SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} E \left[ \exp \left\{\lambda \left(\sum_ {i = 1} ^ {n} X _ {i}\right) \right\} \right] \\ = E \left[ \Pi_ {i = 1} ^ {n} \exp \left\{\lambda X _ {i} \right\} \right] \\ = \int_ {0} ^ {\infty} \Pi_ {i = 1} ^ {n} \exp \{\lambda X _ {i} \} d \mu \\ \leq \Pi_ {i = 1} ^ {n} | | \exp \{\lambda X _ {i} \} | | _ {\frac {\sum_ {k = 1} ^ {n} \sigma_ {k}}{\sigma_ {i}}} \\ = \Pi_ {i = 1} ^ {n} \left(\int_ {0} ^ {\infty} \exp \left\{\lambda X _ {i} \right\} ^ {\frac {\sum_ {k = 1} ^ {n} \sigma_ {k}}{\sigma_ {i}}} d \mu\right) ^ {\frac {\sigma_ {i}}{\sum_ {k = 1} ^ {n} \sigma_ {k}}} \\ = \Pi_ {i = 1} ^ {n} \left(E _ {Y _ {i}} \left[ \exp \left\{\lambda X _ {i} \frac {\sum_ {k = 1} ^ {n} \sigma_ {k}}{\sigma_ {i}} \right\} \right] ^ {\frac {\sigma_ {i}}{\sum_ {k = 1} ^ {n} \sigma_ {k}}} \right. \\ \leq \Pi_ {i = 1} ^ {n} [ \exp \{\frac {1}{2} \sigma_ {i} ^ {2} \lambda^ {2} \frac {\left(\sum_ {k = 1} ^ {n} \sigma_ {k}\right) ^ {2}}{\sigma_ {i} ^ {2}} \} ] ^ {\frac {\sigma_ {i}}{\sum_ {k = 1} ^ {n} \sigma_ {k}}} \\ = \left[ \exp \left\{\frac {1}{2} \lambda^ {2} \left(\sum_ {k = 1} ^ {n} \sigma_ {k}\right) ^ {2} \right\} \right] ^ {\sum_ {i = 1} ^ {n} \frac {\sigma_ {i}}{\sum_ {k = 1} ^ {n} \sigma_ {k}}} \\ = \exp \left\{\frac {1}{2} \lambda^ {2} \left(\sum_ {k = 1} ^ {n} \sigma_ {k}\right) ^ {2} \right\} \\ \end{array}
$$
where the first inequality is by Lemma 3 and the second inequality follows the definition of sub-Gaussian random variables.
Lemma 5. Suppose that random variables  $X_{1}, X_{2}, \ldots, X_{n}$  are independent sub-Gaussian distributed with variance proxy  $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{n}$ , respectively. Then we have that the sum of these sub-Gaussian random variables,  $\sum_{i=1}^{n} X_{i}$ , is again sub-Gaussian with variance proxy  $\sum_{i=1}^{n} \sigma_{i}^{2}$ .
Proof. For any  $\lambda > 0$  note that
$$
\begin{array}{l} E \left[ \exp \left\{\lambda \left(\sum_ {i = 1} ^ {n} X _ {i}\right) \right\} \right] \\ = E \left[ \Pi_ {i = 1} ^ {n} \exp \left\{\lambda X _ {i} \right\} \right]. \\ \end{array}
$$
Since  $X_{1}, X_{2}, \ldots, X_{n}$  are independent random variables, we further have
$$
\begin{array}{l} E \left[ \exp \left\{\lambda \left(\sum_ {i = 1} ^ {n} X _ {i}\right) \right\} \right] \\ = \Pi_ {i = 1} ^ {n} E [ \exp \lambda X _ {i} ] \\ \leq \Pi_ {i = 1} ^ {n} \exp \left\{\frac {1}{2} \lambda^ {2} \sigma_ {i} ^ {2} \right\} \\ = \exp \left\{\frac {1}{2} \lambda^ {2} \sum_ {i = 1} \sigma_ {i} ^ {2} \right\} \\ \end{array}
$$
where the inequality is by the definition of sub-Gaussian random variables.
