SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbf {g} _ {t} = \frac {t - 1}{t + 1} \mathbf {g} _ {t - 1} + \frac {2}{t + 1} \frac {\nabla L (c \widetilde {\mathbf {w}} _ {t - 1} + c t \mathbf {w} _ {t - 1})}{L (c \widetilde {\mathbf {w}} _ {t - 1} + c t \mathbf {w} _ {t - 1})} = \frac {t - 1}{t + 1} \mathbf {g} _ {t - 1} + \frac {2}{t + 1} \frac {\nabla L (c \widetilde {\mathbf {w}} _ {t - 1} + c t \| \mathbf {g} _ {t - 1} \| _ {*} \mathbf {s} _ {t - 1})}{L (c \widetilde {\mathbf {w}} _ {t - 1} + c t \| \mathbf {g} _ {t - 1} \| _ {*} \mathbf {s} _ {t - 1})},
$$
$$
\mathbf{s}_{t} = \operatorname *{-argmax}_{\| \mathbf{s}\| \leq 1} - \mathbf{s}^{\top}\mathbf{g}_{t} = \operatorname *{argmin}_{\| \mathbf{s}\| \leq 1}\mathbf{s}^{\top}\mathbf{g}_{t},
$$
$$
\widetilde {\mathbf {w}} _ {t} = \widetilde {\mathbf {w}} _ {t - 1} + t \| \mathbf {g} _ {t} \| _ {*} \mathbf {s} _ {t},
$$
and let  $\beta_{t,3} = \frac{t - 1}{t + 1}$ ,  $\beta_{t,4} = \frac{\lambda}{4}$ ,  $\beta_{t,4}^{\prime} = \frac{\lambda t\|\mathbf{g}_{t - 1}\|_{*}}{4}$ ,  $\beta_{t,3}^{\prime} = \frac{2}{(t + 1)L(\beta_{t,4}\mathbf{v}_{t - 1} + \beta_{t,4}^{\prime}\mathbf{s}_{t - 1})}$ ,  $\eta_t = t\| \mathbf{g}_t\|_*$ . Finally, we focus on the regret bound. For the w-player, note that  $\Phi (\mathbf{w})$  is  $\lambda$ -strongly convex with respect to  $\| \cdot \|$ . Thus, based on Lemma 3, we have
$$
\sum_ {t = 1} ^ {T} \alpha_ {t} h _ {t} (\mathbf {w} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} h _ {t} (\mathbf {w}) \leq - \sum_ {t = 1} ^ {T} \frac {\lambda (t - 1)}{4} \| \mathbf {w} _ {t} - \mathbf {w} _ {t - 1} \| ^ {2}. \tag {52}
$$
On the other hand, note that  $c = \frac{\lambda}{4}$ , so based on Lemma 8, we have
$$
\begin{array}{l} \sum_ {t = 1} ^ {T} \alpha_ {t} \ell_ {t} (\mathbf {p} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} \ell_ {t} (\mathbf {p}) \leq \frac {4 \log n}{\lambda} + \frac {\lambda}{8} \sum_ {t = 1} ^ {T} t ^ {2} \| \mathbf {A w} _ {t} - \mathbf {A w} _ {t - 1} \| _ {\infty} ^ {2} \\ = \frac {4 \log n}{\lambda} + \frac {\lambda}{8} \sum_ {t = 1} ^ {T} t ^ {2} \left(\max  _ {i \in [ n ]} | y ^ {(i)} \mathbf {x} ^ {(i) \top} (\mathbf {w} _ {t} - \mathbf {w} _ {t - 1}) |\right) ^ {2} \tag {53} \\ \leq \frac {4 \log n}{\lambda} + \frac {\lambda}{8} \sum_ {t = 1} ^ {T} t ^ {2} \| \mathbf {w} _ {t} - \mathbf {w} _ {t - 1} \| ^ {2}. \\ \end{array}
$$
It is easy to verify that  $\frac{t^2}{8} \leq \frac{t(t - 1)}{4}$  for  $t \geq 2$ . So to summarize we get
$$
C _ {T} = \frac {8 \log n}{\lambda T ^ {2}}.
$$
The proof can be finished by plugging in Theorem 1.
