SlowGuess/NeurIPS_2023_OCR huggingface dataset:
-  $G' := G(V \setminus \{v\}, E \setminus \{(u, \ell) \mid u \in \mathcal{N}(v) \wedge \ell \in \mathcal{N}(u)\})$ ,  $G'$  is created by removing  $v$  from  $G$  as well as all the edges incident to neighbors of  $v$ .
-  $G_0 \coloneqq G'(V \cup \{u'|u \in \mathcal{N}(v)\}, E \cup \{(u', u)|u \in \mathcal{N}(v)\})$ ,  $G_0$  is created from  $G'$  by adding the copy nodes  $u'|_{\forall u \in \mathcal{N}(v)}$  and the edges  $(u', u)$  for all  $u \in \mathcal{N}(v)$  (each vertex  $u$  is connected with its copy  $u'$ ).  $G_0$  represents the situation where vertex  $v$  is not selected in the MVC, but its neighbors are.
-  $G_{1} = G(V \cup \{v'\}, E \setminus \{(u, v) | u \in \mathcal{N}(v)\} \cup \{(v', v)\})$ ,  $G_{1}$  is created from  $G$  by removing all edges incident to  $v$  and adding a copy node  $v'$  that is then connected to  $v$ .  $G_{1}$  represents the situation where  $v$  is selected to be part of the MVC.
2In the trivial case where  $G$  is a graph with only connected components of two vertices  $(d(v)\leq 1$  for all  $v\in V)$ , the size of the minimum vertex cover is  $|G| = |E(G)|$ .
Corollary 1 is based on the fact that for any node  $v \in V$  either  $v$  or all of its neighbors  $\mathcal{N}(v)$  lie in the optimal solution. The first case is captured through  $G_{1}$ . Notice that  $G_{1}$  is constructed by  $G$  by removing all the incident edges to  $v \in V$  since they are already covered by  $v$ . The new copy node  $v'$  is added and connected to  $v$  so as to encode that either node  $v$  or its copy  $v'$  should be selected in the minimum vertex cover of  $G_{1}$ . Symmetrically, the graph  $G_{0}$  is constructed by repeating the latter process for each  $u \in \mathcal{N}(v)$ .
As in Algorithm 1, a graph comparing function  $\mathrm{CMP}_{\mathrm{MVC}}(G,G^{\prime})$  induces the following recursive algorithm for computing a vertex cover. The latter is formally described in Algorithm 3. By Corollary 1 in case  $\mathrm{CMP}_{\mathrm{MVC}}(G,G^{\prime}) = \mathbb{I}[\left|\mathrm{MVC}(G)\right| > \left|\mathrm{MVC}(G^{\prime})\right|]$  then Algorithm 3 always computes a minimum vertex cover.
Algorithm 3 Comparator-Induced Algorithm for the MVC problem
1: function  $\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(G(V,E))$  ▷ Algorithm  $\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(G)$  takes a graph  $G$  as input  
2: if  $\forall v\in V:d(v)\leq 1$  then ▷  $G$  is composed of isolated nodes and isolated edges  
3:  $S\gets \emptyset$   
4: for each edge  $(v,v^{\prime})\in E$  do  
5:  $S\gets S\cup \{v\}$  ▷ Select one of the endpoints of each isolated edge  
6: end for  
7: return  $S$   
8: end if  
9: pick a vertex  $v\in V$  with  $d(v)\geq 1$  uniformly at random.  