Proposition 1. Under  $E$ - $R$ , for any  $1 > \delta, \epsilon > 0$ , and any fixed  $t, t \geq L_{s1}$ , the maintained matrix  $P_t$  satisfies
$$
\left| \left| P _ {t} - c E \right| \right| _ {\infty} \leq \delta
$$
with probability  $1 - \frac{\epsilon}{T}$ . This implies that with probability at least  $1 - \epsilon$  for any  $t \geq L_{s_1}$ , we have
$$
\left| \left| P _ {t} - c E \right| \right| _ {\infty} \leq \delta .
$$
Proof. We start with the convergence rate of matrix  $P_{t}$  for fixed  $t$ .
We recall that in E-R, the indicator function  $X_{i,j}^{s}$  for edge  $(i,j)$  at time step  $s$  follows a Bernoulli distribution with mean value  $c$ . This implies that  $\{X_{i,j}^{s}\}_{s}$  are i.i.d. random variables which allows us to use the Chernoff-Hoeffding inequality
$$
P (| \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - c | > \delta) \leq 2 \exp \{- 2 t \delta^ {2} \}.
$$
For the probability term, we note that for any  $t \geq L_{s_1}$ ,
$$
2 \exp \{- 2 t \delta^ {2} \} \leq \frac {\epsilon}{T}
$$
since  $t \geq L_{s_1} \geq \frac{\ln\frac{T}{2c}}{2\delta^2}$  by the choice  $L_{s_1}$  of the burn-in period in setting 1.
As a result, the maintained matrix  $P_{t}$  satisfies with probability at least  $1 - \frac{\epsilon}{T}$  that
$$
\begin{array}{l} \left| \left| P _ {t} - c E \right| \right| _ {\infty} \\ = \max _ {i, j} \left| \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - c \right| \\ \leq \delta \\ \end{array}
$$
which concludes the first part of the statement.
Subsequently, consider the probability  $P(||P_t - cE||_\infty < \delta, \forall t > L_{s_1})$ . We obtain
$$
\begin{array}{l} P \left(\left| \left| P _ {t} - c E \right| \right| _ {\infty} <   \delta , \forall t > L _ {s _ {1}}\right) \\ = 1 - P \left(\cup_ {t \geq L _ {s 1}} | | P _ {t} - c E | | _ {\infty} <   \delta\right) \\ \geq 1 - \sum_ {t \geq L _ {s _ {1}}} P \left(\left| \left| P _ {t} - c E \right| \right| _ {\infty} <   \delta\right) \\ \geq 1 - \left(T - L _ {s _ {1}}\right) \frac {\epsilon}{T} \geq 1 - \epsilon \\ \end{array}
$$
where the first inequality uses the Bonferroni's inequality.
This completes the second part of the statement.
We next pin down the Markov chain governing Algorithm 1. Its states compound to all connected graphs if  $G$  and  $G'$  are connected, then the transition probability is defined by
$$
\pi (G ^ {\prime} | G) = \left\{ \begin{array}{l} 0 \text {i f} | E (G ^ {\prime}) \Delta E (G) | > 1 \\ \frac {2}{M (M - 1)} \text {i f} | E (G ^ {\prime}) \Delta E (G) | = 1 \\ 1 - \frac {2 \alpha (G)}{M (M - 1)} \text {i f} G ^ {\prime} = G. \end{array} \right.
$$
Here  $\Delta$  denotes the symmetric difference and  $\alpha(G)$  is the number of all connected graph that differ with  $G$  by at most one edge. Algorithm 1 is a random walk in the Markov chain denoted as  $CG - MC$ . The intriguing question is if the stationary distribution corresponds to the uniform distribution on all connected graphs on  $M$  nodes and if it is rapidly mixing. The next paragraph gives affirmative answers.
Proposition 2. In  $CG - MC$ , for any time step  $n \geq 1$  and initial connected graph  $G^{init}$ , we have
$$
\left| \left| \pi^ {n} (\cdot | G ^ {\text {i n i t}}) - \pi^ {*} (\cdot) \right| \right| _ {T V} \leq 2 \left(p ^ {*}\right) ^ {n}
$$
where  $p^* = p^*(M) < 1$  and  $\pi^*$  is the uniform distribution on all connected graphs.
