SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} y f \left(\mathcal {T} _ {\text {e n d}}, \mathbf {x}\right) = \sum_ {j \in [ 2 m ]} \phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle\right) \\ \geq \sum_ {j \in \Gamma_ {p}} \left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle - \sum_ {j: (- 1) ^ {j} \neq y} \phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle\right) \\ \geq \eta \sum_ {j \in \Gamma_ {p}} \mathcal {T} _ {1} (n - k) (\gamma - \rho) - \eta \sum_ {j \in [ 2 n ]} B _ {j} (1, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho) + O (\eta) \\ \geq 0. 9 9 \eta m \mathcal {T} _ {1} (n - k) - \eta B (1, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho)) + O (\eta) \\ \end{array}
$$
using  $|\Gamma_p| \geq 0.99$  (Assumption 3). From Lemma C.7
$$
\mathcal {T} _ {1} = \frac {1}{1 . 0 3 \eta m [ 1 + (\gamma + \rho) (n - k) ]} + O (1),
$$
furthermore, combining the assumptions  $100k < n$ ,  $\eta$  sufficiently small, and  $\gamma + \rho < \frac{1}{n - k}$  with Lemma C.2 we see
$$
\begin{array}{l} B (1, \mathcal {T} _ {\text {e n d}}) \leq \frac {2 k}{1 - 4 k (n - k) (\gamma + \rho) ^ {2}} \left[ \frac {1}{\eta} + 2 (n - k) (\gamma + \rho) \left(\frac {1}{\eta}\right) \right] + O (n) \\ \leq \frac {n}{1 0 \eta}. \\ \end{array}
$$
Here we also use that  $\ell(0, \mathbf{x}_i) \leq 1 + m\lambda_w = 1 + O(\eta)$  for all  $i$ .
Combining these inequalities it follows that
$$
\begin{array}{l} y f \left(\mathcal {T} _ {\text {e n d}}, \mathbf {x}\right) \geq \frac {0 . 9 9}{1 . 0 3} \cdot \frac {n - k}{2} (\gamma - \rho) - \frac {n}{1 0} (\gamma + \rho) + O (\eta) \\ \geq \frac {n}{3} - \frac {3 n}{2 5} + O (\eta) > 0, \\ \end{array}
$$
again using  $100k < n$ ,  $\eta$  sufficiently small, and  $\gamma + \rho < \frac{1}{n - k}$ .
C.6 Proof of Theorem 3.1
Theorem C.14 (Theorem 3.1). Let Assumption 2 hold with  $\rho = \gamma /5$ . There exists a sufficiently small step-size  $\eta$  such that with probability at least  $1 - \delta$  over the randomness of the dataset and network initialization the following hold.
1. There exists a positive constant  $C$  such that the training process terminates at an iteration  $\mathcal{T}_{end} \leq \frac{Cn}{\eta}$ .
2. For all  $i\in [2n]$  then  $\ell (\mathcal{T}_{end},\mathbf{x}_i) = 0$
3. There exists a positive constant  $c$  such that the generalization error satisfies
$$
\mathbb {P} \left(\operatorname {s g n} (f \left(\mathcal {T} _ {\text {e n d}}, \mathbf {x}\right)) \neq y\right) \leq \exp \left(- c d \gamma^ {2}\right).
