SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {P} _ {(\mathbf {x}, y) \sim \mathcal {D}} \big (y \neq \operatorname {s i g n} \left(f \left(\mathbf {W} ^ {(t)}, \mathbf {x}\right)\right) \big) \\ = \mathbb {P} _ {(\mathbf {x}, y) \sim \mathcal {D}} (y f (\mathbf {W} ^ {(t)}, \mathbf {x}) \leq 0) \\ = \mathbb {P} _ {(\mathbf {x}, y) \sim \mathcal {D}} (y f (\mathbf {W} ^ {(t)}, \mathbf {x}) \leq 0, y \neq \widehat {y}) + \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} (y f (\mathbf {W} ^ {(t)}, \mathbf {x}) \leq 0, y = \widehat {y}) \tag {71} \\ = p \cdot \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} \big (\widehat {y} f (\mathbf {W} ^ {(t)}, \mathbf {x}) \geq 0 \big) + (1 - p) \cdot \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} \big (\widehat {y} f (\mathbf {W} ^ {(t)}, \mathbf {x}) \leq 0 \big) \\ \leq p + \mathbb {P} _ {(\mathbf {x}, \hat {y}, y) \sim \mathcal {D}} \left(\widehat {y} f \left(\mathbf {W} ^ {(t)}, \mathbf {x}\right) \leq 0\right), \\ \end{array}
$$
where in the second equation we used the definition of  $\mathcal{D}$  in Definition 2.1. It therefore suffices to provide an upper bound for  $\mathbb{P}_{(\mathbf{x},\widehat{y})\sim \mathcal{D}}\big(\widehat{y} f(\mathbf{W}^{(t)},\mathbf{x})\leq 0\big)$ . To achieve this, we write  $\mathbf{x} = (\widehat{y}\pmb {\mu},\pmb {\xi})$  and get
$$
\begin{array}{l} \widehat {y} f (\mathbf {W} ^ {(t)}, \mathbf {x}) = \frac {1}{m} \sum_ {j, r} \widehat {y} j [ \sigma (\langle \mathbf {w} _ {j, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle) + \sigma (\langle \mathbf {w} _ {j, r} ^ {(t)}, \boldsymbol {\xi} \rangle) ] \\ = \frac {1}{m} \sum_ {r} [ \sigma (\langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle) + (P - 1) \sigma (\langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \boldsymbol {\xi} \rangle) ] \\ - \frac {1}{m} \sum_ {r} [ \sigma (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle) + (P - 1) \sigma (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \boldsymbol {\xi} \rangle) ]. \tag {72} \\ \end{array}
$$
The inner product with  $j = \widehat{y}$  can be bounded as
$$
\begin{array}{l} \langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle = \langle \mathbf {w} _ {\widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \rangle + \gamma_ {\widehat {y}, r} ^ {(t)} + \frac {1}{(P - 1)} \sum_ {i = 1} ^ {n} \bar {\rho} _ {\widehat {y}, r, i} ^ {(t)} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \langle \boldsymbol {\xi} _ {i}, \widehat {y} \boldsymbol {\mu} \rangle \\ + \frac {1}{(P - 1)} \sum_ {i = 1} ^ {n} \underline {{\rho}} _ {\widehat {y}, r, i} ^ {(t)} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \langle \boldsymbol {\xi} _ {i}, \widehat {y} \boldsymbol {\mu} \rangle \\ \geq \langle \mathbf {w} _ {\widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \rangle + \gamma_ {\widehat {y}, r} ^ {(t)} - \frac {\sqrt {2 \log (6 n / \delta)}}{P - 1} \cdot \sigma_ {p} \| \boldsymbol {\mu} \| _ {2} \cdot (\sigma_ {p} ^ {2} d / 2) ^ {- 1} \left[ \sum_ {i = 1} ^ {n} \bar {\rho} _ {\widehat {y}, r, i} ^ {(t)} + \sum_ {i = 1} ^ {n} \left| \underline {{\rho}} _ {\widehat {y}, r, i} ^ {(t)} \right| \right] \\ = \left\langle \mathbf {w} _ {\widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \right\rangle + \gamma_ {\widehat {y}, r} ^ {(t)} - \Theta \left(\sqrt {\log (n / \delta)} \cdot \left(P \sigma_ {p} d\right) ^ {- 1} \| \boldsymbol {\mu} \| _ {2}\right) \cdot \Theta (\mathrm {S N R} ^ {- 2}) \cdot \gamma_ {\widehat {y}, r} ^ {(t)} \\ = \left\langle \mathbf {w} _ {\widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \right\rangle + \left[ 1 - \Theta \left(\sqrt {\log (n / \delta)} \cdot P \sigma_ {p} / \| \boldsymbol {\mu} \| _ {2}\right) \right] \gamma_ {\widehat {y}, r} ^ {(t)} \\ = \left\langle \mathbf {w} _ {\widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \right\rangle + \Theta \left(\gamma_ {\widehat {y}, r} ^ {(t)}\right) \\ = \Omega (1), \tag {73} \\ \end{array}
$$
where the inequality is by Lemma B.1; the second equality is obtained by plugging in the coefficient orders we summarized at (70); the third equality is by the condition  $\mathrm{SNR} = \| \pmb{\mu}\|_2 / P\sigma_p\sqrt{d}$ ; the fourth equality is due to  $\| \pmb{\mu}\|_2^2 \geq C \cdot P^2\sigma_p^2\log(n/\delta)$  in Condition 3.1, so for sufficiently large constant  $C$  the equality holds; the last equality is by Lemma D.7. Moreover, we can deduce in a similar manner that
$$
\begin{array}{l} \langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle = \langle \mathbf {w} _ {- \widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \rangle - \gamma_ {- \widehat {y}, r} ^ {(t)} + \sum_ {i = 1} ^ {n} \overline {{\rho}} _ {- \widehat {y}, r, i} ^ {(t)} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \langle \boldsymbol {\xi} _ {i}, - \widehat {y} \boldsymbol {\mu} \rangle \\ + \sum_ {i = 1} ^ {n} \underline {{\rho}} _ {- \widehat {y}, r, i} ^ {(t)} \cdot \| \pmb {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \langle \pmb {\xi} _ {i}, \widehat {y} \pmb {\mu} \rangle \\ \leq \left\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \right\rangle - \gamma_ {- \widehat {y}, r} ^ {(t)} \tag {74} \\ + \sqrt {2 \log (6 n / \delta)} \cdot \sigma_ {p} \| \boldsymbol {\mu} \| _ {2} \cdot (\sigma_ {p} ^ {2} d / 2) ^ {- 1} \left[ \sum_ {i = 1} ^ {n} \bar {\rho} _ {- \widehat {y}, r, i} ^ {(t)} + \sum_ {i = 1} ^ {n} | \underline {{\rho}} _ {- \widehat {y}, r, i} ^ {(t)} | \right] \\ = \langle \mathbf {w} _ {- \widehat {y}, r} ^ {(0)}, \widehat {y} \boldsymbol {\mu} \rangle - \Theta (\gamma_ {- \widehat {y}, r} ^ {(t)}) \\ = - \Omega (1) <   0, \\ \end{array}
$$
where the second equality holds based on similar analyses as in (73).
Denote  $g(\pmb{\xi})$  as  $\sum_{r}\sigma (\langle \mathbf{w}_{- \widehat{y}, r}^{(t)}, \pmb{\xi} \rangle)$ . According to Theorem 5.2.2 in Vershynin (2018), we know that for any  $x \geq 0$  it holds that
$$
\mathbb {P} (g (\boldsymbol {\xi}) - \mathbb {E} g (\boldsymbol {\xi}) \geq x) \leq \exp \left(- \frac {c x ^ {2}}{\sigma_ {p} ^ {2} \| g \| _ {\mathrm {L i p}} ^ {2}}\right), \tag {75}
$$
where  $c$  is a constant. To calculate the Lipschitz norm, we have
$$
