SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \rangle \leq - \frac {\gamma (n - 2 k) - \rho n}{m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (\eta) i f i \not \sim j.
$$
Furthermore for  $i\in \mathcal{S}_T$
$$
\ell \left(\mathcal {T} _ {1}, \mathbf {x} _ {i}\right) \leq \frac {2 \rho n}{1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho)} + O (\eta).
$$
$$
\mathcal {T} _ {1} = \frac {1}{\eta m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (1).
$$
Proof. Let  $t < \mathcal{T}_0$ . By Lemma E.2 and Lemma B.2 we can bound for  $i \in S_F$  and  $i \sim j$  as follows,
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle - \eta T _ {i j} (2, t) - \eta (\gamma - \rho) B _ {j} ^ {(i)} (2, t) + \eta (\gamma + \rho) G _ {j} (2, t) \\ = \eta t ((\gamma + \rho) (n - k) - (\gamma - \rho) (k - 1) - 1) + O (\eta) \\ = \eta t (\gamma (n - 2 k) + \rho n - 1 + (\gamma - \rho)) + O (\eta). \\ \end{array}
$$
Similarly, for  $i\in \mathcal{S}_F$ $i\not\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + \eta T _ {i j} (2, t) + \eta (\gamma + \rho) B _ {j} ^ {(i)} (2, t) - \eta (\gamma - \rho) G _ {j} (2, t) \\ = - \eta t ((\gamma - \rho) (n - k) - (\gamma + \rho) k) + O (\eta) \\ = - \eta t (\gamma (n - 2 k) - \rho n) + O (\eta). \\ \end{array}
$$
For  $i\in \mathcal{S}_T,i\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + \eta T _ {i j} (2, t) + \eta (\gamma + \rho) G _ {j} ^ {(i)} (2, t) - \eta (\gamma - \rho) G _ {j} (2, t) \\ = \eta t (1 + (\gamma + \rho) (n - k - 1) - (\gamma - \rho) k) + O (\eta) \\ = \eta t (1 + \gamma (n - 2 k) + \rho n - (\gamma + \rho)) + O (\eta) \\ \end{array}
$$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + \eta T _ {i j} (2, t) + \eta (\gamma - \rho) G _ {j} ^ {(i)} (2, t) - \eta (\gamma + \rho) G _ {j} (2, t) \\ = \eta t (1 + (\gamma - \rho) (n - k - 1) - (\gamma + \rho) k) + O (\eta) \\ = \eta t (1 + \gamma (n - 2 k) - \rho n - (\gamma - \rho)) + O (\eta). \\ \end{array}
$$
Lastly, for  $i\in S_T,i\not\sim j$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + \eta T _ {i j} (2, t) - \eta (\gamma - \rho) G _ {j} ^ {(i)} (2, t) + \eta (\gamma + \rho) G _ {j} (2, t) \\ = - \eta t ((\gamma - \rho) (n - k) - (\gamma + \rho) k) + O (\eta) \\ = - \eta t (\gamma (n - 2 k) - \rho n) + O (\eta) \\ \end{array}
$$
Therefore, for  $i\in S_T$
$$
\begin{array}{l} f (t, \mathbf {x} _ {i}) = y _ {i} \sum_ {j = 1} ^ {2 m} (- 1) ^ {j} \phi (\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle) \\ = \sum_ {j \sim i} \left\langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \right\rangle \\ \end{array}
$$
from which we conclude
$$
\eta m t (1 + \gamma (n - 2 k) - \rho n - (\gamma - \rho)) + O (\eta) \leq f (t, \mathbf {x} _ {i}) \leq \eta m t (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho)) + O (\eta).
$$
Therefore, as long as
$$
\eta m t (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho)) + O (\eta) <   1, \tag {9}
$$
then  $\ell(t, \mathbf{x}_i) > 0$ . Let  $\mathcal{T}_1$  be the largest value of  $t$  satisfying (9) and  $t < \mathcal{T}_0$ . We see that
From this, the bounds claimed follow.
