SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Simulated Annealing. Finally, procedure SA in Algorithm 3 describes our Simulated Annealing based scheme for CMMAP. The basic principle behind this approach is to consider some neighboring state  $\mathbf{x}_M^{\prime}$  of the current state  $\mathbf{x}_M$ , and probabilistically decides between moving to state  $\mathbf{x}_M^{\prime}$  or staying in the current state. The probability of making the transition from  $\mathbf{x}_M$  to  $\mathbf{x}_M^{\prime}$  is specified by an acceptance probability function  $P(\mathbf{x}_M^{\prime},\mathbf{x}_M,T)$  that depends on the scores of the two states as well as a global time-varying parameter  $T$  called temperature. We chose  $P(\mathbf{x}_M^{\prime},\mathbf{x}_M,T) = e^{\frac{\Delta}{T}}$ ,
Algorithm 3 Local Search for Credal Marginal MAP
1: procedure  $\mathrm{SHC}(\mathcal{C},\mathbf{X}_M\subseteq \mathbf{X},p_{flip})$
2: initialize  $\mathbf{x}_M^* \gets \emptyset$ , best  $\gets -\infty$
3: for all iterations  $i = 1 \dots N$  do
4: initialize  $\mathbf{x}_M$  randomly
5: for all flips  $j = 1 \dots M$  do
6: sample randomly  $p\in (0,1)$
7: let  $\mathcal{N}$  be  $\mathbf{x}_M$ 's neighbors
8: if  $(p\leq p_{flip})$  then
9: select random neighbor  $\mathbf{x}_M^{\prime}\in \mathcal{N}$
0: else
1: for all neighbor  $\mathbf{x}_{M}^{\prime}\in \mathcal{N}$  do
2: compute score  $(\mathbf{x}_M^r)$
3: select highest score neighbor  $\mathbf{x}_M''\in \mathcal{N}$
4: if score  $(\mathbf{x}_M^{\prime \prime})\leq score(\mathbf{x}_M)$  then
5: select random neighbor  $\mathbf{x}_M^{\prime}\in \mathcal{N}$
6: else
7: select  $\mathbf{x}_M^{\prime}\gets \mathbf{x}_M^{\prime \prime}$
8: if score  $(\mathbf{x}_M^{\prime}) >$  best then
9:  $\mathbf{x}_M^* \gets \mathbf{x}_M'$
20: best  $\leftarrow$  score(xM)
21:  $\mathbf{x}_M\gets \mathbf{x}_M^{\prime}$
22: return  $\mathbf{x}_M^*$
23: procedure  $\mathrm{TS}(\mathcal{C},\mathbf{X}_M\subseteq \mathbf{X})$
24: initialize  $\mathbf{x}_M = \emptyset$ , best  $\leftarrow -\infty$
25: for all iterations  $i = 1 \dots N$  do
26: initialize  $\mathbf{x}_M$  randomly
27:  $\mathcal{T}\gets \emptyset$
28: for all flips  $j = 1 \dots M$  do
29:  $\mathcal{T}\gets \mathcal{T}\cup \{\mathbf{x}_M\}$
30: let  $\mathcal{N}$  be  $\mathbf{x}_M$ 's neighbors.