Proof. Based on the definition of  $\pi^{*}$ , we have
$$
\pi^ {*} = \frac {1}{\# \{c o n n e c t e d : g r a p h s \}}.
$$
Therefore, there exists a constant  $0 < C_f < 1$  such that for any two connected graphs  $G, G'$  with  $|E(G)\Delta E(G')| = 1$  we have
$$
\pi (G | G ^ {\prime}) \geq C _ {f} \pi^ {*}.
$$
In essence  $C_f = \frac{1}{\pi^*}\min_{G,G'}\pi (G|G') < 1$
If  $G = G'$ , then there are two possible cases. First, if  $\alpha(G) < \frac{(M(M - 1)}{2}$ , then  $\pi(G|G) > \frac{2}{M(M - 1)} > \pi^*$  and  $\pi(G|G) > 0$ . Otherwise, we have  $\pi(G|G) = 0$ . In other words, the set  $G \notin \{G': \pi(G'|G) \leq \pi^*(G'), \pi(G'|G) > 0\}$ .
This implies that for  $G' \in \{G': \pi(G'|G) \leq \pi^*(G'), \pi(G'|G) > 0\}$ , we have  $|E(G)\Delta E(G')| = 1$  and subsequently  $\pi(G|G') \geq C_f\pi^*$ .
We start with the one-step transition and obtain
$$
\begin{array}{l} \left\| \pi (\cdot | G) - \pi^ {*} (\cdot) \right\| _ {T V} \\ = 2 \sup  _ {A} | \int_ {A} (\pi (G ^ {\prime} | G) - \pi^ {*} (G ^ {\prime})) d _ {G ^ {\prime}} | \\ \leq 2 \int_ {\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) - \pi^ {*} \left(G ^ {\prime}\right) \leq 0 \right\}} \left(- \pi \left(G ^ {\prime} \mid G\right) + \pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} \\ \leq 2 \int_ {\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) = 0 \right\}} \left(- \pi \left(G ^ {\prime} \mid G\right) + \pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} + \\ 2 \int_ {\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) > 0, \pi \left(G ^ {\prime} \mid G\right) - \pi^ {*} \left(G ^ {\prime}\right) \leq 0 \right\}} (- \pi \left(G ^ {\prime} \mid G\right) + \pi^ {*} \left(G ^ {\prime}\right)) d _ {G ^ {\prime}} \\ = 2 \int_ {\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) = 0 \right\}} \left(\pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} + 2 \left(1 - C _ {f}\right) \int_ {\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) > 0, \pi \left(G ^ {\prime} \mid G\right) - \pi^ {*} \left(G ^ {\prime}\right) \leq 0 \right\}} \left(\pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} \\ \leq 2 P \left(\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) = 0 \right\}\right) + 2 \left(1 - C _ {f}\right) \left(1 - P \left(\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) = 0 \right\}\right)\right) \\ \leq 2 (1 - \frac {1}{\# \{c o n n e c t e d \quad g r a p h s \}}) + 2 (1 - C _ {f}) (1 - (1 - \frac {1}{\# \{c o n n e c t e d \quad g r a p h s \}})) \\ \dot {=} 2 p ^ {\prime} + 2 \left(1 - C _ {f}\right) \left(1 - p ^ {\prime}\right) = 2 \left(p ^ {\prime} + \left(1 - C _ {f}\right) \left(1 - p ^ {\prime}\right)\right) \\ \stackrel {\cdot} {=} 2 p ^ {*} \\ \end{array}
$$
where we denote the term  $1 - \frac{1}{\# \{ \text{connected graphs} \}}$  and the term  $p' + (1 - C_f)(1 - p')$  by  $p'$  and  $p^*$ , respectively. It is worth noting that  $p^* = p^*(M)$  and  $p^* < 1$  since  $p', C_f < 1$ . Here the third inequality uses the above argument on the graphs in the set  $\{G': \pi(G'|G) \leq \pi^*(G'), \pi(G'|G) > 0\}$ , and the last inequality uses the following result. By definition,
$$
\begin{array}{l} P \left(\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) = 0 \right\}\right) \\ = 1 - P \left(\left\{G ^ {\prime}: \pi \left(G ^ {\prime} \mid G\right) > 0 \right\}\right) \\ \leq 1 - P \left(\left\{G ^ {\prime}: | E (G) \Delta E \left(G ^ {\prime}\right) | = 1 \right\}\right) \\ = 1 - \frac {\alpha (G)}{\# \{c o n n e c t e d g r a p h s \}} \leq 1 - \frac {1}{\# \{c o n n e c t e d g r a p h s \}} \\ \end{array}
$$
where the last inequality uses  $\alpha (G)\geq 1$  by the definition of  $\alpha (G)$
Suppose at time step  $n$ , the result holds, i.e. for any  $G$
$$
\left| \left| \pi^ {n} (\cdot | G) - \pi^ {*} (\cdot) \right| \right| _ {T V} \leq 2 \left(p ^ {*}\right) ^ {n}.