$$
Proof. Under Assumption 3 Statement 1 and 2 follow from Lemma C.12, Note the bound in Statement 1 comes from Lemma C.2 applied from iteration 0 to iteration  $\mathcal{T}_{\mathrm{end}}$ , using  $4k(n - k)(\gamma + \rho)^2 < 4/99$  and  $\ell(0, \mathbf{x}_i) = 1 + O(\eta)$  for all  $i \in [2n]$ . With regards to Statement 3, from Lemma C.13 if  $|\langle \mathbf{n}, \mathbf{n}_\ell \rangle| < \frac{\rho}{1 - \gamma}$  for all  $l \in [2n]$  it follows that  $\operatorname{sgn}(f(\mathcal{T}_{\mathrm{end}}, \mathbf{x})) = y$ . Therefore, as  $\frac{\rho}{1 - \rho} > \rho$  and analogous to Lemma A.1, under Assumption 3 there exists a positive constant  $c$  such that
$$
\begin{array}{l} \mathbb {P} (\operatorname {s g n} (f (\mathcal {T} _ {\mathrm {e n d}}, \mathbf {x})) \neq y) \leq \mathbb {P} \left(\bigcup_ {l = 1} ^ {2 n} \left(| \langle \mathbf {n}, \mathbf {n} _ {\ell} \rangle | \geq \frac {\rho}{1 - \gamma}\right)\right) \\ \leq 2 n \mathbb {P} (\mathbf {n} \in \operatorname {C a p} (\mathbf {e} _ {1}, \gamma / 5)) \\ \leq 2 n \exp \left(- \frac {d \gamma^ {2}}{1 5}\right) \\ \leq \exp (- c d \gamma^ {2}). \\ \end{array}
$$
Finally, under Assumption 2 then Assumption 3 holds with probability at least  $1 - \delta$  by Lemma C.1.
Appendix D Non-benign overfitting
Assumption 4. With  $\delta, \rho \in (0,1)$  we assume the following conditions on the data and model hyperparameters.
1.  $n\geq 1$
2.  $m\geq \log_2(\frac{4n}{\delta})$
3.  $d \geq \max \left\{3, 3\rho^{-2}\ln \left(\frac{4n^2}{\delta}\right)\right\}$ ,
4.  $k\geq 0$
5.  $\gamma \leq \frac{1}{6\sqrt{dn}}$
6.  $\eta < \frac{1}{2mn}$
7.  $\lambda_w < \eta$
In our analysis we require two additional assumptions on the training sample and activations at initialization.
Assumption 5. Let  $\rho \in (0,1)$  satisfy  $\rho \leq \min \left\{\frac{1 - \gamma}{4n},\frac{1}{2n - 1} -\gamma \right\}$  and in addition to the conditions detailed in Assumption 5, assume the following two conditions hold.
1. For all  $i \in [2n]$  there exists a  $j \in [2m]$  such that  $(-1)^{j} = y_{i}$  and  $i \in \mathcal{A}_{j}^{(0)}$ .
2. For all  $i,l\in [2n],i\neq l|\langle \mathbf{n}_i,\mathbf{n}_l\rangle |\leq \frac{\rho}{1 - \gamma}.$
Note under these assumptions that  $\gamma < \frac{1}{24n^2}$ , this implies  $\frac{1}{2n - 1} > \gamma$  and  $\rho \leq \min \left\{\frac{1 - \gamma}{4n}, \frac{1}{2n - 1} - \gamma\right\}$  if  $\rho \leq \frac{1}{5n}$ . As demonstrated in the following Lemma, these additional two conditions hold with high probability over the randomness of the initialization and training set.
Lemma D.1. The additional conditions of Assumption 5 hold with probability at least  $1 - \delta$ .
Proof. Using Lemma A.7, then as long as  $m \geq \log_2\left(\frac{4n}{\delta}\right)$  the probability the first condition does not hold is at most  $\delta / 2$ . Using Lemma A.1, and observing  $\frac{\rho}{1 - \gamma} > \rho$ , then as long as
$$
d \geq \max \left\{3, 3 \rho^ {- 2} \ln \left(\frac {4 n ^ {2}}{\delta}\right) \right\}
$$
the probability that the second condition does not hold is also at most  $\delta /2$ . Using the union bound we conclude that both properties hold with probability at least  $\delta$ .
