SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| \boldsymbol {\theta} _ {k} - \boldsymbol {\theta} _ {\tau} \right\| + \lambda_ {k} - \lambda_ {\tau} \leq \sum_ {t = \tau} ^ {k - 1} \left\| \boldsymbol {\theta} _ {t + 1} - \boldsymbol {\theta} _ {t} \right\| + \lambda_ {t + 1} - \lambda_ {t} \\ \leq \sum_ {t = \tau} ^ {k - 1} \left(L _ {h} ^ {\prime} \alpha_ {t} + L _ {g} ^ {\prime} \eta_ {t}\right) \left(\left\| \boldsymbol {\theta} _ {t} \right\| + \lambda_ {t} + 1\right) \\ \leq \left(\left\| \boldsymbol {\theta} _ {\tau} \right\| + \lambda_ {\tau} + 1\right) \sum_ {t = \tau} ^ {k - 1} \left(L _ {h} ^ {\prime} \alpha_ {t} + L _ {g} ^ {\prime} \eta_ {t}\right) \prod_ {i = 0} ^ {t - \tau} \left(L _ {h} ^ {\prime} \alpha_ {\tau + i} + L _ {g} ^ {\prime} \eta_ {\tau + i} + 1\right). \tag {52} \\ \end{array}
$$
Therefore, the following inequality holds
$$
\begin{array}{l} \left\| \boldsymbol {\theta} _ {k} - \boldsymbol {\theta} _ {0} \right\| + \left| \lambda_ {k} - \lambda_ {0} \right| \\ \leq \left\| \boldsymbol {\theta} _ {k} - \boldsymbol {\theta} _ {\tau} \right\| + \left| \lambda_ {k} - \lambda_ {\tau} \right| + \left\| \boldsymbol {\theta} _ {\tau} - \boldsymbol {\theta} _ {0} \right\| + \left| \lambda_ {\tau} - \lambda_ {0} \right| \\ \leq 2 \left(L _ {h} ^ {\prime} \alpha_ {0} + L _ {q} ^ {\prime} \eta_ {0}\right) \tau \left(\left\| \boldsymbol {\theta} _ {0} \right\| + \left| \lambda_ {0} \right| + 1\right) \\ + \left(\| \boldsymbol {\theta} _ {\tau} \| + \lambda_ {\tau} + 1\right) \sum_ {t = \tau} ^ {k - 1} \left(L _ {h} ^ {\prime} \alpha_ {t} + L _ {g} ^ {\prime} \eta_ {t}\right) \prod_ {i = 0} ^ {t - \tau} \left(L _ {h} ^ {\prime} \alpha_ {\tau + i} + L _ {g} ^ {\prime} \eta_ {\tau + i} + 1\right) \\ \leq 2 \left(L _ {h} ^ {\prime} \alpha_ {0} + L _ {g} ^ {\prime} \eta_ {0}\right) \tau \left(\left\| \boldsymbol {\theta} _ {0} \right\| + \left| \lambda_ {0} \right| + 1\right) \\ \end{array}
$$
$$
\begin{array}{l} + \left(2 \left(L _ {h} ^ {\prime} \alpha_ {0} + L _ {g} ^ {\prime} \eta_ {0}\right) \tau + 1\right) \sum_ {t = \tau} ^ {k - 1} \left(L _ {h} ^ {\prime} \alpha_ {t} + L _ {g} ^ {\prime} \eta_ {t}\right) \prod_ {i = 0} ^ {t - \tau} \left(L _ {h} ^ {\prime} \alpha_ {\tau + i} + L _ {g} ^ {\prime} \eta_ {\tau + i} + 1\right) \left(\left\| \boldsymbol {\theta} _ {0} \right\| + \left| \lambda_ {0} \right| + 1\right) \\ \leq \left(\frac {1}{2} + \frac {3}{2} \sum_ {t = \tau} ^ {k - 1} \left(L _ {h} ^ {\prime} \alpha_ {t} + L _ {g} ^ {\prime} \eta_ {t}\right) \prod_ {i = 0} ^ {t - \tau} \left(L _ {h} ^ {\prime} \alpha_ {\tau + i} + L _ {g} ^ {\prime} \eta_ {\tau + i} + 1\right)\right) \left(\| \boldsymbol {\theta} _ {0} \| + | \lambda_ {0} | + 1\right), \\ \end{array}
$$
with the last equality holds when  $(L_h^\prime \alpha_0 + L_g^\prime \eta_0)\tau \leq 1 / 4$ . When  $\sum_{t = \tau}^{k - 1}(L_h^\prime \alpha_t + L_g^\prime \eta_t)\prod_{i = 0}^{t - \tau}(L_h^\prime \alpha_{\tau +i} + L_g^\prime \eta_{\tau +i} + 1)$  is non-increasing with  $k$ , then we can set  $c_{1}$  as  $c_{1} := \frac{1}{2} +\frac{3}{2}.(L_h^\prime \alpha_\tau +L_g^\prime \eta_\tau)(L_h^\prime \alpha_\tau +L_g^\prime \eta_\tau +1)$ . This completes the proof.
Provided Lemma 9, we have the following lemma related with local linearization of  $\mathbf{Q}$  functions and the original  $\mathbf{Q}$  functions.
Lemma 10 (Lemma 5.2 in [13]). There exists a constant  $c_{1}$  such that
$$
\mathbb {E} \left[ \| h (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) - h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) \| ^ {2} | \mathcal {F} _ {k - \tau} \right] \leq \mathcal {O} \left(c _ {1} ^ {3} (\| \boldsymbol {\theta} _ {0} \| + | \lambda_ {0} | + 1) ^ {3} \cdot m ^ {- 1 / 2}\right).
$$
Lemma 10 indicates that if the updated parameter is always bounded in a ball with the initialized one as the center and a fixed radius, the local linearized function  $f_{0}(\cdot)$  in (18) and the original neural network approximated function  $f(\cdot)$  in (6) have bounded gap, which tends to be zero as the width of hidden layer  $m$  grow large. For interested readers, please refer to [13] for detailed proofs of this lemma.