10:  $G^{\prime}\coloneqq G(V\setminus \{v\},E\setminus \{(u,\ell)\mid u\in \mathcal{N}(v)\land \ell \in \mathcal{N}(u)\})$   
11:  $G_0\coloneqq G'(V\cup \{u'|u\in \mathcal{N}(v)\} ,E\cup \{(u',u)|u\in \mathcal{N}(v)\}$   
12:  $G_{1} = G(V\cup \{v'\} ,E\setminus \{(u,v)|u\in \mathcal{N}(v)\} \cup \{(v',v)\})$   
13: if  $\mathrm{CMP}_{\mathrm{MVC}}(G_0,G_1) = 0$  then  
14:  $G\gets G_0$  ▷ Remove vertex  $v$  and put it's neighbors in the MVC  
15: else  
16:  $G\gets G_1$  ▷ Put vertex  $v$  in the MVC and remove the edges to its neighbors  
17: end if  
18: return  $\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(G)$   
19: end function
By adjusting the notion of consistency (see Definition 3) in the context of minimum vertex cover as  $\mathrm{CMP}_{\mathrm{MVC}}(G,G^{\prime}) = 0$  if and only if  $\mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(G)\right|\right]\leq \mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(G^{\prime})\right|\right]$ , one can establish that, if  $\mathrm{CMP}_{\mathrm{MVC}}$  is a consistent comparator,  $\mathcal{A}^{\mathrm{CMP}_{\mathrm{MVC}}}(\cdot)$  always outputs a minimum vertex cover. In Appendix E, we present our experimental results produced after the self-training of a consistent in the context of Minimum Vertex Cover.
Additional information related to the hyper-parameters, baselines, and datasets are depicted in this section.
C.1 Training Setup and Baselines
Training setup: The sizes of the linear layers at each iteration are as follows:  $96 \times 32$ ,  $96 \times 32$ , and  $96 \times 32$ . After this module, 4 linear layers follow, with the following sizes:  $96 \times 32$ ,  $32 \times 32$ ,  $32 \times 32$ , and  $64 \times 1$ . Furthermore, the output of the last GEM iteration has a skip connection into the last dense layer of the model, which is also why the final layer has 64 input neurons. Every linear layer in the entire network is followed by layer normalization [Ba et al., 2016b]. Each model is trained for 300 epochs total with batch size 32 using Adam optimizer [Kingma and Ba, 2014] and a learning rate of 0.001. We split the graph datasets into  $80\%$  for training and  $20\%$  for testing, except
3Notice that for the MVC, we have a  $>$  sign in the function instead of a  $<$  sign like in the MIS. This comes from the fact that the recursive algorithm for the MVC in corollary 1 aims to find a minimum-sized set, instead of the maximum like in Theorem 1.
for COLLAB and RB datasets where the number of test graphs is much bigger than the number of train graphs. From the training data, a validation set is constructed by dedicating  $20\%$  of the data in the buffer to it. Considering the baseline methods, we trained them for 300 epochs, and in the case of reinforcement learning, for 300 episodes. A precise description of the Hyper-parameters values is reported in Table 3.
Baselines: Besides the methods specified in Sec. 6, in Appendix D we employ the IMDB dataset [Morris et al., 2020] and graphs coming from the Erdos-Renyi [Erdős et al., 1960], Barabasi-Albert [Albert and Barabási, 2002] and Watts-Strogatz [Watts and Strogatz, 1998] distributions. Moreover, in Appendix E we use two datasets (RB200 and RB500) that resemble the RB dataset employed in Sec. 6, but the two datasets have different numbers of nodes. As mentioned in Sec. 6 and in Appendix E, we use traditional baselines, such as the Greedy MIS employed in Sec. 6 and Greedy MVC employed in Appendix E. Greedy MIS iteratively chooses the node with the minimum degree to be part of the independent set and removes its neighbors from the graph, while Greedy MVC considers the node with the highest degree as part of the vertex cover set, still removing the neighbors of the node. We also implement Simple Local Search, which, given a time limit, randomly adds a node to the independent set and removes conflicting nodes. The largest set in the time period is used as the solution of the algorithm.
Hardware: Any experiment with a reported execution time has been performed on a laptop with Windows 10 and Python 3.10. The device uses a CUDA-enabled NVIDIA GeForce GTX 1060 Max-Q GPU with 6GB VRAM, and an Intel(R) Core(TM) I7-7700HQ CPU with a  $2.80\mathrm{GHz}$  clock speed, and 16GB of RAM. Our method is implemented in PyTorch.
C.2 Datasets
Two important parameters are considered for the analysis of the datasets: the size and the density of the graphs. The first parameter is measured in terms of total number of nodes. The second one reflects how sparse a graph is and it is measured as the ratio between the total number of edges of the graph and the total number of edges if the graph would have been complete.