D.1 Proof of Lemma 3.7
Lemma D.2 (Lemma 3.7). In addition to Assumption 5, assume also that  $\ell(t_0, \mathbf{x}_i) \leq a$  for all  $i \in [2n]$ . Then
$$
T (t _ {0}, t) \leq \frac {2 n}{1 - (2 n - 1) (\gamma + \rho)} \left(\frac {a}{\eta} + 3 m\right).
$$
Proof. From Lemma B.5,  $\phi (\rho -\gamma)\leq \rho +\gamma$  , and the assumption on  $a$
$$
T _ {i} (t _ {0}, t) \leq \frac {a}{\eta} + 3 m + (\gamma + \rho) T ^ {(i)} (t _ {0}, t).
$$
If we sum over  $i$ , we get
$$
T (t _ {0}, t) \leq \frac {2 n a}{\eta} + 6 m n + (2 n - 1) (\gamma + \rho) T (t _ {0}, t),
$$
D.2 Supporting lemmas
Lemma D.3. If Assumption 5 holds, then the training process converges to zero loss.
Proof. By Lemma D.2, there is an upper bound on the number of updates independent of iteration. This can only happen if there is some iteration after which we make no updates. In turn, this can only happen if every point is either at zero loss or activates no neurons. We prove by induction that every point activates a neuron each iteration  $t = 0$ . Consider an arbitrary point  $i \in [2n]$ , at  $t = 0$  the induction hypothesis is true by Statement 1 of Assumption 5. Suppose the induction hypothesis is true at iteration  $t$ , we consider the following two cases separately in order to show the induction hypothesis also holds at iteration  $t + 1$ .
1. If  $\ell(t, \mathbf{x}_i) > 0$ , then by assumption we can choose a  $j \in [2m]$  such that  $\phi(\langle \mathbf{w}_j^{(t)}, \mathbf{x}_i \rangle) > 0$ . We bound
$$
\begin{array}{l} \phi (\langle \mathbf {w} _ {j} ^ {(t + 1)}, \mathbf {x} _ {i} \rangle) \geq \phi (\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle) + \eta - \eta (\gamma + \rho) T _ {j} ^ {(i)} (t, t + 1) \\ \geq \phi \left(\left\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \right\rangle\right) + \eta \left[ 1 - (\gamma + \rho) (2 n - 1) \right] \\ > 0, \\ \end{array}
$$
which follows as  $\gamma +\rho <  \frac{1}{2n - 1}$  (Assumption 5).
2. If  $\ell(t, \mathbf{x}_i) = 0$ , then
$$
\begin{array}{l} y _ {i} f (t, \mathbf {x} _ {i}) = y _ {i} \sum_ {j = 1} ^ {2 m} (- 1) ^ {j} \phi (\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle) \\ \leq \sum_ {j: (- 1) ^ {j} = y _ {i}} \phi \left(\left\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \right\rangle\right) \\ \end{array}
$$
is bounded below by 1. This means that there is some  $j$  such that  $\phi (\langle \mathbf{w}_j^{(t)},\mathbf{x}_i\rangle)\geq \frac{1}{m}$ . We bound
$$
\begin{array}{l} \phi \left(\left\langle \mathbf {w} _ {j} ^ {(t + 1)}, \mathbf {x} _ {i} \right\rangle\right) \geq \phi \left(\left\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \right\rangle\right) - \eta (\gamma + \rho) T _ {j} ^ {(i)} (t, t + 1) \\ \geq \frac {1}{m} - \eta (\gamma + \rho) (2 n - 1) \\ > 0 \\ \end{array}
$$
as  $\eta <  \frac{1}{2mn}$  (Assumption 5).