31: initialize  $\mathbf{x}_M''\gets \emptyset, b\gets -\infty$
32: for all neighbor  $\mathbf{x}_M^{\prime}\in \mathcal{N}$  do
33: if  $\mathbf{x}_M^{\prime}\notin \mathcal{T}$  and score  $(\mathbf{x}_M^{\prime}) > b$  then
44:  $\mathbf{x}_M^{\prime \prime}\gets \mathbf{x}_M^{\prime}$
35:  $b\gets score(\mathbf{x}_M^{\prime})$
36: if  $\mathbf{x}_{M}^{\prime \prime} = \emptyset$  then
37: select random neighbor  $\mathbf{x}_M^{\prime}\in \mathcal{N}$
38: else
39:  $\mathbf{x}_M^{\prime}\gets \mathbf{x}_M^{\prime \prime}$
40: if score  $(\mathbf{x}_M^{\prime}) >$  best then
41:  $\mathbf{x}_M^* \gets \mathbf{x}_M^*$
42: best  $\leftarrow$  score(xM)
43:  $\mathbf{x}_M\gets \mathbf{x}_M^{\prime}$
44: if  $size(\mathcal{T}) \geq S$  then
45: prune  $\mathcal{T}$  until size  $(\mathcal{T}) < S$
46: return  $\mathbf{x}_M^*$
47: procedure SA  $(\mathcal{C},\mathbf{X}_M\subseteq \mathbf{X},T_{init},\sigma)$
48: initialize  $\mathbf{x}_M^*$  randomly
49: best  $\leftarrow$  score(xM*)
50: for all iterations  $i = 1 \dots N$  do
51: set  $\mathbf{x}_M\gets \mathbf{x}_M^*,T\gets T_{init}$
52: for all flips  $j = 1 \dots M$  do
53: let  $\mathcal{N}$  be  $\mathbf{x}_M$ 's neighbors
54: select random neighbor  $\mathbf{x}_M^{\prime}\in \mathcal{N}$
55:  $\Delta \gets \log score(\mathbf{x}_M^{\prime}) - \log score(\mathbf{x}_M)$
56: if  $\Delta > 0$  then
57:  $\mathbf{x}_M\gets \mathbf{x}_M^{\prime}$
58: else
59: sample randomly  $p\in (0,1)$
60: if  $p < e^{\frac{\Delta}{T}}$  then
61:  $\mathbf{x}_M\gets \mathbf{x}_M^{\prime}$
62: if score  $(\mathbf{x}_M) >$  best then
63:  $\mathbf{x}_M^* \gets \mathbf{x}_M$
64: best  $\leftarrow$  score(xM)
65:  $T\gets T*\sigma$
66: return  $\mathbf{x}_M^*$
where  $\Delta = \log \overline{P} (\mathbf{x}_M^{\prime}) - \log \overline{P} (\mathbf{x}_M)$ . At each iteration, the temperature is decreased using a cooling schedule  $\sigma < 1$ . Like SHC and TS, algorithm SA implements a random restarts strategy.
Theorem 4 (complexity). Given a credal network  $\mathcal{C}$ , the complexity of algorithms SHC, TS and SA is time  $O(N \cdot M \cdot P)$  and space  $O(n)$ , where  $n$  is the number of variables,  $N$  is the number of iterations,  $M$  is the maximum number of flips allowed per iteration, and  $P$  bounds the complexity of approximating the probability of evidence in  $\mathcal{C}$ .
We evaluate the proposed algorithms for CMMAP on random credal networks and credal networks derived from real-world applications. All competing algorithms were implemented in  $\mathrm{C + + }$  and the experiments were run on a 32-core machine with 128GB of RAM running Ubuntu Linux 20.04.