Real-world datasets and RB: Our work utilizes real-world datasets that simulate social network scenarios, like IMDB, COLLAB and TWITTER. The RB dataset that we employ consists of the same graphs utilized in the research conducted by Karalias and Loukas [2020], whereas the RB200 and RB500 datasets align with those employed in the work by Wang and Li [2023]. Notably, RB200 and RB500 were generated by configuring a small hyper-parameter  $\rho = 0.25$ , making the instances within these datasets challenging, as specified by Wang and Li [2023].
Graph distribution: In Appendix D, we conduct testing on well-known graph distributions, specifically the Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz distributions [Erdős et al., 1960, Albert and Barabási, 2002, Watts and Strogatz, 1998], abbreviated as ER, BA, and WS in Table 4. The objective is to evaluate the performance of our model on datasets where the instance densities differ from those of real-world datasets.
Special dataset: The Special dataset has two nodes,  $u$  and  $v$ , which are connected to a set of nodes  $I$ . The size of set  $I$  is denoted as  $n$ . Moreover, the nodes within  $I$  are all independent of each other, since there are no connections between any pair of nodes  $i$  and  $j$  within  $I$ . Additionally, all nodes in  $I$  are connected to every node in a clique of nodes called  $C$ , which has a total of  $n + a$  nodes. When employing the Greedy MIS heuristic, nodes  $u$ ,  $v$ , and one node from  $C$  are selected as part of the MIS (Maximum Independent Set), while the MIS itself in reality corresponds to  $I$ , which has a dimensionality of  $n$ . Notice that the Greedy MIS heuristic performs poorly in these instances.
Table 3: Experimental setting employed in the training process.
Name	Value
Learning rate	0.001
Optimizer	Adam
Output layer size (D)	32
Number of GEM layers (K)	3
Number of fully connected layers (L)	4
Batch size	32
Training epochs	300
Table 4: Statistics of the employed datasets. For each dataset, it is reported the average number of nodes and density of the graphs, and the total number of graphs employed for both training and testing (denoted as 'Train' and 'Test' respectively).
IMDB	COLLAB	TWITTER	RB	SPECIAL	RB200	RB500	ER/BA/WS
Nodes	19.77	74.49	131.76	216.67	106.89	197.28	540.79	158.10
Density	0.51	0.52	0.205	0.218	0.530	0.205	0.179	0.105
Train	800	600	777	200	160	400	400	400
Test	200	1000	196	400	40	100	100	100
D Additional Experiments on MIS
Besides the datasets of Table 1, we also performed experiments in the IMDB dataset and in three known graph distributions: Erdos Renyi, Barabasi Albert and Watts Strogatz. Table 5 shows that all methods, except for the random comparator, always achieve optimal results for the IMDB dataset, since the size of the IMDB instances is extremely small, as depicted by Table 4. Moreover, it is worth noticing that, on the ER dataset, our method outperforms the greedy heuristic in the mixed roll-outs case.
Table 5: Test set approximation ratios (higher is better) on one dataset made with real-world instances (IMDB) and three datasets made by three known graph distributions (ER/BA/WS). We report the average approximation ratio along with the standard deviation.
IMDB	ER	BA	WS
Comparator (Normal Roll-outs)	1.000	0.930 ± 0.068	0.922 ± 0.089	0.823 ± 0.119
Comparator (Mixed Roll-outs)	1.000	0.954 ± 0.058	0.942 ± 0.055	0.831 ± 0.116
Greedy MIS	1.000	0.950 ± 0.045	0.959 ± 0.045	0.937 ± 0.058
Random Comparator	0.874 ± 0.261	0.584 ± 0.250	0.490 ± 0.251	0.610 ± 0.185
Simple Local Search (10s)	1.000	0.670 ± 0.184	0.533 ± 0.189	0.707 ± 0.141
SCIP 8.0.3 (1s)	1.000	0.908 ± 0.147	0.919 ± 0.193	0.926 ± 0.150
SCIP 8.0.3 (5s)	1.000	0.944 ± 0.067	0.969 ± 0.044	0.977 ± 0.054
Gurobi 10.0 (0.5s)	1.000	0.945 ± 0.074	0.916 ± 0.128	0.990 ± 0.030
Gurobi 10.0 (1s)	1.000	0.960 ± 0.061	0.956 ± 0.101	0.997 ± 0.014
Gurobi 10.0 (5s)	1.000	0.988 ± 0.026	0.998 ± 0.014	0.999 ± 0.004
E Experiments on Minimum Vertex Cover
Experiments for the MVC problem are performed over the RB200 and RB500 Xu et al. [2007], which were specifically designed to generate hard instances. In addition, we evaluate our results by comparing them with the methodologies proposed in *Erdos' GNN Karalias* and Loukas [2020], *RUN-CSP Toenshoff* et al. [2019], and *Meta-EGN* Wang and Li [2023]. Specifically, the results presented in Table 6 are sourced from Wang and Li [2023], with the exception of the comparator and Greedy MVC, and all experiments were conducted using the same datasets as in their study.[Wang and Li, 2023].
