymbol {x}, \boldsymbol {x} ^ {\prime}} | \ell_ {\boldsymbol {W}} (\boldsymbol {x}) - \ell_ {\boldsymbol {W} ^ {\prime}} (\boldsymbol {x} ^ {\prime}) | \\ \leqslant \max  _ {\boldsymbol {W}, \boldsymbol {x}} | \ell_ {\boldsymbol {W}} (\boldsymbol {x}) | \\ \leqslant \frac {1}{2} \max  _ {\boldsymbol {W}, \boldsymbol {x}} \left(\| \mathcal {E} \left(\boldsymbol {X} _ {r}\right) \| _ {2} + \| \boldsymbol {W} ^ {\intercal} \mathcal {E} \left(\boldsymbol {X} _ {\bar {r}}\right) \| _ {2}\right) ^ {2} \\ \leqslant \frac {1}{2} \max  _ {\boldsymbol {W}, \boldsymbol {x}} \left(\sqrt {\rho_ {\operatorname* {m a x}}} + \| \boldsymbol {W} ^ {\intercal} \| _ {\infty , 2}\right) ^ {2} \\ \leqslant \frac {1}{2} \max  _ {\boldsymbol {W}, \boldsymbol {x}} (\sqrt {\rho_ {\operatorname* {m a x}}} + \| \boldsymbol {W} \| _ {1, 2}) ^ {2} \\ \leqslant \frac {1}{2} \max  _ {\boldsymbol {W}, \boldsymbol {x}} (\sqrt {\rho_ {\operatorname* {m a x}}} + \sqrt {\rho_ {[ n ]}} \| \boldsymbol {W} \| _ {F}) ^ {2} \\ \leqslant \frac {1}{2} \max  _ {\boldsymbol {W}, \boldsymbol {x}} (\sqrt {\rho_ {\operatorname* {m a x}}} + \sqrt {\rho_ {[ n ]}} \| \boldsymbol {W} \| _ {B, 2, 1}) ^ {2} \\ \leqslant \frac {1}{2} \left(\sqrt {\rho_ {\operatorname* {m a x}}} + \sqrt {\rho_ {[ n ]}} \bar {B}\right) ^ {2} \\ := B _ {\rho}. \\ \end{array}
$$
Define  $\varepsilon_{\mathrm{max}}\coloneqq \max (\sqrt{\varepsilon},\varepsilon)$ . Therefore, we find that
$$
\sup  _ {\mathbb {Q} \in \mathcal {A} _ {\varepsilon} ^ {D} (\tilde {\mathbb {P}} _ {m})} \mathbb {E} _ {\mathbb {Q}} \ell_ {\boldsymbol {W}} (\boldsymbol {X}) \leqslant \mathbb {E} _ {\tilde {\mathbb {P}} _ {m}} \ell_ {\boldsymbol {W}} (\boldsymbol {X}) + C \varepsilon_ {\max } B _ {\rho}.
$$
Similar to the Wasserstein robust risk, we observe that the following results hold for any  $\mathbb{Q} \in \mathcal{A}_{\varepsilon}^{D}(\tilde{\mathbb{P}}_{m})$ .
With probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2}{2|\mathcal{S}_r|^2}\right)$ , we have
$$
\Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) \geqslant \Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) - C \varepsilon_ {\max } | \mathcal {S} _ {r} | ^ {\frac {1}{2}} - t.
$$
With probability at least  $1 - 2|\mathcal{S}_r^c ||\mathcal{S}_r|\exp \left(-\frac{mt^2}{2\rho_{\max}^2|\mathcal{S}_r|^2}\right)$
$$
\left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right\| _ {B, 1, \infty} \leqslant C \varepsilon_ {\max } \rho_ {\max } | \mathcal {S} _ {r} | + t.
$$
With probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2}{2|\mathcal{S}_r|^2}\right)$
$$
\left\| \mathbf {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \mathbf {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \right\| _ {\infty , \infty} \leqslant C \varepsilon_ {\max } | \mathcal {S} _ {r} | + t.
$$
With probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2(\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{32|\mathcal{S}_r|^3}\right) - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$  and  $\varepsilon_{\mathrm{max}}\leqslant C\min (\frac{t\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{8|\mathcal{S}_r|\sqrt{|\mathcal{S}_r|}},\frac{\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}),$
$$
\left\| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {\infty , \infty} \leqslant t.
$$
With probability at least  $1 - \mathcal{O}\big(\exp \big(-\frac{Cm}{\rho_{\max}^2|S_r|^3} +\log |S_r^c | + \log |S_r|\big)\big)$  and  $\varepsilon_{\mathrm{max}}\leqslant \frac{C}{\rho_{\mathrm{max}}|S_r|^{3 / 2}},$
Thanks to the boundedness of the error term  $e$ , we have similar conclusions to Lemma 12 if  $\varepsilon_{\max} \leqslant \frac{\mu}{\sigma}$  holds.
In such wise, the properties in Theorem 9 hold with the same condition on  $\lambda_B^*$  and the condition on  $\varepsilon_{\mathrm{max}}$  that  $\varepsilon_{\mathrm{max}} \leqslant \frac{\bar{C}\mu}{\sigma\rho_{\mathrm{max}}\rho_{[n]}^{3/2}}$ . Since we set  $\varepsilon = \frac{\varepsilon_0}{m}$  and define  $\varepsilon_{\mathrm{max}} := \max(\sqrt{\varepsilon}, \varepsilon)$ , the condition on  $\varepsilon_{\mathrm{max}}$  implies that
$$
m \geqslant \max (\frac {\varepsilon_ {0} C ^ {2} \sigma^ {2} \rho_ {\mathrm {m a x}} ^ {2} \rho_ {[ n ]} ^ {3}}{\mu^ {2}}, \frac {\varepsilon_ {0} C \sigma \rho_ {\mathrm {m a x}} \rho_ {[ n ]} ^ {3 / 2}}{\mu}).
$$
The final sample complexity becomes
$$
m = \mathcal {O} (\frac {C (\varepsilon_ {0} + \log (n / \delta) + \log \rho_ {[ n ]}) \sigma^ {2} \rho_ {\max} ^ {4} \rho_ {[ n ]} ^ {3}}{\min (\mu^ {2} , 1)}).
$$
D More Empirical Results
Table 2 lists the complete experimental results.