We consider the two exact algorithms denoted by CVE and DFS, as well as the four approximation schemes denoted by SHC, TS, SA and  $\mathrm{CMBE}(i)$ , respectively. The local search algorithms used  $N = 10$  iterations and  $M = 10,000$  maximum flips per iteration, and they all used the approximate L2U algorithm with 10 iterations [29] to evaluate the MAP assignments during search. Furthermore, for SHC we set the flip probability  $p_{flip}$  to 0.2, TS used a taboo list of size 100, while for SA we set
n	Q	w*	SHC
time (#)	W	TS
time (#)	W	SA
time (#)	W	CMBE(2)
time (#)	W
random
100	20	25	32.69	100	23.08	100	6.47	100	225.28 (70)	1
40	37	163.05	100	79.11	100	14.78	100	327.67 (43)	0
60	23	421.93	100	185.41	100	29.99	100	224.93 (7)	0
150	30	39	254.32	100	141.03	100	24.08	100	294.48 (43)	0
60	57	1143.78	100	531.45	100	70.06	100	555.98 (14)	0
90	66	2811.47	100	1259.79	100	139.78	75	925.41 (2)	0
200	50	58	1044.79	100	490.38	100	72.09	100	276.98 (38)	0
100	86	3496.77 (32)	32	2143.79	100	211.47	14	927.31 (1)	0
150	69	3601.67 (16)	16	3550.99 (17)	17	339.34	72	- (0)	0
grid
100	20	25	31.35	100	22.77	100	4.66	100	0.07	2
40	37	155.34	100	79.83	100	10.51	100	3.85	0
60	23	358.81	100	168.79	100	19.18	100	28.76	0
144	30	36	219.49	100	121.86	100	21.02	100	0.34	0
60	53	878.63	100	426.70	100	54.77	100	1.03	0
90	26	2109.47	100	958.13	100	102.93	73	27.79	0
196	50	55	817.52	100	382.46	100	58.13	100	0.68	0
100	56	3045.54 (94)	94	1453.39	100	147.11	13	51.01 (98)	0
150	22	3601.25 (23)	23	3011.47 (93)	93	190.26	3	41.27 (99)	2
k-tree
100	20	25	68.25	100	44.25	100	10.48	100	221.18 (55)	0
40	37	307.91	100	151.97	100	23.19	100	163.59 (8)	0
60	23	650.72	100	306.26	100	40.19	100	- (0)	0
150	30	28	443.33	100	245.71	100	44.58	100	492.55 (26)	0
60	47	1647.29	100	724.01	100	106.71	100	14.68 (1)	0
90	51	2917.01 (84)	84	1541.91	100	192.76	82	- (0)	0
200	50	45	1306.43	100	660.59	100	108.36	100	1199.83	0
100	64	3376.59 (54)	54	1917.95	100	266.24	21	- (0)	0
1500	48	3602.98 (4)	4	3334.49 (59)	59	344.96	38	- (0)	0
Table 1: Results on random, grid and  $k$ -tree credal networks. Mean CPU times in seconds, number of instance solved (#) and number of wins (W). Time limit 1 hour, 8GB of RAM.
the initial temperature and cooling schedule to  $T_{init} = 100$  and  $\sigma = 0.9$ , respectively. For CMBE(i) we set the i-bound  $i$  to 2 and used the same L2U algorithm to evaluate the solution found. All competing algorithms were allocated a 1 hour time limit and 8GB of memory per problem instance.
In all our experiments, we report the CPU time in seconds, the number of problems solved within the time/memory limit and the number of times an algorithm converged to the best possible solution. The latter is called the number of wins and is meant to be a measure of solution quality for the respective algorithm. We also record the number of variables  $(n)$ , the number (or percentage) of MAP variables (Q) and the constrained induced widths  $(w^{*})$ . The best performance points are highlighted.
6.1 Random Credal Networks
For our purpose, we generated random credal networks,  $m$ -by-  $m$  grid networks as well as  $k$ -tree networks. Specifically, for the random networks, we varied the number of variables  $n \in \{100, 150, 200\}$ , for grids, we choose  $m \in \{10, 14, 16\}$ , and for  $k$ -trees we selected  $k = 2$  and the number of variables  $n \in \{100, 150, 200\}$ , respectively. In all cases, the maximum domain size was set to 2 and the conditional credal sets were generated uniformly at random as probability intervals such that the difference between the lower and upper probability bounds was at most 0.3.
First, we note that the exact algorithms CVE and DFS could only solve very small problems with up to 10 variables and 5 MAP variables. The main reason for the poor performance of these algorithms is the extremely large size of the intermediate potentials generated during the variable elimination procedure which causes the algorithms to run out of memory or time on larger problems. Therefore, we omit their evaluation hereafter.
