SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\frac {\eta_ {T}}{\alpha_ {T}} \left(f \left(y _ {T + 1}\right) - f \left(x ^ {*}\right)\right) + \mathbf {D} _ {\psi} \left(x ^ {*}, x _ {T + 1}\right) \leq 2 R _ {1} ^ {2}
$$
$$
\begin{array}{l} f \left(y _ {T + 1}\right) - f \left(x ^ {*}\right) \leq \frac {2 R _ {1} ^ {2} \alpha_ {T}}{\eta_ {T}} = 6 R _ {1} ^ {2} c \gamma^ {2} L \alpha_ {T} ^ {2} \\ = 6 \max  \left\{1 0 ^ {4} L \gamma^ {2} R _ {1} ^ {2} (T + 1) ^ {- 2}; 6 R _ {1} (T + 1) ^ {- 1} (2 6 T) ^ {\frac {1}{p}} \gamma^ {\frac {p - 1}{p}} \sigma \right\}. \\ \end{array}
$$
2. Following the similar steps to the proof of Theorem D.1, and noticing that  $(c_{t})$  is a increasing sequence, we obtain the convergence rate.