Lemma D.4. Suppose that at epoch  $\tau$  every point is at zero loss. Then
$$
\eta T (0, \tau) \geq n + O (\eta).
$$
Proof. If  $\ell (\tau ,\mathbf{x}_i) = 0$  for all  $i\in [2n]$ , then  $y_{i}f(\tau ,\mathbf{x}_{i})\geq 1$  for all  $i\in [2n]$ . We bound
$$
\begin{array}{l} y _ {i} f (\tau , \mathbf {x} _ {i}) \leq \sum_ {j: (- 1) ^ {j} = y _ {i}} \left(\phi \left(\left\langle \mathbf {w} _ {j} ^ {(\tau)}, \mathbf {x} _ {i} \right\rangle\right) - \phi \left(\left\langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} _ {i} \right\rangle\right)\right) + O (\eta) \\ \leq \sum_ {j: (- 1) ^ {j} = y _ {i}} \eta \left[ T _ {i j} (0, \tau) + (\gamma + \rho) T _ {j} ^ {(i)} (0, \tau) \right] + O (\eta) \\ \leq \eta T _ {i} (0, \tau) + \eta (\gamma + \rho) T ^ {(i)} (0, \tau) + O (\eta). \\ \end{array}
$$
Summing over  $i \in [2n]$  we see that
$$
2 n \leq \eta [ 1 + (2 n - 1) (\gamma + \rho) ] T (0, \tau) \leq 2 \eta T (0, \tau) + O (\eta),
$$
Lemma D.5. Let  $y \sim U(\{-1, 1\})$  and consider a clean test point  $\mathbf{x} = y(\sqrt{\gamma}\mathbf{v} + \sqrt{1 - \gamma}\mathbf{n})$ , where  $\mathbf{n} \sim U(\mathbb{S}^d \cap \operatorname{span}(\mathbf{v})^\perp)$ . If Assumption 5 holds, then
$$
\mathbb {P} (y f (\mathcal {T} _ {e n d}, \mathbf {x}) <   0) \geq \frac {1}{8}.
$$
Proof. Observe by symmetry of the distributions of both  $y$  and  $\mathbf{n}$  that  $-\mathbf{x}$  is identically distributed to  $\mathbf{x}$  and furthermore that the labels of  $\mathbf{x}$  and  $-\mathbf{x}$  are opposite. As a result, if  $y(f(\mathcal{T}_{\mathrm{end}}, \mathbf{x}) - f(\mathcal{T}_{\mathrm{end}}, -\mathbf{x})) < 0$  then at least one of  $y(f(\mathcal{T}_{\mathrm{end}}, \mathbf{x}) < 0$  or  $-y(f(\mathcal{T}_{\mathrm{end}}, -\mathbf{x}) < 0$ , in turn implying at least one of them is misclassified. By construction,  $\langle \mathbf{w}_j^{(t)}, \mathbf{x} \rangle > 0$  iff  $\langle \mathbf{w}_j^{(t)}, -\mathbf{x} \rangle < 0$ , therefore
$$
\begin{array}{l} y \left(f \left(\mathcal {T} _ {\text {e n d}}, \mathbf {x}\right) - f \left(\mathcal {T} _ {\text {e n d}}, - \mathbf {x}\right)\right) = y \sum_ {j = 1} ^ {2 m} (- 1) ^ {j} \left(\phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle\right) - \phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, - \mathbf {x} _ {i} \right\rangle\right)\right) \\ = \sum_ {j = 1} ^ {2 m} y (- 1) ^ {j} \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \rangle . \\ \end{array}
$$
Unwinding the GD update to a neuron we have
$$
\mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)} = \mathbf {w} _ {j} ^ {(0)} + \eta \sum_ {i = 1} ^ {2 n} T _ {i j} (0, \mathcal {T} _ {\text {e n d}}) (- 1) ^ {i + j} \mathbf {x} _ {i}.