E.3 Proof of Lemma 3.11
Lemma E.4 (Lemma 3.11). Let Assumption 7 hold. Suppose at iteration  $t_0$  the following conditions are satisfied.
a.  $\ell (t_0,\mathbf{x}_i)\leq a$  for all  $i\in S_T$
b.  $\phi (\langle \mathbf{w}_j^{(t_0)},\mathbf{x}_i\rangle)\leq b$  for all  $i\in S_F$  and  $i\sim j$
c. For all iterations  $\tau$  satisfying  $t_0\leq \tau \leq t$  it holds that  $i\in \mathcal{A}_j^{(\tau)}$  only if  $i\sim j$
d. For all iterations  $\tau$  satisfying  $t_0\leq \tau \leq t$ ,  $i\in \mathcal{A}_j^{(\tau)}$  if  $i\sim j$  and  $i\in S_T$ .
Then for  $j \in [2m]$  and  $p \in \{-1, 1\}$  we have
$$
\begin{array}{l} B _ {j} (t _ {0}, \tau) \leq \frac {k}{\eta} \left(\frac {3 b}{2} + \frac {2 a}{m}\right), \\ \sum_ {j \sim s} G _ {j} (t _ {0}, t) \leq \frac {1}{\gamma \eta} \left(\frac {3 a}{2} + m b\right). \\ \end{array}
$$
Proof. Consider an arbitrary neuron  $j \in [2m]$ , using Lemma B.3 and assumption (b) we bound for  $t < \tau \leq t, i \in S_F$ , and  $i \sim j$
$$
\phi (\langle \mathbf {w} _ {j} ^ {(\tau)}, \mathbf {x} _ {i} \rangle) \leq b - \eta (T _ {i j} (t _ {0}, \tau) - (\gamma + \rho) G _ {j} (t _ {0}, \tau) - 1).
$$
As  $\phi (\langle \mathbf{w}_j^{(\tau)},\mathbf{x}_i\rangle)\geq 0$  in general we may conclude that
$$
T _ {i j} (t _ {0}, \tau) \leq \frac {b}{\eta} + (\gamma + \rho) G _ {j} (t _ {0}, \tau) + 1.
$$
Summing over all  $i\in \mathcal{S}_F$  such that  $i\sim j$  then by assumption (c)
$$
\sum_{\substack{i\in \mathcal{S}_{F}\\ i\sim j}}T_{ij}(t_{0},\tau) = B_{j}(t_{0},\tau).
$$
Combining these expressions it follows that
$$
B _ {j} (t _ {0}, \tau) \leq \frac {k b}{\eta} + k (\gamma + \rho) G _ {j} (t _ {0}, \tau) + k.
$$
As the number of clean updates on a pair of neurons  $\ell$  and  $j$  with  $\ell \sim j$  is the same by assumptions (c) and (d), then we may rewrite this bound as
$$
B _ {j} \left(t _ {0}, \tau\right) \leq \frac {k b}{\eta} + \frac {k (\gamma + \rho)}{m} \sum_ {\ell \sim j} G _ {\ell} \left(t _ {0}, t\right) + k. \tag {10}
$$
Let  $s \in [2m]$  be arbitrary, we proceed to bound  $\sum_{j \sim s} G_j(t_0, t)$ . Using Lemma B.2, for  $i \in S_T$  and  $i \sim j$
$$
\langle \mathbf {w} _ {j} ^ {(\tau - 1)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(t)}, \mathbf {x} _ {i} \rangle + \eta (T _ {i j} (t _ {0}, \tau - 1) + (\gamma - \rho) G _ {j} ^ {(i)} (t _ {0}, \tau - 1) - (\gamma + \rho) B _ {j} (t _ {0}, \tau - 1)).