\begin{array}{l} | g (\boldsymbol {\xi}) - g \left(\boldsymbol {\xi} ^ {\prime}\right) | = \left| \sum_ {r = 1} ^ {m} \sigma \left(\left\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \boldsymbol {\xi} \right\rangle\right) - \sum_ {r = 1} ^ {m} \sigma \left(\left\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \boldsymbol {\xi} ^ {\prime} \right\rangle\right) \right| \\ \leq \sum_ {r = 1} ^ {m} \left| \sigma \big (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \pmb {\xi} \rangle \big) - \sigma \big (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \pmb {\xi} ^ {\prime} \rangle \big) \right| \\ \leq \sum_ {r = 1} ^ {m} | \langle \mathbf {w} _ {- \widehat {\gamma}, r} ^ {(t)}, \pmb {\xi} - \pmb {\xi} ^ {\prime} \rangle | \\ \leq \sum_ {r = 1} ^ {m} \left| \left| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \right| \right| _ {2} \cdot \left| \left| \boldsymbol {\xi} - \boldsymbol {\xi} ^ {\prime} \right| \right| _ {2}, \\ \end{array}
$$
where the first inequality is by triangle inequality, the second inequality is by the property of ReLU; and the last inequality is by Cauchy-Schwartz inequality. Therefore, we have
$$
\left\| g \right\| _ {\mathrm {L i p}} \leq \sum_ {r = 1} ^ {m} \left\| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \right\| _ {2}, \tag {76}
$$
and since  $\langle \mathbf{w}_{-\widehat{y},r}^{(t)},\pmb {\xi}\rangle \sim \mathcal{N}\big(0,\| \mathbf{w}_{-\widehat{y},r}^{(t)}\| _2^2\sigma_p^2\big)$ , we can get
$$
\mathbb {E} g (\pmb {\xi}) = \sum_ {r = 1} ^ {m} \mathbb {E} \sigma (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \pmb {\xi} \rangle) = \sum_ {r = 1} ^ {m} \frac {\| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \| _ {2} \sigma_ {p}}{\sqrt {2 \pi}} = \frac {\sigma_ {p}}{\sqrt {2 \pi}} \sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \| _ {2}.
$$
Next we seek to upper bound the 2-norm of  $\mathbf{w}_{j,r}^{(t)}$ . First, we tackle the noise section in the decomposition, namely:
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} \cdot \| \pmb {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \pmb {\xi} _ {i} \right\| _ {2} ^ {2} \\ = \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t) ^ {2}} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {- 2} + 2 \sum_ {1 \leq i _ {1} <   i _ {2} \leq n} \rho_ {j, r, i _ {1}} ^ {(t)} \rho_ {j, r, i _ {2}} ^ {(t)} \cdot \| \boldsymbol {\xi} _ {i _ {1}} \| _ {2} ^ {- 2} \cdot \| \boldsymbol {\xi} _ {i _ {2}} \| _ {2} ^ {- 2} \cdot \langle \boldsymbol {\xi} _ {i _ {1}}, \boldsymbol {\xi} _ {i _ {2}} \rangle \\ \leq 4 \sigma_ {p} ^ {- 2} d ^ {- 1} \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} ^ {2} + 2 \sum_ {1 \leq i _ {1} <   i _ {2} \leq n} \left| \rho_ {j, r, i _ {1}} ^ {(t)} \rho_ {j, r, i _ {2}} ^ {(t)} \right| \cdot (1 6 \sigma_ {p} ^ {- 4} d ^ {- 2}) \cdot (2 \sigma_ {p} ^ {2} \sqrt {d \log (6 n ^ {2} / \delta)}) \\ = 4 \sigma_ {p} ^ {- 2} d ^ {- 1} \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} ^ {2} + 3 2 \sigma_ {p} ^ {- 2} d ^ {- 3 / 2} \sqrt {\log (6 n ^ {2} / \delta)} \Big [ \bigg (\sum_ {i = 1} ^ {n} | \rho_ {j, r, i} ^ {(t)} | \bigg) ^ {2} - \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} ^ {2} \Big ] \\ = \Theta (\sigma_ {p} ^ {- 2} d ^ {- 1}) \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} ^ {2} + \widetilde {\Theta} (\sigma_ {p} ^ {- 2} d ^ {- 3 / 2}) \left(\sum_ {i = 1} ^ {n} | \rho_ {j, r, i} ^ {(t)} |\right) ^ {2} \\ \leq \left[ \Theta (\sigma_ {p} ^ {- 2} d ^ {- 1} n ^ {- 1}) + \widetilde {\Theta} (\sigma_ {p} ^ {- 2} d ^ {- 3 / 2}) \right] \left(\sum_ {i = 1} ^ {n} \left| \overline {{\rho}} _ {j, r, i} ^ {(t)} \right| + \sum_ {i = 1} ^ {n} \left| \underline {{\rho}} _ {j, r, i} ^ {(t)} \right|\right) ^ {2} \\ \leq \Theta (\sigma_ {p} ^ {- 2} d ^ {- 1} n ^ {- 1}) \left(\sum_ {i = 1} ^ {n} \overline {{\rho}} _ {j, r, i} ^ {(t)}\right) ^ {2}, \\ \end{array}
