SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} b _ {2} = \frac {\frac {\mathbb {E} \left[ | | w _ {1} - \hat {w} _ {\theta_ {1}} ^ {*} | | ^ {2} \right]}{T} + C _ {w , 1} + \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} C _ {w , 1} + \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {2} C _ {w , 1} \epsilon^ {2}}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {1}} \\ + \frac {\frac {\sqrt {C _ {w , 1}} \alpha^ {\theta^ {2}} H _ {w} U _ {g} ^ {2}}{\sqrt {2} (1 - \gamma) ^ {2}} + 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {2} \epsilon^ {2} + \frac {4 L _ {w} ^ {2}}{(1 - \gamma) ^ {2}} \alpha^ {\theta^ {2}} U _ {g} ^ {2}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\min}} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\min}} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {1}} \\ = O \left(\frac {1}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right), \\ \end{array}
$$
$$
\begin{array}{l} b _ {3} = \frac {\sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} C _ {w , 1} + c _ {2} \epsilon^ {2} \left(\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}\right)} + \frac {\alpha^ {\theta^ {2}} H _ {w} U _ {g} ^ {2}}{\sqrt {2} (1 - \gamma) ^ {2}} \sqrt {\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\operatorname* {m i n}}} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {1}} \\ = O \left(\frac {1}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right). \\ \end{array}
$$
Now we can apply Small Gain Theorem, where we conclude
$$
y \leq O \left(\frac {1}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right).
$$
$$
x \leq a _ {1} y + a _ {2} = O \left(\frac {1}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right).
$$
G.3 Small Gain Analysis in the Markov Sampling Case
$$
\begin{array}{l} \left(\frac {\alpha_ {\theta}}{2} - \alpha_ {\theta} ^ {2} H _ {V}\right) \Delta_ {V} \leq \frac {\mathbb {E} \left[ V _ {\theta_ {1}} ^ {*} - V _ {\theta_ {T + 1}} ^ {*} \right]}{T} + c _ {1} \left(\alpha_ {\theta} ^ {2} H _ {V} + \frac {\alpha_ {\theta}}{2}\right) \Delta_ {Q} \\ + c _ {2} \left(\alpha_ {\theta} ^ {2} H _ {V} + \frac {\alpha_ {\theta}}{2}\right) \epsilon^ {2} + \frac {2 H _ {V}}{(1 - \gamma) ^ {2}} \alpha^ {\theta 2} U _ {g} ^ {2}, \\ \end{array}
$$
$$
C _ {w, 1} = \tilde {O} \left(\frac {1}{\sqrt {T}} \cdot \frac {1}{\sqrt {m}}\right) + O \left(\frac {1}{\sqrt {T}} \epsilon\right) + O \left(\frac {1}{\sqrt {T}} \epsilon^ {2}\right) + O \left(\frac {1}{T}\right),
$$
$$
C _ {w, 2} = O \left((\epsilon + 1) \frac {(\log T) ^ {2}}{T}\right) + \tilde {O} \left((\epsilon + 1) \left(\frac {1}{\sqrt {m}} + 1\right) \frac {\log T}{T}\right).
$$
$$
a _ {1} = \frac {c _ {1} \left(\alpha_ {\theta} {} ^ {2} H _ {V} + \frac {\alpha_ {\theta}}{2}\right)}{\frac {\alpha_ {\theta}}{2} - \alpha_ {\theta} {} ^ {2} H _ {V}} = O (1),
$$
$$
a _ {2} = \frac {\frac {\mathbb {E} \left[ V _ {\theta_ {1}} ^ {*} - V _ {\theta_ {T + 1}} ^ {*} \right]}{T} + c _ {2} \left(\alpha_ {\theta} ^ {2} H _ {V} + \frac {\alpha_ {\theta}}{2}\right) \epsilon^ {2} + \frac {2 H _ {V}}{(1 - \gamma) ^ {2}} \alpha^ {\theta^ {2}} U _ {g} ^ {2}}{\frac {\alpha_ {\theta}}{2} - \alpha_ {\theta} ^ {2} H _ {V}} = O \left(\frac {1}{\sqrt {T}}\right) + O \left(\epsilon^ {2}\right),
$$
$$
b _ {1} = \frac {\frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} + 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\min }} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\min }} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {1}} = O (1),
$$
$$
\begin{array}{l} b _ {2} = \frac {\frac {\mathbb {E} \left[ | | w _ {1} - \hat {w} _ {\theta_ {1}} ^ {*} | | ^ {2} \right]}{T} + C _ {w , 1} + C _ {w , 2} + \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} C _ {w , 1} + \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {2} C _ {w , 1} \epsilon^ {2}} + \frac {\sqrt {C _ {w , 1}} \alpha^ {\theta^ {2}} H _ {w} U _ {g} ^ {2}}{\sqrt {2} (1 - \gamma) ^ {2}}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\mathrm {m i n}}} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\mathrm {m i n}}} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta^ {2}} L _ {w} ^ {2} c _ {1}} \\ + \frac {2 \alpha^ {w} U _ {f} \left(\frac {\sqrt {2} \alpha^ {\theta} L _ {w} U _ {g}}{1 - \gamma} + \frac {\alpha^ {\theta 2} H _ {w} U _ {g} ^ {2}}{\sqrt {2} (1 - \gamma) ^ {2}}\right) + 2 \alpha^ {\theta 2} L _ {w} ^ {2} c _ {2} \epsilon^ {2} + \frac {4 L _ {w} ^ {2}}{(1 - \gamma) ^ {2}} \alpha^ {\theta 2} U _ {g} ^ {2}}{2 \alpha^ {w} - \frac {1}{\sqrt {2}} \alpha^ {\theta} L _ {w} \left(\frac {1}{\lambda_ {\min}} - 2 \alpha^ {w}\right) - \sqrt {2} \alpha^ {\theta} L _ {w} \sqrt {c _ {1} \left(\frac {1}{\lambda_ {\min}} - 2 \alpha^ {w}\right)} - 2 \alpha^ {\theta 2} L _ {w} ^ {2} c _ {1}} \\ = O \left(\frac {\tau_ {\mathrm {m i x}} ^ {2}}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right), \\ \end{array}
$$
$$
y = O \left(\frac {(\log T) ^ {2}}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right),
$$
$$
x \leq a _ {1} y + a _ {2} = O \left(\frac {(\log T) ^ {2}}{\sqrt {T}}\right) + O (\epsilon) + \tilde {O} \left(\frac {1}{\sqrt {m}}\right).
$$