$$
Then we consider the transition kernel at the  $n + 1$  step. Note that
$$
\begin{array}{l} \left| \left| \pi^ {n + 1} (\cdot | G) - \pi^ {*} (\cdot) \right| \right| _ {T V} \\ = 2 \sup  _ {A} \left| \int_ {A} \left(\pi^ {n + 1} \left(G ^ {\prime} \mid G\right) - \pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} \right| \\ \leq 2 \sup  _ {A} \left| \int_ {S} \int_ {A} \left(\pi^ {n} \left(G ^ {\prime} \mid S\right) - \pi^ {*} \left(G ^ {\prime}\right)\right) \left(\pi (S \mid G) - \pi^ {*} (S)\right) d _ {G ^ {\prime}} d _ {S} \right| \\ = 2 \sup  _ {A} | \int_ {S} (\pi (S | G) - \pi^ {*} (S)) (\int_ {A} (\pi^ {n} (G ^ {\prime} | S) - \pi^ {*} (G ^ {\prime})) d _ {G ^ {\prime}}) d _ {S} | \\ \leq 2 \cdot \frac {1}{2} | | \pi^ {n} (\cdot | S) - \pi^ {*} (\cdot) | | _ {T V} \cdot \frac {1}{2} | | \pi (\cdot | G) - \pi^ {*} (\cdot) | | _ {T V} \\ = \frac {1}{2} | | \pi^ {n} (\cdot | S) - \pi^ {*} (\cdot) | | _ {T V} | | \pi (\cdot | G) - \pi^ {*} (\cdot) | | _ {T V} \\ \leq \frac {1}{2} \cdot 2 (p ^ {*}) ^ {n} \cdot 2 p ^ {*} = 2 (p ^ {*}) ^ {n + 1} \\ \end{array}
$$
where the second inequality is by using the definition of  $||\cdot||_{TV}$  and the last inequality holds by the results in the basis step and the induction step, respectively. The first inequality requires more arguments as follows. Consider the integral
$$
\begin{array}{l} \int_ {S} \int_ {A} (\pi^ {n} (G ^ {\prime} | S) - \pi^ {*} (G ^ {\prime})) (\pi (S | G) - \pi^ {*} (S)) d _ {G ^ {\prime}} d _ {S} \\ = \int_ {S} \int_ {A} \left(\pi^ {n} \left(G ^ {\prime} \mid S\right) - \pi^ {*} \left(G ^ {\prime}\right)\right) \pi (S \mid G) d _ {G ^ {\prime}} d _ {S} - \left(\pi^ {n} \left(G ^ {\prime} \mid S\right) - \pi^ {*} \left(G ^ {\prime}\right)\right) \pi^ {*} (S) d _ {G ^ {\prime}} d _ {S} \\ = \int_ {S} \int_ {A} \pi^ {n} \left(G ^ {\prime} \mid S\right) \pi (S \mid G) d _ {G ^ {\prime}} d _ {S} - \int_ {S} \int_ {A} \pi^ {*} \left(G ^ {\prime}\right) \pi^ {n} (S \mid G) d _ {G ^ {\prime}} d _ {S} - \\ \int_ {S} \int_ {A} \pi^ {n} \left(G ^ {\prime} \mid S\right) \pi^ {*} (S) d _ {G ^ {\prime}} d _ {S} + \int_ {S} \int_ {A} \pi^ {*} \left(G ^ {\prime}\right) \pi^ {*} (S) d _ {G ^ {\prime}} d _ {S} \\ \geq \int_ {A} \pi^ {n + 1} \left(G ^ {\prime} | G\right) d _ {G ^ {\prime}} - \int_ {A} \pi^ {*} \left(G ^ {\prime}\right) d _ {G ^ {\prime}} - \\ \int_ {S} \pi^ {*} (S) d _ {S} + \int_ {S} \pi^ {*} (G ^ {\prime}) d _ {G ^ {\prime}} \\ = \int_ {A} \pi^ {n + 1} \left(G ^ {\prime} | G\right) d _ {G ^ {\prime}} - \int_ {A} \pi^ {*} (S) d _ {S} = \int_ {A} \left(\pi^ {n + 1} \left(G ^ {\prime} | G\right) - \pi^ {*} \left(G ^ {\prime}\right)\right) d _ {G ^ {\prime}} \\ \end{array}
$$
where the results hold by exchanging the orders of the integrals as a result of Funibi's Theorem and the inequality uses the fact that  $\int_{A}\pi^{n}(G^{\prime}|S)d_{G^{\prime}}\leq 1$
This completes the proof by concluding the mathematical induction.