$$
By assumptions (c) and (d)
$$
\begin{array}{l} y _ {i} f (\tau - 1, \mathbf {x} _ {i}) = \sum_ {j \sim i} \phi \left(\left\langle \mathbf {w} _ {j} ^ {(\tau - 1)}, \mathbf {x} _ {i} \right\rangle\right) \\ \geq \sum_ {j \sim i} \left(\langle \mathbf {w} _ {j} ^ {(t _ {0})}, \mathbf {x} _ {i} \rangle + \eta T _ {i j} (t _ {0}, \tau - 1) \right. \\ \left. + \eta (\gamma - \rho) G _ {j} ^ {(i)} (t _ {0}, \tau - 1) - \eta (\gamma + \rho) B _ {j} (t _ {0}, \tau - 1)\right) \\ \geq (1 - a) + \eta (1 - (\gamma - \rho)) T _ {i} (t _ {0}, \tau - 1) \\ + \left(\sum_ {j \sim i} \left((\gamma - \rho) G _ {j} (t _ {0}, \tau - 1) - (\gamma + \rho) B _ {j} (t _ {0}, \tau - 1)\right)\right). \\ \end{array}
$$
Note either  $T_{i}(t_{0},\tau) = T_{i}(t_{0},\tau -1)$  or  $\ell (\tau ,\mathbf{x}_i) > 0$ . Consider the case where the latter holds, then
$$
\eta ((1 - (\gamma - \rho)) T _ {i} (t _ {0}, \tau - 1) + \sum_ {j \sim i} ((\gamma - \rho) G _ {j} (t _ {0}, \tau - 1) - (\gamma + \rho) B _ {j} (t _ {0}, \tau - 1))) <   a.
$$
Furthermore, suppose  $\tau' \leq \tau$  is the first iteration before  $\tau$  such that  $G_{j}(\tau', \tau) = 0$  and let  $i \in S_{T}$ ,  $i \sim s$  be a point that makes an update at iteration  $\tau' - 1$ . Using the above bound it follows that
$$
\eta ((1 - (\gamma - \rho)) T _ {i} (t _ {0}, \tau^ {\prime} - 1) + \sum_ {j \sim i} ((\gamma - \rho) G _ {j} (t _ {0}, \tau^ {\prime} - 1) - (\gamma + \rho) B _ {j} (t _ {0}, \tau^ {\prime} - 1))) <   a.
$$
$$
\sum_ {j \sim i} ((\gamma - \rho) G _ {j} (t _ {0}, \tau^ {\prime} - 1) - (\gamma + \rho) B _ {j} (t _ {0}, \tau^ {\prime} - 1)) <   \frac {a}{\eta}
$$
$$
\sum_ {j \sim s} ((\gamma - \rho) G _ {j} (t _ {0}, \tau^ {\prime}) - (\gamma + \rho) B _ {j} (t _ {0}, \tau^ {\prime})) <   \frac {a}{\eta} - 2 (n - k) (\gamma - \rho).
$$
By the construction of  $\tau^{\prime}$  it follows that
$$
\sum_ {j \sim s} ((\gamma - \rho) G _ {j} (t _ {0}, \tau) - (\gamma + \rho) B _ {j} (t _ {0}, \tau)) <   \frac {a}{\eta} - 2 (n - k) (\gamma - \rho).
$$
From this we get the bound
$$
\sum_ {j \sim s} G _ {j} \left(t _ {0}, t\right) \leq \frac {1}{\gamma - \rho} \left(\frac {a}{\eta} + (\gamma + \rho) \sum_ {j \sim s} B _ {j} \left(t _ {0}, t\right)\right) + O (1). \tag {11}
$$
Combining (10) summed over  $j\sim s$  with (11), then
$$
\begin{array}{l} \sum_ {j \sim s} G _ {j} (t _ {0}, t) \leq \frac {1}{\gamma - \rho} \left(\frac {a}{\eta} + (\gamma + \rho) \left(\frac {k m b}{\eta} + k (\gamma + \rho) \sum_ {j \sim s} G _ {j} (t _ {0}, t) + k\right)\right) + O (1) \\ \leq \frac {1}{\gamma - \rho - k (\gamma + \rho) ^ {2}} \left(\frac {a}{\eta} + \frac {k m b (\gamma + \rho)}{\eta}\right) + O (1) \\ \end{array}
$$
$$
B _ {j} (t _ {0}, \tau) \leq \frac {k b}{\eta} + \frac {k (\gamma + \rho)}{m (\gamma - \rho - k (\gamma + \rho) ^ {2})} \left(\frac {a}{\eta} + \frac {k m b (\gamma + \rho)}{\eta}\right) + O (1).