F Regret Bounds for OCO Algorithms
In this section, we provide standard regret bounds for Follow-The-Leader $^{+}$  (FTL $^{+}$ ), Optimistic Follow-The-Leader (OptimisticFTL), Follow-The-Regularized-Leader (FTRL), Follow-The-Regularized-Leader $^{+}$ (FTRL $^{+}$ ), and Optimistic Follow-The-Regularized-Leader (Optimistic FTRL). Lemmas 3, 4, 5 and 6 are based on Lemma of 3 of Wang et al. (2021b), and Lemmas 7 and 8 comes from Theorems 6.8 and 7.35 of Orabona (2019).
Lemma 3. Consider a weighted online learning problem with a series of  $\lambda$ -strongly functions  $f_{1},\ldots ,f_{T}$ , and a series of corresponding parameters  $\alpha_{1},\ldots ,\alpha_{T}$ . The  $FTL^{+}$ algorithm, given by
$$
\mathbf{z}_{t} = \operatorname *{argmin}_{\mathbf{z}\in \mathbb{R}^{d}}\sum_{i = 1}^{t}\alpha_{i}f_{i}(\mathbf{z}),
$$
achieves the following regret bound:
$$
\forall \mathbf {z} \in \mathbb {R} ^ {d}, \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \leq - \sum_ {t = 1} ^ {T} \left(\frac {\lambda \sum_ {s = 1} ^ {t - 1} \alpha_ {s}}{2}\right) \| \mathbf {z} _ {t} - \mathbf {z} _ {t - 1} \| ^ {2}.
$$
Lemma 4. Consider a weighted online learning problem with a series of  $\lambda$ -strongly functions  $f_{1},\ldots ,f_{T}$ , and a series of corresponding parameters  $\alpha_{1},\ldots ,\alpha_{T}$ . Let  $\widehat{\mathbf{z}}_t = \mathrm{argmin}_{\mathbf{z}\in \mathbb{R}^d}\sum_{i = 1}^{t - 1}\alpha_ih_i(\mathbf{z})$ . Then, the Optimistic FTL algorithm, given by
$$
\mathbf{z}_{t} = \operatorname *{argmin}_{\mathbf{z}\in \mathbb{R}^{d}}\sum_{i = 1}^{t - 1}\alpha_{i}f_{i}(\mathbf{z}) + \alpha_{t}f_{t - 1}(\mathbf{z}),
$$
achieves the following regret  $\forall \mathbf{z}\in \mathbb{R}^d$  ..
$$
\begin{array}{l} \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \\ \leq \sum_ {t = 1} ^ {T} \alpha_ {t} \left(f _ {t} (\mathbf {z} _ {t}) - f _ {t} \left(\widehat {\mathbf {z}} _ {t + 1}\right) - f _ {t - 1} (\mathbf {z} _ {t}) + f _ {t - 1} \left(\widehat {\mathbf {z}} _ {t + 1}\right)\right) - \sum_ {t = 1} ^ {T} \frac {\lambda \left(\sum_ {i = 1} ^ {t} \alpha_ {i}\right)}{2} \| \mathbf {z} _ {t} - \widehat {\mathbf {z}} _ {t + 1} \| ^ {2}. \\ \end{array}
$$
Lemma 5. Consider a weighted online learning problem with a series of convex functions  $f_{1},\ldots ,f_{T}$  and a series of corresponding parameters  $\alpha_{1},\ldots ,\alpha_{T}$ . Let  $R(\mathbf{z})$  be a 1-strongly convex regularizer wrt  $\| \cdot \|$ . The FTRL algorithm, given by
$$
\mathbf{z}_{t} = \operatorname *{argmin}_{\mathbf{z}\in \mathbb{R}^{d}}\sum_{i = 1}^{t - 1}\alpha_{i}f_{i}(\mathbf{z}) + R(\mathbf{z}),
$$
$$
\forall \mathbf {z} \in \mathbb {R} ^ {d}, \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \leq R (\mathbf {z}) + 2 \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \| \nabla f _ {t} (\mathbf {z} _ {t}) \| _ {*} ^ {2}.