Proposition 3. For any  $1 > \delta, \epsilon > 0$ , we obtain that for setting 2, for any fixed  $t \geq L_{s_2}$ , the maintained matrix  $P_t$  satisfies with probability  $1 - 2\frac{\epsilon}{T}$
Meanwhile, the graph generated by Algorithm 3 converges to the stationary distribution with
$$
| | \pi^ {t} (\cdot | G) - \pi^ {*} (\cdot) | | _ {T V} \leq \frac {1 \delta}{5},
$$
where  $\pi^{*}$  is the uniform distribution on all connected graphs.
Proof. Suppose we run the rapidly mixing markov chain for a time period of length  $\tau_{1} = \frac{\ln\frac{\zeta}{2}}{\ln p^{*}}$  where  $\zeta < \frac{\delta}{5}$ . By applying Proposition 2, we obtain that for any time step  $t > \tau_{1}$ ,
$$
\left| \left| \pi^ {t} (\cdot | G) - f (\cdot) \right| \right| _ {T V} \leq 2 \left(p ^ {*}\right) ^ {\tau_ {1}}.
$$
For the second phase, we reset the counter of time step and denote the starting point  $t = \tau_1 + 1$  as  $t = 1$ . Based on the definition of  $P_t$ , we have  $P_t = \left(\frac{\sum_{s=1}^t X_{i,j}^s}{t}\right)_{1 \leq i \neq j \leq M}$  where  $X_{i,j}^s$  is the indicator function for edge  $(i,j)$  on Graph  $G_s$  that follows the distribution  $\pi^{\tau_1 + s}(G,\cdot)$ . Let us denote  $Y_{i,j}^s$  the indicator function for edge  $(i,j)$  on Graph  $G_s^{obj}$  following the distribution  $\pi^*(\cdot)$  and  $P_t^{obj} = \left(\frac{\sum_{s=1}^t Y_{i,j}^s}{t}\right)_{1 \leq i \neq j \leq M}$ .
By the Chernoff-Hoeffding inequality and specifying  $\zeta = 2(p^{*})^{\tau_{1}}$ , we derive
$$
P \left(\left| E \left[ Y _ {i, j} ^ {1} \right] - \frac {\sum_ {s = 1} ^ {t} Y _ {i , j} ^ {s}}{t} \right| \geq \zeta\right) \leq 2 \exp \left\{- 2 t \zeta^ {2} \right\}, \tag {3}
$$
$$
| | P _ {t} ^ {o b j} - c E | | _ {\infty} \leq \zeta
$$
holds with probability  $1 - 2\exp \{-2t\zeta^2\}$ .
Consider the difference between  $P_{t}$  and  $P_{t}^{obj}$  and we obtain that
$$
\begin{array}{l} P _ {t} - P _ {t} ^ {o b j} \\ = (\frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s} - \sum_ {s = 1} ^ {t} Y _ {i , j} ^ {s}}{t}) _ {1 \leq i \neq j \leq M} \\ = \left(\frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - E \left[ X _ {i, j} ^ {1} \right] + E \left[ X _ {i, j} ^ {1} - Y _ {i, j} ^ {1} \right] + E \left[ Y _ {i, j} ^ {1} \right] - \frac {\sum_ {s = 1} ^ {t} Y _ {i , j} ^ {s}}{t}\right) _ {1 \leq i \neq j \leq M} \\ \end{array}
$$
where the last term is bounded by (3).