Among the neural approaches, Table 6 highlights similar results for the two datasets, and our model achieves the best result for the RB500 dataset. Moreover, as depicted by the last rows of Table 6, optimal solvers do not always reach the optimal value over the graphs of the datasets, since, as mentioned before, RB instances are known to be hard.
Our model, benefiting from dynamic programming techniques, possesses the ability to decompose complex instances into more manageable sub-instances. As a result, it shows good skills in addressing the inherent complexity of the RB200 and RB500 datasets, surpassing other methods in this regard.
Table 6: Test set approximation ratio (lower is better) for different methods on real-world datasets. We present the performance of different algorithms on the minimum vertex cover problem (MVC). For each cell in the table, the average approximation ratio is reported together with the standard deviation. The time limit of solvers is reported next to the name. If the standard deviation is not reported, then the algorithm achieves always the optimal solution over all graphs of the dataset. The best performances are reported in bold.
Method (↓) Dataset (→)	RB200	RB500
CMP	1.031 ± 0.006	1.015 ± 0.004
Erdos' GNN	1.031 ± 0.004	1.021 ± 0.002
Meta-EGN	1.028 ± 0.005	1.016 ± 0.002
RUN-CSP	1.124 ± 0.001	1.062 ± 0.005
Greedy MVC	1.027 ± 0.007	1.014 ± 0.003
Random CMP	1.063 ± 0.027	1.031 ± 0.015
SCIP 8.0.3 (1s)	1.017 ± 0.104	1.025 ± 0.018
SCIP 8.0.3 (5s)	1.016 ± 0.007	1.011 ± 0.003
Gurobi 10.0 (0.5s)	1.009 ± 0.006	1.013 ± 0.004
Gurobi 10.0 (1s)	1.002 ± 0.003	1.012 ± 0.003
Gurobi 10.0 (5s)	1.000 ± 0.001	1.004 ± 0.003
F Additional Background Information
In our work, we use Gurobi as a matter of comparison for our results. Since Gurobi is an Integer Linear Programming (ILP) solver, we believe it is useful to briefly revise the ILP formulations of the MIS and MVC problems. In particular, given a graph  $G(V,E)$  with  $n = |V|$ , we use binary decision variables  $X = (x_{i})_{1\leq i\leq n}\in \{0,1\}^{n}$  for each vertex  $i\in V$  such that  $x_{i} = 1$  if vertex  $i$  is considered as part of the solution and  $x_{i} = 0$  otherwise. In ILP, the goal is to assign values to every  $x_{i}$  variable such that a function  $F(G,X)$ , called energy function, is either maximized or minimized under a set of constraints.
F.1 Integer Linear Programming for the Maximum Independent Set
In MIS, the energy function is defined as the sum of every binary decision variable:  $F(G,X) = \sum_{i\in V}x_i$ . Since every edge  $(i,j)$  can have at most one of its nodes in the independent set (otherwise the independent condition is violated), the sum of decision variables of  $i$  and  $j$  is at most one:
$$
\text {m a x i m i z e} \quad \sum_ {i \in V} x _ {i}
$$
$$
\text {s u b j e c t} \quad x _ {i} + x _ {j} \leq 1 \quad \forall (i, j) \in E \tag {3}
$$
$$
\text {a n d} \quad x _ {i} \in \{0, 1 \} \quad \forall i \in V.