$$
Using  $\rho \leq \frac{\gamma}{5}$ ,  $\eta$  sufficiently small and  $\gamma \leq \frac{1}{36k}$  (Assumption 7) these bounds simplify to
$$
B _ {j} (t _ {0}, \tau) \leq \frac {k}{\eta} \left(\frac {3 b}{2} + \frac {2 a}{m}\right),
$$
$$
\sum_ {j \sim s} G _ {j} (t _ {0}, t) \leq \frac {1}{\gamma \eta} \left(\frac {3 a}{2} + m b\right).
$$
E.4 Late training
Lemma E.5. Under Assumption 7 the training process terminates at an iteration  $\mathcal{T}_{\text{end}}$  satisfying
$$
\ell (\mathcal {T} _ {e n d}, \mathbf {x} _ {i}) = 0
$$
for all  $i\in \mathcal{S}_T$  and
$$
\phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {e n d}\right)}, \mathbf {x} _ {i} \right\rangle\right) = 0
$$
for all  $i\in \mathcal{S}_F$  and  $j\in [2m]$
Proof. By Lemma E.3, at iteration  $t = \mathcal{T}_1$  with
$$
a = \frac {2 \rho n}{1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho)} + O (\eta),
$$
$$
b = \frac {\gamma (n - 2 k) + \rho n - 1 + (\gamma - \rho)}{m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (\eta)
$$
then the first two conditions of Lemma E.4 are satisfied. Next, using Lemma B.2, we see by induction on  $t \geq \mathcal{T}_1$  that if
$$
B _ {j} (\mathcal {T} _ {1}, t) <   \frac {\gamma (n - 2 k) - \rho n}{\eta m (\gamma + \rho) (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (1)
$$
then for  $i \not\sim j$  and  $\mathcal{T}_1 \leq \tau \leq t$ ,
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(\tau)}, \mathbf {x} _ {i} \rangle \leq \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \rangle + \eta (\gamma + \rho) B _ {j} \left(\mathcal {T} _ {1}, \tau\right) \\ \leq \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \rangle + \eta (\gamma + \rho) B _ {j} \left(\mathcal {T} _ {1}, t\right) \\ <   0 \\ \end{array}
$$
and for  $i\in S_T,i\sim j$  , and  $\mathcal{T}_1\leq \tau \leq t$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {(\tau)}, \mathbf {x} _ {i} \rangle \geq \langle \mathbf {w} _ {j} ^ {(\mathcal {T} _ {1})}, \mathbf {x} _ {i} \rangle - \eta (\gamma + \rho) B _ {j} (\mathcal {T} _ {1}, \tau) \\ \geq \left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {1}\right)}, \mathbf {x} _ {i} \right\rangle - \eta (\gamma + \rho) B _ {j} \left(\mathcal {T} _ {1}, t\right) \\ \geq \frac {1 - (\gamma - \rho)}{m (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (\eta) \\ > 0 \\ \end{array}
$$
for  $\eta$  sufficiently small. Thus we have shown under an additional assumption on  $B_{j}(\mathcal{T}_{1},t)$  that with  $t_0 = \mathcal{T}_1$  and  $a$  and  $b$  as defined above, then all four conditions of Lemma E.4 are satisfied. As a result GD converges or terminates as long as
$$
k \left(\frac {3 b}{2 \eta} + \frac {2 a}{\eta m}\right) <   \frac {\gamma (n - 2 k) - \rho n}{\eta m (\gamma + \rho) (1 + \gamma (n - 2 k) + \rho n - (\gamma - \rho))} + O (1)
$$
$$
k (\gamma + \rho) \left(\frac {3}{2} (\gamma (n - 2 k) - 1 + (\gamma - \rho)) + \frac {1 1}{2} \rho n\right) <   \gamma (n - 2 k) - \rho n + O (\eta).