For the second quantity  $E[(X_{i,j}^{s} - Y_{i,j}^{s})]$ , we have that for any  $s, i, j$
$$
\begin{array}{l} E [ (X _ {i, j} ^ {s} - Y _ {i, j} ^ {s}) | G _ {s}, G _ {s} ^ {o b j} ] \\ = 1 _ {G _ {s} \text {c o n t a i n s e d g e} (i, j) \& G _ {s} ^ {o b j}} \text {d o e s n o t c o n t a i n s e d g e} (i, j) - 1 _ {(G _ {s} \text {d o e s n o t c o n t a i n s e d g e} (i, j) \& G _ {s} ^ {o b j})} \text {c o n t a i n s e d g e} (i, j)) \\ \end{array}
$$
and subsequently by the law of total expectation, we further obtain
$$
\begin{array}{l} E [ (X _ {i, j} ^ {s} - Y _ {i, j} ^ {s}) ] \\ = E \left[ E \left[ \left(X _ {i, j} ^ {s} - Y _ {i, j} ^ {s}\right) \mid G _ {s}, G _ {s} ^ {o b j} \right] \right] \\ = E \left[ 1 _ {G _ {s} \text {c o n t a i n s e d g e} (i, j) \& G _ {s} ^ {o b j}} \text {d o e s n o t c o n t a i n s e d g e} (i, j) - 1 _ {\left(G _ {s} \text {d o e s n o t c o n t a i n s e d g e} (i, j) \& G _ {s} ^ {o b j} \text {c o n t a i n s e d g e} (i, j)\right)} \right] \\ = E \left[ 1 _ {G _ {s} \text {c o n t a i n s e d g e} (i, j)} \right] \cdot E \left[ 1 _ {G _ {s} ^ {o b j} \text {d o e s n o t c o n t a i n e d g e} (i, j)} \right] - \\ \end{array}
$$
[ E[1_{(G_s \text{ does not contain edge } (i,j))} \cdot E[1_{G_s^{obj} \text{ contains edge } (i,j)})] \text{ since } G_s \text{ and } G_s^{obj} \text{ are independent.} ]
$$
\begin{array}{l} = \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {\tau_ {1} + s} (S | G) d S \cdot \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S - \\ \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {\tau_ {1} + s} (S | G) d S \cdot \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \\ = \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {\tau_ {1} + s} (S | G) d S \cdot \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S - \\ \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \cdot \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S + \\ \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \cdot \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S - \\ \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {\tau_ {1} + s} (S | G) d S \cdot \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \\ = \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S \left(\int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \left(\pi^ {\tau_ {1} + s} (S | G) - \pi^ {*} (S)\right) d S\right) + \\ \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \left(\int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \left(- \pi^ {\tau_ {1} + s} (S | G) + \pi^ {*} (S)\right) d S\right) \\ \leq \int_ {A} 1 _ {S \text {d o e s n o t c o n t a i n e d g e} (i, j)} \pi^ {*} (S) d S \cdot | | \pi^ {\tau_ {1} + s} (\cdot | G) - \pi^ {*} (\cdot) | | _ {T V} + \\ \int_ {A} 1 _ {S \text {c o n t a i n s e d g e} (i, j)} \pi^ {*} (S) d S \cdot | | \pi^ {\tau_ {1} + s} (\cdot | G) - \pi^ {*} (\cdot) | | _ {T V} \\ = \left\| \pi^ {\tau_ {1} + s} (\cdot | G) - \pi^ {*} (\cdot) \right\| _ {T V} \\ \leq 2 \left(p ^ {*}\right) ^ {\tau_ {1}}. \\ \end{array}
$$
In like manner, we achieve
$$
\begin{array}{l} E [ (- X _ {i, j} ^ {s} + Y _ {i, j} ^ {s}) ] \\ \leq 2 \left(p ^ {*}\right) ^ {\tau_ {1}}. \tag {4} \\ \end{array}
$$
We now proceed to the analysis on the first term. Though  $\{X_{i,j}^{s}\}_{s}$  are neither independent or identically distributed random variables, the difference  $\frac{\sum_{s=1}^{t} X_{i,j}^{s}}{t} - E[X_{i,j}^{1}]$  can be upper bounded by the convergence property of  $\pi^n$ . Note that  $X_{i,j}^{s}$  is only different from  $X_{i,j}^{s+1}$  when edge  $(i,j)$  is sampled at time step  $s$  and the generated graph is accepted.