$$
F.2 Integer Linear Programming for the Minimum Vertex Cover
Differently from MIS, the ILP formulation for the MVC consists in minimizing the objective function  $F(G,X) = \sum_{i\in V}x_{i}$ . Since every edge of the graph has to have at least one of the two nodes in the vertex cover set (otherwise the edge wouldn't be covered), the sum of the decision variables of the two nodes defining the edge has to be greater (or equal) to one:
$$
\text {m i n i m i z e} \quad \sum_ {i \in V} x _ {i}
$$
$$
\text {s u b j e c t} \quad x _ {i} + x _ {j} \geq 1 \quad \forall (i, j) \in E \tag {4}
$$
In this section, we perform an ablation study on several parameters of the comparator function (without using mixed roll-outs) on the COLLAB dataset. The following parameters will be tweaked:
-  $D$ : The dimensionality of the initial node embeddings, and the output size of the layers in the network.
-  $K$ : The number of iterations in the Graph Embedding Module.
-  $L$  : The number of fully connected layers in the network.
We experiment linearly with the following parameters:  $D \in [8, 16, 32, 48, 64]$ ,  $K \in [1, 2, 3, 4, 5]$ , and  $L \in [2, 3, 4, 5]$ , with a base configuration of  $D = 32$ ,  $K = 3$ , and  $L = 4$ . This means that when testing with  $D = 48$ , the model configuration will be  $D = 48$ ,  $K = 3$ , and  $L = 4$ . The meaning of these parameters can be found in greater detail in Sec. 4.2.
All models were trained for 200 epochs, with the experiments performed on a single GPU with 6GB RAM.
Table 7: Model Performance for a different value of  $D$  . The best performance is highlighted in bold.
Parameter (D)	8	16	32	48	64
Model Performance	0.976 ± 0.053	0.972 ± 0.050	0.990 ± 0.049	0.980 ± 0.055	0.984 ± 0.048
Table 8: Model Performance for a different value of  $K$  . The best performance is highlighted in bold.
Parameter (K)	1	2	3	4	5
Model Performance	0.912 ± 0.097	0.944 ± 0.068	0.990 ± 0.049	0.992 ± 0.037	0.960 ± 0.059
Table 9: Model Performance for a different value of  $L$  . The best performance is highlighted in bold.
Parameter (L)	2	3	4	5
Model Performance	0.922 ± 0.084	0.987 ± 0.057	0.990 ± 0.049	0.993 ± 0.046
From Tables 7 to 9, we can see that a value of  $D = 32$ ,  $K = 3$ , and  $L = 4$  is a reasonable choice for the model architecture, also taking into account runtimes during inference for a model with smaller parameters. Interestingly, the model performance starts to degrade for  $K = 5$ , possibly due to over-smoothing.
Designing neural networks for combinatorial optimization (CO) problems is still in its infancy, and as such there are a lot of exciting questions. Nonetheless, delving even more in this direction might potentially impact our society. Important fields like medicine or biology require the solution of CO problems in a short time since the execution time can be critical for the health of a patient. Therefore, tackling CO problems with deep learning techniques can bring important benefits to society, but can also be used for malicious purposes. Therefore, we do encourage the community to consider those challenges and perspectives.
I Omitted Proofs
Theorem 1. Let a graph  $G(V, E) \in \mathcal{G}$ . Then for any vertex  $v \in V$  with  $d(v) \geq 1$ ,
$$
\left| \operatorname {M I S} (G) \right| = \max  \left(\left| \operatorname {M I S} \left(G / \mathcal {N} (v)\right) \right|, \left| \operatorname {M I S} \left(G / \{v \}\right) \right|\right).
$$
Proof. Let  $G(V, E)$  be a graph and  $\mathrm{MIS}(G)$  be its maximum independent set. We want to show that for any vertex  $v \in V$  with  $d(v) \geq 1$ , the size of  $\mathrm{MIS}(G)$  can be obtained by either removing  $v$  or removing its neighbors  $\mathcal{N}(v)$ .
Consider two cases:
$v\in \mathrm{MIS}(G)$
In this case, if we remove  $\mathcal{N}(v)$  from  $G$ , the resulting graph is denoted as  $G' = G \setminus \mathcal{N}(v)$ . Since the neighbors of  $v$  cannot be in the maximum independent set, removing it does not affect the size of  $\mathrm{MIS}(G)$ . Therefore,  $\mathrm{MIS}(G) = \mathrm{MIS}(G')$ .