$$
Lemma 6. Consider a weighted online learning problem with a series of convex functions  $f_{1},\ldots ,f_{T}$  and a series of corresponding parameters  $\alpha_{1},\dots,\alpha_{T}$ . Let  $R(\mathbf{z})$  be a 1-strongly convex regularizer wrt  $\| \cdot \|$ . The FTRL+ algorithm, given by
$$
\mathbf{z}_{t} = \operatorname *{argmin}_{\mathbf{z}\in \mathbb{R}^{d}}\eta \sum_{i = 1}^{t}\alpha_{i}f_{i}(\mathbf{z}) + R(\mathbf{z}),
$$
$$
\forall \mathbf {z} \in \mathcal {Z}, \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} \left(\mathbf {z} _ {t}\right) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \leq \frac {R (\mathbf {z})}{\eta} - \sum_ {t = 1} ^ {T} \frac {1}{2 \eta} \| \mathbf {z} _ {t} - \mathbf {z} _ {t - 1} \| ^ {2}. \tag {54}
$$
Lemma 7. Consider a weighted online learning problem with a series of convex functions  $f_{1},\ldots ,f_{T}$  and a series of corresponding parameters  $\alpha_{1},\ldots ,\alpha_{T}$ . Let  $\Phi (\mathbf{z})$  be a  $\beta$ -strongly convex regularizer wrt  $\| \cdot \|$ . Let  $\mathbf{z}_0\in \mathbb{R}^d$  be the initial point. The OMD algorithm, given by
$$
\nabla \Phi (\mathbf {z} _ {t}) = \nabla \Phi (\mathbf {z} _ {t - 1}) - \eta \alpha_ {t - 1} \nabla f _ {t - 1} (\mathbf {z} _ {t - 1}),
$$
$$
\forall \mathbf {z} \in \mathbb {R} ^ {d}, \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \leq \frac {D _ {\Phi} (\mathbf {z} ; \mathbf {z} _ {0})}{\eta} + \frac {1}{\beta} \sum_ {t = 1} ^ {T} \eta \alpha_ {t} ^ {2} \| \nabla f _ {t} (\mathbf {z} _ {t}) \| _ {*} ^ {2}.
$$
Lemma 8. Consider a weighted online learning problem with a series of convex functions  $f_{1},\ldots ,f_{T}$  and a series of corresponding parameters  $\alpha_{1},\dots,\alpha_{T}$ . Let  $R(\mathbf{z})$  be a 1-strongly convex regularizer wrt  $\| \cdot \|$ , and  $\psi_t(\mathbf{z})$  be the optimism term in round  $t$ . The Optimistic FTRL algorithm, given by
$$
\mathbf {z} _ {t} = \underset {\mathbf {z} \in \mathbb {R} ^ {d}} {\operatorname {a r g m i n}} \sum_ {i = 1} ^ {t - 1} \alpha_ {i} f _ {i} (\mathbf {z}) + \alpha_ {t} \psi_ {t} (\mathbf {z}) + \frac {1}{\eta} R (\mathbf {z}),
$$
$$
\forall \mathbf {z} \in \mathcal {Z}, \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z} _ {t}) - \sum_ {t = 1} ^ {T} \alpha_ {t} f _ {t} (\mathbf {z}) \leq \frac {R (\mathbf {z})}{\eta} + \sum_ {t = 1} ^ {T} \left[ \frac {\| \alpha_ {t} \nabla f _ {t} (\mathbf {z} _ {t}) - \alpha_ {t} \nabla \psi_ {t} (\mathbf {z} _ {t}) \| _ {*} ^ {2}}{2 / \eta}. \right] \tag {55}
$$