$$
\begin{array}{l} P \left(G _ {s + 1} \mid G _ {s}, G _ {s - 1}, \dots , G _ {1}\right) \\ = P \left(G _ {s + 1} \mid G _ {s}\right). \\ \end{array}
$$
Meanwhile, we can write  $X_{i,j}^{s+1} = 1_{X_{i,j}^{s+1}=1} = 1_{G_{s+1}\text{contains edge }(i,j)}$  and similarly,  $X_{i,j}^{s} = 1_{X_{i,j}^{s}=1} = 1_{G_{s}\text{contains edge }(i,j)}$ . Denote event  $E$  as  $E = \{\text{connected graph } G \text{ contains edge }(i,j)\}$ . This gives us
$$
\begin{array}{l} X _ {i, j} ^ {s + 1} = 1 _ {G _ {s + 1} \in E}, \\ X _ {i, j} ^ {s} = 1 _ {G _ {s} \in E}. \\ \end{array}
$$
Furthermore, the quantity  $\int_{A}1_{S\in E}\pi^{*}(S)dS$  can be simplified as
$$
\begin{array}{l} E [ Y _ {i, j} ^ {1} ] = E [ 1 _ {G _ {s} ^ {o b j}} \in E ] \\ = \int_ {A} 1 _ {S \in E} \pi^ {*} (S) d S \\ \end{array}
$$
since  $G_{s}^{obj}$  follows a distribution with density  $\pi^{*}(\cdot)$ .
A new Hoeffding lemma for markov chains has been recently shown as follows in [Fan et al., 2021]. Let  $a(\lambda) = \frac{1 + \lambda}{1 - \lambda}$  where  $\lambda$  is the spectrum of the Markov chain  $CG - MC$  and by the Theorem 2.1 in [Fan et al., 2021], we obtain that
$$
P (| \sum_ {s = 1} ^ {t} 1 _ {G _ {s} \in E} - t E [ Y _ {i, j} ^ {1} ] | > t \zeta) \leq 2 \exp \{- 2 a (\lambda) ^ {- 1} t \zeta^ {2} \}
$$
$$
i. e. P \left(\left| \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - E \left[ Y _ {i, j} ^ {1} \right] \right| > \zeta\right) \leq 2 \exp \{- 2 a (\lambda) ^ {- 1} t \zeta^ {2} \} \tag {5}
$$
since  $X_{i,j}^{s} = 1_{G_{s} \in E}$  satisfies  $0 \leq 1_{G_{s} \in E} \leq 1$ , i.e. the values are within the range of [0, 1].