$$
$$
\langle \mathbf {x} _ {i}, \mathbf {x} \rangle = y (- 1) ^ {i} (\gamma + (1 - \gamma) \beta (i) \langle \mathbf {n} _ {i}, \mathbf {n} \rangle))
$$
$$
\begin{array}{l} \sum_ {j = 1} ^ {2 m} y (- 1) ^ {j} \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \rangle = \sum_ {j = 1} ^ {2 m} y (- 1) ^ {j} \langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} \rangle + \eta \sum_ {j = 1} ^ {2 m} \sum_ {i = 1} ^ {2 n} y (- 1) ^ {j} T _ {i j} (0, \mathcal {T} _ {\text {e n d}}) (- 1) ^ {i + j} \langle \mathbf {x} _ {i}, \mathbf {x} \rangle \\ \leq 2 m \lambda_ {w} + \eta \sum_ {i = 1} ^ {2 n} T _ {i} (0, \mathcal {T} _ {\text {e n d}}) (\gamma + (1 - \gamma) \beta (i) \langle \mathbf {n} _ {i}, \mathbf {n} \rangle)) \\ = 2 m \lambda_ {w} + \eta \left(T (0, \mathcal {T} _ {\mathrm {e n d}}) \gamma + \left\langle \mathbf {n}, (1 - \gamma) \sum_ {i = 1} ^ {2 n} T _ {i} (0, \mathcal {T} _ {\mathrm {e n d}}) \beta (i) \mathbf {n} _ {i} \right\rangle\right) \\ \stackrel {d} {=} 2 m \lambda_ {w} + \eta (T (0, \mathcal {T} _ {\mathrm {e n d}}) \gamma - \| \mathbf {z} \| (1 - \gamma) \langle \mathbf {n}, \mathbf {u} \rangle), \\ \end{array}
$$
where the final equality follows from symmetry of the noise distribution,  $\mathbf{z} := \sum_{i=1}^{2n} T_i(0, \mathcal{T}_{\mathrm{end}})\mathbf{n}_i$  and  $\mathbf{u} = \frac{\mathbf{z}}{\|\mathbf{z}\|}$ . Observe
$$
\begin{array}{l} \| \mathbf {z} \| ^ {2} = \sum_ {i, l = 1} ^ {2 n} T _ {i} (0, \mathcal {T} _ {\text {e n d}}) T _ {\ell} (0, \mathcal {T} _ {\text {e n d}}) \langle \mathbf {n} _ {i}, \mathbf {n} _ {\ell} \rangle \\ \geq \sum_ {i} ^ {2 n} T _ {i} ^ {2} (0, \mathcal {T} _ {\text {e n d}}) - \frac {\rho}{1 - \gamma} \sum_ {i \neq \ell} T _ {i} (0, \mathcal {T} _ {\text {e n d}}) T _ {\ell} (0, \mathcal {T} _ {\text {e n d}}) \\ = \frac {1 - \gamma + \rho}{1 - \gamma} \sum_ {i} ^ {2 n} T _ {i} ^ {2} (0, \mathcal {T} _ {\text {e n d}}) - \frac {\rho}{1 - \gamma} T ^ {2} (0, \mathcal {T} _ {\text {e n d}}) \\ \geq \left(\frac {1}{2 n} - \frac {\rho}{1 - \gamma}\right) T ^ {2} (0, \mathcal {T} _ {\text {e n d}}). \\ \end{array}
$$
where the final inequality follows from Jensen's inequality. By assumption  $4n\rho < 1 - \gamma$ , and  $10m\lambda_w \leq n\gamma \leq \frac{\sqrt{n}}{6\sqrt{d}}$ , furthermore trivially  $(1 - \gamma) > 0.8$ . Conditioning on the event  $\langle \mathbf{n}, \mathbf{u} \rangle > 0$  which holds with probability  $1/2$ , these inequalities in combination with Lemma D.4 give
$$
\begin{array}{l} \sum_ {j = 1} ^ {2 m} y (- 1) ^ {j} \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \rangle \leq 2 m \lambda_ {w} + n \gamma - \frac {\sqrt {n}}{2} (1 - \gamma) \langle \mathbf {n}, \mathbf {u} \rangle \\ \leq \frac {1}{5} \sqrt {\frac {n}{d}} - \frac {4}{1 0} \sqrt {n} \langle \mathbf {n}, \mathbf {u} \rangle . \\ \end{array}
$$
Therefore, if  $\langle \mathbf{n},\mathbf{u}\rangle >0$  then the condition
$$
\langle \sqrt {d} \mathbf {n}, \mathbf {u} \rangle \geq \frac {1}{2} \tag {8}
$$