$$
This is true by Assumption 7, as
$$
\begin{array}{l} k (\gamma + \rho) \left(\frac {3}{2} (\gamma (n - 2 k) - 1 + (\gamma - \rho)) + \frac {1 1}{2} \rho n\right) \leq \frac {1}{3 0} \left(\frac {3}{2} \gamma (n - 2 k) - \frac {3}{2} + \frac {1}{3 6} + \frac {1}{2}\right) \\ \leq \frac {1}{2 0} \gamma (n - 2 k) - \frac {1}{2} \\ <   \gamma (n - 2 k) - \rho n, \\ \end{array}
$$
where above we used  $\gamma \leq \min \left\{\frac{1}{36k},\frac{1}{36}\right\}$  and  $\rho \leq \min \left\{\frac{\gamma}{5},\frac{n}{11}\right\}$ .
Lemma E.6. Assume Assumption 7 holds. Let  $y \in \{-1,1\}$  be drawn uniformly at random and  $\mathbf{x} := y\sqrt{\gamma}\mathbf{v} + \sqrt{1 - \gamma}\mathbf{n}$  where  $\mathbf{n} \sim \mathrm{Uniform}(\mathcal{S}^{d-1} \cap \mathrm{span}\{\mathbf{v}\}^{\perp})$ . Suppose that  $|\langle \mathbf{n},\mathbf{n}_{\ell}\rangle| < \frac{\rho}{1 - \gamma}$  for all  $l \in [2n]$ , then  $yf(\mathcal{T}_{end},\mathbf{x}) > 0$ .
Proof. We proceed as in the proof of Lemma C.13. Following the same steps as in (5), for any  $j \in [2m]$
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \rangle = \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + (- 1) ^ {j} \eta \sum_ {\ell = 1} ^ {2 n} T _ {\ell j} (1, \mathcal {T} _ {\text {e n d}}) y _ {\ell} \langle \mathbf {x} _ {\ell}, \mathbf {x} _ {i} \rangle \\ = \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + (- 1) ^ {j} y \eta \sum_ {\ell = 1} ^ {2 n} T _ {\ell j} (2, t) (- 1) ^ {\ell} y \beta (\ell) \langle \mathbf {x} _ {\ell}, \mathbf {x} \rangle \\ = \left\langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \right\rangle + (- 1) ^ {j} y \eta \sum_ {\ell = 1} ^ {2 n} T _ {\ell j} (2, t) \lambda_ {i \ell} ^ {\prime}, \\ \end{array}
$$
Recall, from Lemma E.2, for any  $j \in [2m]$  then  $G_{j}(2,\mathcal{T}_{\mathrm{end}}) \geq G_{j}(2,\mathcal{T}_{1}) = (\mathcal{T}_{1} - 2)(n - k)$ . As a consequence, for  $j \in \Gamma_p$  we have
$$
\begin{array}{l} \langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \rangle \geq \langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \rangle + \eta G _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma - \rho) - \eta B _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho) \\ \geq O (\eta) + \mathcal {T} _ {1} (n - k) (\gamma - \rho) - \eta B _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho). \\ \end{array}
$$
For  $j$  such that  $(-1)^j = y$  then
$$
\begin{array}{l} \phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle\right) \leq \phi \left(\left\langle \mathbf {w} _ {j} ^ {(2)}, \mathbf {x} _ {i} \right\rangle - \eta G _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma - \rho) + \eta B _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho)\right) \\ \leq O (\eta) + \eta B _ {j} (2, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho). \\ \end{array}