-  $v \notin \operatorname{MIS}(G)$
In this case, we can remove  $v$  from  $G$  to obtain the graph  $G' = G \setminus \{v\}$ . Since  $v$  is not in the maximum independent set, removing it does not affect the size of  $\mathrm{MIS}(G)$ . Therefore,  $\mathrm{MIS}(G) = \mathrm{MIS}(G')$ .
By considering these two cases, we have shown that for any vertex  $v \in V$ , the maximum independent set  $\mathrm{MIS}(G)$  can be obtained by either removing  $v$  or removing its neighbors  $\mathcal{N}(v)$ . Thus, we can express the size of the maximum independent set as follows:
$$
\left| \operatorname {M I S} (G) \right| = \max  \left(\left| \operatorname {M I S} \left(G / \mathcal {N} (v)\right) \right|, \left| \operatorname {M I S} (G / \{v \}) \right|\right).
$$
Theorem 2. Let a consistent comparator  $\mathrm{CMP}:\mathcal{G}\times \mathcal{G}\mapsto \{0,1\}$ . Then the algorithm  $\mathcal{A}^{\mathrm{CMP}}(\cdot)$  always computes a Maximum Independent Set,  $\mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}}(G)\right|\right] = |\mathrm{MIS}(G)|$  for all  $G\in \mathcal{G}$ .
Proof. Let us consider a consistent comparator CMP. We will establish Theorem 2 with an induction on the number of edges  $i$ .
- Induction Basis  $(i = 0)$ : Let  $G(V,E) \in \mathcal{G}[0]$  then  $\mathcal{A}^{\mathrm{CMP}}(G) = V = \mathrm{MIS}(G)$ .
- Induction Hypothesis:  $\mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}}(G)\right|\right] = |\mathrm{MIS}(G)|$  for all  $G\in \mathcal{G}[j]$  with  $j\leq i$
- Induction Step: Let  $G \in \mathcal{G}[i + 1]$  and consider a node  $v$  with degree  $d(v) \geq 1$ . Consider also the graphs  $G_0 := G / \{v\}$  and  $G_1 := G / \{\mathcal{N}(v)\}$ . Both  $G_0$  and  $G_1$  admit less than  $i$  edges and thus by the inductive hypothesis,  $\mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}}(G_0)\right|\right] = |\mathrm{MIS}(G_0)|$  and  $\mathbb{E}\left[\left|\mathcal{A}^{\mathrm{CMP}}(G_1)\right|\right] = |\mathrm{MIS}(G_1)|$ . Hence  $\mathrm{CMP}(G_0, G_1) = 0$  if and only if  $|\mathrm{MIS}(G_0)| \geq |\mathrm{MIS}(G_1)|$ . As a result,  $|\mathcal{A}_{\mathrm{CMP}}(G)| = \max(|\mathrm{MIS}(G_0)|, |\mathrm{MIS}(G_1)|)$  and Theorem 1 implies that  $|\mathcal{A}_{\mathrm{CMP}}(G)| = |\mathrm{MIS}(G)|$ .
J Additional Experiments on Consistency
In an attempt to gain insight into the decisions of the comparator, we measure something that we refer to as the consistency value of a comparator. The consistency of a comparator is the proportion of graph pairs in which the equation of consistency from definition 3 is fulfilled.
Fig. 3 shows the consistency values as the training progresses for three different benchmark datasets. Overall, the consistency curves have an increasing behavior, indicating an increase in the consistency of the decision that a comparator makes as training progresses.
Consistency of the comparator for three datasets
Figure 3: The figure highlights the behavior of the consistency of the comparator for the three datasets: COLLAB, TWITTER, and RB. The consistency (y-axis) represents the confidence of the comparator during the generation of the recursive tree. A consistency value close to 1 indicates high confidence in the comparator. Moreover, the consistency is calculated during the training process at the end of every iteration (x-axis), where an iteration corresponds to a total of 10 epochs. The consistency value associated with iteration 0 is obtained at the beginning of iteration 1 and indicates the consistency of the random comparator.