$$
where for the first inequality, we used Lemma B.1; for the second inequality, we used the definition of  $\overline{\rho},\underline{\rho}$ ; for the second to last equation, we plugged in coefficient orders. We can thus upper bound the 2-norm of  $\mathbf{w}_{j,r}^{(t)}$  as:
$$
\begin{array}{l} \| \mathbf {w} _ {j, r} ^ {(t)} \| _ {2} \leq \| \mathbf {w} _ {j, r} ^ {(0)} \| _ {2} + \gamma_ {j, r} ^ {(t)} \cdot \| \boldsymbol {\mu} \| _ {2} ^ {- 1} + \frac {1}{P - 1} \left\| \sum_ {i = 1} ^ {n} \rho_ {j, r, i} ^ {(t)} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {- 2} \cdot \boldsymbol {\xi} _ {i} \right\| _ {2} \\ \leq \| \mathbf {w} _ {j, r} ^ {(0)} \| _ {2} + \gamma_ {j, r} ^ {(t)} \cdot \| \boldsymbol {\mu} \| _ {2} ^ {- 1} + \Theta (P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {- 1 / 2}) \cdot \sum_ {i = 1} ^ {n} \bar {\rho} _ {j, r, i} ^ {(t)} \\ = \Theta \left(\sigma_ {0} \sqrt {d}\right) + \Theta \left(P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {- 1 / 2}\right) \cdot \sum_ {i = 1} ^ {n} \bar {\rho} _ {j, r, i} ^ {(t)}, \tag {77} \\ \end{array}
$$
where the first inequality is due to the triangle inequality, and the equality is due to the following comparisons:
$$
\begin{array}{l} \frac {\gamma_ {j , r} ^ {(t)} \cdot \| \boldsymbol {\mu} \| _ {2} ^ {- 1}}{\Theta (P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {- 1 / 2}) \cdot \sum_ {i = 1} ^ {n} \bar {\rho} _ {j , r , i} ^ {(t)}} = \Theta (P ^ {- 1} \sigma_ {p} d ^ {1 / 2} n ^ {1 / 2} \| \boldsymbol {\mu} \| _ {2} ^ {- 1} \mathrm {S N R} ^ {2}) \\ = \Theta (P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {1 / 2} \| \boldsymbol {\mu} \| _ {2}) \\ = O (1) \\ \end{array}
$$
based on the coefficient order  $\sum_{i=1}^{n} \overline{\rho}_{j,r,i}^{(t)} / \gamma_{j,r}^{(t)} = \Theta(\mathrm{SNR}^{-2})$ , the definition  $\mathrm{SNR} = \|\boldsymbol{\mu}\|_2 / (\sigma_p \sqrt{d})$  and the condition for  $d$  in Condition 3.1; and also  $\|\mathbf{w}_{j,r}^{(0)}\|_2 = \Theta(\sigma_0 \sqrt{d})$  based on Lemma D.7. With this and (73), we analyze the key component in (80):
$$
\begin{array}{l} \frac {\sum_ {r} \sigma (\langle \mathbf {w} _ {\widehat {y} , r} ^ {(t)} , \widehat {y} \boldsymbol {\mu} \rangle)}{(P - 1) \sigma_ {p} \sum_ {r = 1} ^ {m} \left\| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \right\| _ {2}} \geq \frac {\Theta (1)}{\Theta (\sigma_ {0} \sqrt {d}) + \Theta (P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {- 1 / 2}) \cdot \sum_ {i = 1} ^ {n} \overline {{\rho}} _ {j , r , i} ^ {(t)}} \\ \geq \frac {\Theta (1)}{\Theta \left(\sigma_ {0} \sqrt {d}\right) + O \left(P ^ {- 1} \sigma_ {p} ^ {- 1} d ^ {- 1 / 2} n ^ {1 / 2} \alpha\right)} \tag {78} \\ \geq \min  \left\{\Omega \left(\sigma_ {0} ^ {- 1} d ^ {- 1 / 2}\right), \Omega \left(P \sigma_ {p} d ^ {1 / 2} n ^ {- 1 / 2} \alpha^ {- 1}\right) \right\} \\ \geq 1. \\ \end{array}
$$
It directly follows that
$$