By the result  $E[X_{i,j}^{1}] - \zeta \leq E[Y_{i,j}^{1}] \leq E[X_{i,j}^{1}] + \zeta$ , we obtain
$$
P \left(\left| \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - E \left[ X _ {i, j} ^ {1} \right] \right| > 2 \zeta\right) \leq 2 \exp \{- 2 a (\lambda) ^ {- 1} t \zeta^ {2} \}. \tag {6}
$$
Putting the results (3), (4) and (6) together, we derive
$$
\begin{array}{l} \left\| P _ {t} - P _ {t} ^ {o b j} \right\| _ {\infty} \\ = \max  _ {i, j} | \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - E [ X _ {i, j} ^ {1} ] + E [ X _ {i, j} ^ {1} - Y _ {i, j} ^ {1} ] + E [ Y _ {i, j} ^ {1} ] - \frac {\sum_ {s = 1} ^ {t} Y _ {i , j} ^ {s}}{t} | \\ \leq \max  _ {i, j} \left( \right.\left| \frac {\sum_ {s = 1} ^ {t} X _ {i , j} ^ {s}}{t} - E \left[ X _ {i, j} ^ {1} \right]\right| + \left| E \left[ X _ {i, j} ^ {1} - Y _ {i, j} ^ {1} \right]\right| + \left| E \left[ Y _ {i, j} ^ {1} \right] - \frac {\sum_ {s = 1} ^ {t} Y _ {i , j} ^ {s}}{t} \right| \\ \leq 2 \zeta + \zeta + \zeta = 4 \zeta \\ \end{array}
$$
which holds with probability at least  $1 - 2\exp \{-2a(\lambda)^{-1}t\zeta^2\} -2\exp \{-2t\zeta^2\}$
For the probability term  $1 - 2\exp \{-2a(\lambda)^{-1}t\zeta^2\} -2\exp \{-2t\zeta^2\}$  , we have
$$
2 \exp \{- 2 a (\lambda) ^ {- 1} t \zeta^ {2} \} \leq \frac {\epsilon}{T},
$$
$$
2 \exp \{- 2 t \zeta^ {2} \} \leq \frac {\epsilon}{T}
$$
which holds by
$$
t \geq L _ {s _ {2}} - \tau_ {1} = a (\lambda) \frac {\ln \frac {T}{2 \epsilon}}{2 \zeta^ {2}}.
$$
Therefore, the distance between the empirical matrix and the constant matrix reads as with probability at least  $1 - 2\frac{\epsilon}{T}$ ,
$$
\begin{array}{l} \left\| P _ {t} - c E \right\| _ {\infty} \\ \leq \left| \left| P _ {t} - P _ {t} ^ {o b j} \right| \right| _ {\infty} + \left| \left| P _ {t} ^ {o b j} - c E \right| \right| _ {\infty} \\ \leq 4 \zeta + \zeta = 5 \zeta <   \delta \\ \end{array}
$$
where  $\zeta = 2(p^{*})^{\tau_{1}} < \frac{1}{5}\delta$  by the choice of parameter  $\delta$  and  $\zeta$ . This completes the proof of Proposition 3.
Next, we proceed to explicitly characterize the spectrum of  $CG - MC$  which plays a role in the length of burning period  $L_{s_2}$  and  $L_{s_3}$ .
Proposition 4. In setting 2, the spectral gap  $1 - \lambda$  of  $CG - MC$  satisfies that for  $\theta > 1$ ,
$$
1 - \lambda \geq \frac {1}{2 \frac {\ln \theta}{\ln 2 p ^ {*}} \ln 2 \theta + 1}.
$$
Proof. It is worth noting that for  $G \neq G'$  and  $|E(G)\Delta E(G')| = 1$ , we have
$$
\pi^ {*} (G) \pi (G ^ {\prime} | G) = \pi^ {*} (G ^ {\prime}) \pi (G | G ^ {\prime})
$$
by the fact that  $\pi^{*}(G) = \pi^{*}(G^{\prime})$  and  $\pi (G^{\prime}|G) = \pi (G|G^{\prime}) = \frac{2}{M(M - 1)}$ .
For  $G \neq G'$  and  $|E(G)\Delta E(G')| > 1$  or  $|E(G)\Delta E(G')| = 0$ , we have  $\pi^{*}(G)\pi (G'|G) = \pi^{*}(G')\pi (G|G')$  since  $\pi (G'|G) = \pi (G|G') = 0$ .
For  $G = G'$ , we have  $\pi^{*}(G)\pi (G^{\prime}|G) = \pi^{*}(G^{\prime})\pi (G|G^{\prime})$  by the expression.
As a result,  $CG - MC$  is reversible. Meanwhile, it is ergodic since it has a stationary distribution  $\pi^{*}$  as stated in Proposition 2.
Henceforth, by the result of Theorem 1 in [McNew, 2011] that holds for any ergodic and reversible Markov chain, we have
$$
\frac {1}{2 \ln 2 e} \frac {\lambda_ {2}}{1 - \lambda_ {2}} \leq \tau (e)
$$
where  $\tau(e)$  is the mixing time for an error tolerance  $e$  and  $\lambda_2$  is the second largest eigenvalue of  $CG - MC$ . Choosing  $e > \frac{1}{2}$  immediately gives us