$$
$$
\begin{array}{l} y f (\mathcal {T} _ {\text {e n d}}, \mathbf {x}) = \sum_ {j \in [ 2 m ]} \phi (\langle \mathbf {w} _ {j} ^ {(\mathcal {T} _ {\text {e n d}})}, \mathbf {x} \rangle) \\ \geq \sum_ {j: (- 1) ^ {j} = y} \left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle - \sum_ {j: (- 1) ^ {j} \neq y} \phi \left(\left\langle \mathbf {w} _ {j} ^ {\left(\mathcal {T} _ {\text {e n d}}\right)}, \mathbf {x} \right\rangle\right) \\ \geq \eta \sum_ {j: (- 1) ^ {j} = y} (\mathcal {T} _ {1} - 2) (n - k) (\gamma - \rho) - \eta \sum_ {j \in [ 2 n ]} B _ {j} (2, \mathcal {T} _ {\mathrm {e n d}}) (\gamma + \rho) + O (\eta) \\ \geq \eta m \mathcal {T} _ {1} (n - k) (\gamma - \rho) - \eta B (2, \mathcal {T} _ {\text {e n d}}) (\gamma + \rho)) + O (\eta) \\ \end{array}
$$
Decompose  $B(2, \mathcal{T}_{\mathrm{end}}) = B(2, \mathcal{T}_1) + B(\mathcal{T}_1, \mathcal{T}_{\mathrm{end}})$  and observe from Lemma E.2 that
$$
B (2, \mathcal {T} _ {1}) = 2 k m (\mathcal {T} _ {1} - 2) = 2 k m \mathcal {T} _ {1} + O (1).
$$
From Lemma E.3 and Lemma E.4, using the assumptions  $\rho \leq \frac{n}{11}$ ,  $\eta$  sufficiently small and  $\gamma \leq \min\left\{\frac{k}{36}, \frac{1}{36}\right\}$  then
$$
\begin{array}{l} B \left(\mathcal {T} _ {1}, \mathcal {T} _ {\text {e n d}}\right) \leq 2 m k \left(\frac {3 b}{2 \eta} + \frac {2 a}{\eta m}\right) \\ \leq 2 m k \mathcal {T} _ {1} \left(\frac {3 (\gamma (n - 2 k) + \rho n - 1 + (\gamma - \rho))}{2} + 4 \rho n\right) + O (1) \\ \leq 2 m \mathcal {T} _ {1} \left(\frac {n - 2 k}{2 4} + \frac {1}{2 2} - \frac {1}{2} + \frac {1}{7 2} + \frac {4}{1 1}\right) + O (1) \\ \leq \frac {m \mathcal {T} _ {1} (n - k)}{1 2}. \\ \end{array}
$$
Using the assumption that  $k \leq \frac{n}{100}$  and  $\eta$  is sufficiently small we see that
$$
B (2, \mathcal {T} _ {\text {e n d}}) \leq \frac {m \mathcal {T} _ {1} (n - k)}{9}.
$$
$$
y f \left(\mathcal {T} _ {\text {e n d}}, \mathbf {x}\right) \geq \eta m \mathcal {T} _ {1} (n - k) ((\gamma - \rho) - (\gamma + \rho) / 9) + O (\eta),
$$
then  $yf(\mathcal{T}_{\mathrm{end}}, \mathbf{x})$  is positive provided  $(\gamma - \rho) - (\gamma + \rho) / 9$  is positive and  $\eta$  is sufficiently small. Both these conditions are guaranteed by Assumption 7 and thus the test point is correctly classified.
E.5 Proof of Theorem 3.8
Theorem E.7 (Theorem 3.8). Let Assumption 6 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 we have the following.
1. There exists a positive constant  $C$  such that the training process terminates at an iteration  $\mathcal{T}_{\text{end}} \leq \frac{Cn}{\eta}$ .