\sum_ {r} \sigma \left(\left\langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \right\rangle\right) - \frac {(P - 1) \sigma_ {p}}{\sqrt {2 \pi}} \sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \| _ {2} > 0. \tag {79}
$$
Now using the method in (75) with the results above, we plug (74) into (72) and then (71), to obtain
$$
\begin{array}{l} \mathbb {P} _ {(\mathbf {x}, \hat {y}, y) \sim \mathcal {D}} \left(\hat {y} f \left(\boldsymbol {W} ^ {(t)}, \mathbf {x}\right) \leq 0\right) \\ \leq \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} \left(\sum_ {r} \sigma (\langle \mathbf {w} _ {- \widehat {y}, r} ^ {(t)}, \boldsymbol {\xi} \rangle) \geq (1 / (P - 1)) \sum_ {r} \sigma (\langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \rangle)\right) \\ = \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} \left(g (\boldsymbol {\xi}) - \mathbb {E} g (\boldsymbol {\xi}) \geq (1 / (P - 1)) \sum_ {r} \sigma \left(\left\langle \mathbf {w} _ {\widehat {y}, r} ^ {(t)}, \widehat {y} \boldsymbol {\mu} \right\rangle\right) - \frac {\sigma_ {p}}{\sqrt {2 \pi}} \sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \| _ {2}\right) \\ \leq \exp \left[ - \frac {c \left(\left(1 / (P - 1)\right) \sum_ {r} \sigma (\langle \mathbf {w} _ {\widehat {y} , r} ^ {(t)} , \widehat {y} \boldsymbol {\mu} \rangle) - (\sigma_ {p} / \sqrt {2 \pi}) \sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \| _ {2}\right) ^ {2}}{\sigma_ {p} ^ {2} \left(\sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \| _ {2}\right) ^ {2}} \right] \\ = \exp \left[ - c \left(\frac {\sum_ {r} \sigma \left(\left\langle \mathbf {w} _ {\widehat {y} , r} ^ {(t)} , \widehat {y} \boldsymbol {\mu} \right\rangle\right)}{\left(P - 1\right) \sigma_ {p} \sum_ {r = 1} ^ {m} \left\| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \right\| _ {2}} - 1 / \sqrt {2 \pi}\right) ^ {2} \right] \\ \leq \exp (c / 2 \pi) \exp \left(- 0. 5 c \left(\frac {\sum_ {r} \sigma (\langle \mathbf {w} _ {\widehat {y} , r} ^ {(t)} , \widehat {y} \boldsymbol {\mu} \rangle)}{(P - 1) \sigma_ {p} \sum_ {r = 1} ^ {m} \left\| \mathbf {w} _ {- \widehat {y}, r} ^ {(t)} \right\| _ {2}}\right) ^ {2}\right), \tag {80} \\ \end{array}
$$
where the second inequality is by (79) and plugging (76) into (75), the third inequality is due to the fact that  $(s - t)^2 \geq s^2 / 2 - t^2, \forall s, t \geq 0$ .
And we can get from (78) and (80) that
$$
\begin{array}{l} \mathbb {P} _ {(\mathbf {x}, \widehat {y}, y) \sim \mathcal {D}} \big (\widehat {y} f (\pmb {W} ^ {(t)}, \mathbf {x}) \leq 0 \big) \leq \exp (c / 2 \pi) \exp \left(- 0. 5 c \Bigg (\frac {\sum_ {r} \sigma (\langle \mathbf {w} _ {\widehat {y} , r} ^ {(t)} , \widehat {y} \pmb {\mu} \rangle)}{(P - 1) \sigma_ {p} \sum_ {r = 1} ^ {m} \| \mathbf {w} _ {- \widehat {y} , r} ^ {(t)} \| _ {2}}\right) ^ {2}\left. \right) \\ \leq \exp \left(\frac {c}{2 \pi} - C \min  \left\{\sigma_ {0} ^ {- 2} d ^ {- 1}, P \sigma_ {p} ^ {2} d n ^ {- 1} \alpha^ {- 2} \right\}\right) \\ \leq \exp \left(- 0. 5 C \min  \left\{\sigma_ {0} ^ {- 2} d ^ {- 1}, P \sigma_ {p} ^ {2} d n ^ {- 1} \alpha^ {- 2} \right\}\right) \\ \leq \epsilon , \\ \end{array}
$$
where  $C = O(1)$ , the last inequality holds since  $\sigma_0^2 \leq 0.5Cd^{-1}\log (1 / \epsilon)$  and  $d \geq 2C^{-1}P^{-1}\sigma_p^{-2}n\alpha^2\log (1 / \epsilon)$ .