implies at least one of  $\mathbf{x}$  or  $-\mathbf{x}$  is misclassified. Suppose  $\mathbf{n} \sim U(\mathbb{S}^d \cap \operatorname{span}(\mathbf{v})^\perp)$  is such that (8) holds. Then as  $y \sim U(\{-1, 1\})$  it follows given  $\mathbf{n}$  that either  $\mathbf{x}$  or  $-\mathbf{x}$  are sampled each with equal probability and thus the chance of misclassifying is at least  $1/2$ . As a result, the probability of misclassification is at least
$$
\frac {1}{4} \mathbb {P} \left(\langle \sqrt {d} \mathbf {n}, \mathbf {u} \rangle \geq \frac {1}{2}\right) \geq \frac {1}{8}
$$
as claimed. We note the final inequality above follows by showing that the two spherical caps corresponding to the set of unit vectors  $\mathbf{z}$  satisfying  $\langle \mathbf{n},\mathbf{u}\rangle \geq 1 / (2\sqrt{d})$  account for less than half the area of  $\mathbb{S}^{d - 1}$ , which can be derived from formulas provided in (S, 2011). This inequality has also appeared for instance in (Asi & Duchi, 2019).
D.3 Proof of Theorem 3.6
Theorem D.6 (Theorem 3.6). Assume Assumption 4 holds with  $\rho = \frac{1}{5n}$ . With probability at least  $1 - \delta$  over the randomness of the dataset and network initialization the following hold.
1. There exists a positive constant  $C$  such that the training process terminates at an iteration  $\mathcal{T}_{end} \leq \frac{Cn}{\eta}$ .
2. For all  $i\in [2n]$ $\ell (\mathcal{T}_{end},\mathbf{x}_i) = 0$
3. The generalization error satisfies
Proof. Under Assumption 5 Statement 1 and 2 come from Lemma D.3. The bound on  $\mathcal{T}_{\mathrm{end}}$  comes from Lemma D.2 applied between iterations 0 and  $\mathcal{T}_{\mathrm{end}}$ , using  $\ell(0, \mathbf{x}_i) = 1 + O(\eta)$  for all  $i \in [2n]$  and  $(\gamma + \rho)(2n - 1) < \frac{2}{3}$ . Statement 3 follows from Lemma D.5. We conclude by observing under Assumption 4 that Assumption 5 holds with probability at least  $1 - \delta$ .
Appendix E No-overfitting
Assumption 6. With  $\delta, \rho \in (0,1)$  and  $C$  a generic, positive constant, we assume the following conditions on the data and model hyperparameters.
1.  $n \geq C \log \left(\frac{2m}{\delta}\right)$
2.  $m\geq 2$
3.  $d \geq \left\{3, 3\rho^{-2}\ln \left(\frac{6n^2}{\delta}\right)\right\}$ ,
4.  $k < \frac{n}{100}$
5.  $\frac{3}{n} <  \gamma <  \frac{1}{36}\min \{k^{-1},1\} ,$
6.  $\lambda_w < \eta$
For our analysis we make two additional assumptions.
Assumption 7. Let  $\rho \in (0,1)$  satisfy  $\gamma \geq 5\rho$ . In addition to the assumptions detailed in Assumption 6, suppose the following conditions hold.
1.  $\Gamma = [2m]$
2. For all  $i, l \in [2n]$  such that  $i \neq l$  then  $|\langle \mathbf{n}_i, \mathbf{n}_l \rangle| \leq \frac{\rho}{1 - \gamma}$ .
We remark under these assumptions that for sufficiently large  $n$  then  $\rho$  satisfies the inequality  $\rho < \min \left\{\frac{\gamma(n - 3k) - 2}{n + k}, \frac{\gamma}{5}, \frac{n}{11}\right\}$ . As shown in the following lemma, these two additional conditions hold with high probability over the randomness of the initialization and training set.