2. For all  $i\in S_T$  then  $\ell (\mathcal{T}_{end},\mathbf{x}_i) = 0$  while  $\ell (\mathcal{T}_{end},\mathbf{x}_i) = 1$  for all  $i\in S_F$
3. There exists a positive constant  $c$  such that the generalization error satisfies
Proof. Under Assumption 7, Statements 1 and 2 follow from Lemma E.5. The bound on  $\mathcal{T}_{\mathrm{end}}$  follows from Lemma E.4 applied at  $t_0 = 2$ , indeed the number of iterations cannot exceed the number of updates which, as  $\gamma > \frac{3}{n}$ , is bounded as
$$
B (2, \mathcal {T} _ {\text {e n d}}) + G (2, \mathcal {T} _ {\text {e n d}}) \leq k \left(\frac {4}{\eta m}\right) + \frac {2}{\gamma} \left(\frac {3}{2 \eta}\right) + O (\eta)
$$
Again under Assumption 7 using Lemma E.6 then Statement 3 follows in exactly the same manner as the proof of Statement 3 for Theorem C.14. Finally, Lemma E.1 implies that under Assumption 6 then Assumption 7 holds with probability at least  $1 - \delta$ .
Appendix F Numerical simulations
Reproducibility statement: the code used to generate the following figures can be found at https://github.com/wswartworth/benign_overfitting.
To investigate our theory we train two-layer neural networks with ReLU activations using full-batch gradient descent and a fixed step size. We train on a synthetic binary classification dataset generated as per Section 2.1. Finally, we train using both the hinge and logistic loss.
Figure 1: from left to right, the first row shows the clean, corrupt, and test losses as a function of epoch (or iteration). The second row shows the fraction of clean, corrupt, and test points that are classified correctly. These plots were generated with  $n = 100$ ,  $d = 800$ ,  $k/n = 0.1$ ,  $m = 100$ ,  $\gamma = 0.015$ , and a step size of  $\eta = 0.01$ .
In Figure 1 we call attention to the difference in the training dynamics of hinge loss versus logistic loss. Perhaps the key difference between the hinge loss and logistic loss is that the contributions from any given point do not get smaller as the point approaches 0 loss. Furthermore, unlike with the logistic loss, points can actually attain zero hinge loss after a finite number of epochs. While a point has zero loss it ceases to contribute to the update of the network parameters. As a result, points close to zero hinge loss periodically activate and deactivate giving rise to the chaotic behavior observed as the training loss approaches zero. We emphasize that managing this behavior required a careful analysis distinct from that of prior works analysing the logistic loss.
In Figure 2 we call particular attention to the bottom right plot. Our theory predicts a phase transition between benign overfitting and non-benign overfitting when  $\gamma \approx c / n$ : the phase transition we observe empirically in the bottom-right heatmap suggests this estimate is reasonable. With regard to the hinge loss over the corrupt points, displayed in the top-right heatmap, we observe another phase transition, this time between overfitting and non-overfitting. The top and bottom heatmaps of the left-hand column display the hinge loss over the clean training set and total training set respectively, these appear very similar due to the fact that clean points make up  $95\%$  of the training set. The clean points fail to achieve zero, or close to zero, hinge loss only when  $\gamma$  is small and  $n$  is large. As stated in the caption, in these experiments  $d$  is fixed and thus as  $n$  increases the near-orthogonality condition we require on the noise components in order to prove convergence to zero clean loss is compromised. As a result, when  $\gamma$  is small and the correlations between noise vectors is potentially large it is possible for pairs of points with opposite labels to be significantly correlated.
Figure 2: from left to right in the top row we show the loss on clean training and corrupt training points after training. In the bottom row and again from left to right we show the total loss after training and the test loss on 10000 randomly generated points. For each plot we set  $d = 1000$ ,  $m = 30$ ,  $\eta = 0.005$  and train for 5000 iterations of gradient descent using hinge loss. In each plot we vary  $\gamma$  and  $n$  and hold the fraction of corrupt points constant at 0.05. In the bottom right plot we also graph the curve  $c / n$  for  $c \approx 0.6$