Lemma E.1. There exists a positive constant  $C$  such that if  $n \geq C \log \left(\frac{2m}{\delta}\right)$  then the extra conditions of Assumption 7 hold with probability at least  $1 - \delta$ .
Proof. Using Lemma A.3, there exists a positive constant  $C$  such that for  $n \geq C$  there is a turn exists a constant  $c$  such that the probability the first condition does not hold is at most  $m \exp(-cn)$ . Setting  $\delta \geq 2m \exp(-cn)$  and rearranging, as long as  $n \geq C \log\left(\frac{2m}{\delta}\right)$ , then the probability the first condition does not hold is at most  $\delta/2$ . Using Lemma A.1 and observing  $\frac{\rho}{1 - \gamma} > \rho$ , then under the condition on  $d$  stated in Assumption 7, the probability that the second condition does not hold is also at most  $\frac{\delta}{2}$ . Using the union bound, we therefore conclude that both properties hold with probability at least  $\delta$ .
E.1 Proof of Lemma 3.9
Lemma E.2 (Lemma 3.9). Suppose Assumption 7 holds. Consider an arbitrary  $j \in [2m]$  and iteration  $t$  satisfying  $2 \leq t < \mathcal{T}_0$ . Then  $i \in \mathcal{A}_j^{(t)}$  iff  $i \sim j$ .
Proof. First we establish at iteration  $t = 1$  that for all  $i \in S_T$ ,  $i \in \mathcal{A}_j^{(t)}$  iff  $i \sim j$ . The argument here is similar to that of Lemma C.3. Suppose  $i \sim j$  and  $i \in S_T$ . By Assumption 7 all neurons are in  $\Gamma$ , therefore from the definition of  $\Gamma_p$  for all  $j \in [2m]$  we have  $G_j^{(i)}(0,1)(\gamma - \rho) - B_j^{(i)}(0,1)(\gamma + \rho) \geq \frac{2\lambda_w}{\eta}$ . Using Lemma B.2
On the other hand, if  $i \not\sim j$  then again from Lemma B.2
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} _ {i} \rangle - \eta (T _ {i j} (0, 1) + G _ {j} ^ {(i)} (0, 1) (\gamma - \rho) - B _ {j} ^ {(i)} (0, 1) (\gamma + \rho)) \\ \leq \langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} _ {i} \rangle - \eta (G _ {j} (0, 1) (\gamma - \rho) - B _ {j} (0, 1) (\gamma + \rho)) \\ \leq - \lambda_ {w}. \\ \end{array}
$$
Now we consider  $t = 2$ . If  $i \in S_T$  and  $i \sim j$  then
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle + \eta (T _ {i j} (1, 2) + G _ {j} ^ {(i)} (1, 2) (\gamma - \rho) - B _ {j} ^ {(i)} (1, 2) (\gamma + \rho)) \\ > \eta (1 + (n - k - 1) (\gamma - \rho) - 2 k (\gamma + \rho)) \\ \end{array}
$$
whereas if  $i\in S_T$  and  $i\not\sim j$  then
$$
\begin{array}{l} \left\langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \right\rangle \leq \left\langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \right\rangle - \eta \left(T _ {i j} (1, 2) + G _ {j} ^ {(i)} (1, 2) (\gamma - \rho) - B _ {j} ^ {(i)} (1, 2) (\gamma + \rho)\right) \\ \leq - \eta ((n - k) (\gamma - \rho) - 2 k (\gamma + \rho)). \\ \end{array}
$$
By assumption  $\gamma >\frac{2 + (n + k)\rho}{n - 3k} >\frac{(n + k)\rho}{n - 3k}$  and therefore for  $i\in S_T$  then  $i\in \mathcal{A}_j^{(2)}$  iff  $i\sim j$ . Again using Lemma B.2, for  $i\in S_F$  and  $i\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} _ {i} \rangle - \eta (T _ {i j} (0, 1) - G _ {j} ^ {(i)} (0, 1) (\gamma - \rho) + B _ {j} ^ {(i)} (0, 1) (\gamma + \rho)) \\ > - \lambda_ {w} - \eta \left(1 - \frac {2 \lambda_ {w}}{\eta}\right) \\ > - \eta , \\ \end{array}
$$
and for  $i\in \mathcal{S}_F$  and  $i\not\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(0)}, \mathbf {x} _ {i} \rangle + \eta \left(T _ {i j} (0, 1) - G _ {j} ^ {(i)} (0, 1) (\gamma - \rho) + B _ {j} ^ {(i)} (0, 1) (\gamma + \rho)\right) \\ <   \lambda_ {w} + \eta \left(1 - \frac {2 \lambda_ {w}}{\eta}\right) \\ <   \eta . \\ \end{array}
$$
Therefore, as  $\gamma >\frac{2 + (n + k)\rho}{n - 3k}$  then for  $i\in S_F$  and  $i\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle - \eta (T _ {i j} (1, 2) - G _ {j} ^ {(i)} (1, 2) (\gamma - \rho) + B _ {j} ^ {(i)} (1, 2) (\gamma + \rho)) \\ > - \eta + \eta ((n - k) (\gamma - \rho) - 2 k (\gamma + \rho) - 1) \\ > 0 \\ \end{array}
$$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(1)}, \mathbf {x} _ {i} \rangle + \eta \left(T _ {i j} (1, 2) - G _ {j} ^ {(i)} (1, 2) (\gamma - \rho) + B _ {j} ^ {(i)} (1, 2) (\gamma + \rho)\right) \\ <   \eta - \eta ((n - k) (\gamma - \rho) - 2 k (\gamma + \rho) - 1) \\ <   0. \\ \end{array}
$$
With the base case established we proceed by induction to prove if  $i \in \mathcal{A}_j^{(t-1)}$  iff  $i \sim j$ , then  $i \in \mathcal{A}_j^{(t)}$  iff  $i \sim j$ . By the assumptions on  $\gamma$ , the induction hypothesis and again using Lemma B.2, for  $i \sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(t - 1)}, \mathbf {x} _ {i} \rangle - \eta \left(T _ {i j} (t - 1, t) - G _ {j} ^ {(i)} (t - 1, t) (\gamma - \rho) + B _ {j} ^ {(i)} (t - 1, t) (\gamma + \rho)\right) \\ > \eta ((n - k) (\gamma - \rho) - k (\gamma + \rho) - 1) \\ > 0 \\ \end{array}
$$
and for  $i \not\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(t - 1)}, \mathbf {x} _ {i} \rangle + \eta \left(T _ {i j} (t - 1, t) - G _ {j} ^ {(i)} (t - 1, t) (\gamma - \rho) + B _ {j} ^ {(i)} (t - 1, t) (\gamma + \rho)\right) \\ <   - \eta ((n - k) (\gamma - \rho) - k (\gamma + \rho) - 1) \\ <   0. \\ \end{array}
$$
Therefore, for an epoch  $t$  satisfying  $2 \leq t \leq \mathcal{T}_1$  then  $i \in \mathcal{A}_j^{(t-1)}$  iff  $i \sim j$ .
E.2 Proof of Lemma 3.10
Lemma E.3 (Lemma 3.10). Suppose Assumption 7 holds, then there is an iteration  $\mathcal{T}_1 < \mathcal{T}_0$  such that
$$
\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \rangle \leq \frac {\gamma (n - 2 k) + \rho n - 1 + (\gamma - \rho)}{m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (\eta) i f i \in \mathcal {S} _ {F}, i \sim j
$$
$$
\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \rangle \geq \frac {1 + \gamma (n - 2 k) - \rho n - (\gamma - \rho)}{m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (\eta) i f i \in \mathcal {S} _ {T}, i \sim j
$$