problem	w*	SHC time (#)	W	TS time (#)	W	SA time (#)	W	CMBE(2) time (#)	W
alarm	12	27.32 (10)	10	21.31 (10)	10	4.89 (10)	10	324.23 (10)	0
child	7	3.51 (10)	10	3.69 (10)	10	1.19 (10)	10	0.64 (10)	0
link	239	3655.67 (2)	2	3628.15 (2)	1	1300.14 (10)	8	- (0)	0
insurance	12	64.22 (10)	10	45.77 (10)	10	17.11 (10)	10	97.16 (9)	0
hepar2	25	1734.11 (10)	10	833.82 (10)	10	163.16 (10)	10	- (0)	0
pathfinder	33	2509.89 (10)	10	79.78 (10)	10	93.98 (10)	10	- (0)	0
hailfinder	14	126.60 (10)	10	72.73 (10)	10	12.53 (10)	10	531.52 (10)	0
largefam	402	- (0)	0	- (0)	0	2903.55 (10)	10	- (0)	0
mastermind1	389	- (0)	0	- (0)	0	3600.85 (10)	10	- (0)	0
mastermind2	726	- (0)	0	- (0)	0	3617.17 (5)	5	- (0)	0
mastermind3	1193	- (0)	0	- (0)	0	3650.54 (3)	3	- (0)	0
mildew	12	22.49 (10)	10	15.22 (10)	10	3.15 (10)	10	0.16 (10)	0
munin	175	3615.45 (5)	4	3639.57 (3)	3	652.72 (10)	5	- (0)	0
pedigree1	74	3603.87 (7)	7	3609.44 (8)	8	410.61 (10)	0	1269.45 (8)	0
pedigree7	147	3620.74 (2)	1	3625.51 (2)	1	1689.89 (10)	9	- (0)	0
pedigree9	175	- (0)	0	- (0)	0	1719.92 (10)	6	1128.50 (4)	4
win95pts	28	3612.38 (6)	6	3610.94 (6)	6	821.20 (10)	10	- (0)	0
xandes	75	3619.00 (6)	6	3611.76 (6)	6	3565.62 (10)	3	- (0)	0
xdiabetes	75	3605.08 (9)	9	3603.76 (10)	10	175.76 (10)	0	182.38 (6)	0
zbarley	18	140.93 (10)	10	82.36 (10)	10	18.47 (10)	10	863.64 (1)	0
zpigs	105	3606.83 (4)	4	3603.8 (5)	5	234.84 (10)	5	100.47 (2)	0
zwater	16	207.07 (10)	10	126.87 (10)	10	44.68 (10)	10	- (0)	0
Table 2: Results on real-world credal networks with  $Q = {50}\%$  MAP variables. Mean CPU time in seconds, number of instances solved (#) and number of wins (W). Time limit 1 hour, 8 GB of RAM.
Table 1 summarizes the results obtained on random, grid and  $k$ -tree networks. Each data point represents an average over 100 random problem instances generated for each problem size ( $n$ ) and number of MAP variables ( $Q$ ), respectively. Next to the running time we show the number of instances solved within the time/memory limit (if the number is omitted then all 100 instances were solved). We can see that in terms of running time, CMBE(2) performs best on the grid networks. This is because the intermediate potentials generated during elimination are relatively small size and therefore are processed quickly. However, the algorithm is not able to converge to good quality solutions compared with its competitors. The picture is reversed on the random and  $k$ -tree networks where CMBE(2) is the worst performing algorithm both in terms of running time and solution quality. In this case, the relatively large intermediate potentials cause the algorithm to exceed the time and memory limits on many problem instances and thus impact negatively its performance.
The local search algorithms SHC, TS and SA yield the best performance in terms of solution quality with all three algorithms almost always converging to the best possible solutions on these problem instances. In terms of running time, SA is the fastest algorithm achieving almost one order of magnitude speedup over its competitors, especially for larger numbers of MAP variables (e.g.,  $k$ -trees with  $n = 100$  variables and  $Q = 60$  MAP variables). Algorithms SHC and TS have comparable running times (with SHC being slightly slower than TS) but they are significantly slower than SA. This is due to the significantly larger computational overhead required for evaluating the scores of all the neighbors of the current state, especially when there are many MAP variables.
6.2 Real-World Credal Networks
Table 2 shows the results obtained on a set of credal networks derived from 22 real-world Bayesian networks<sup>1</sup> by converting the probability values in the CPTs into probability intervals such that the difference between the corresponding lower and upper probability bounds was at most 0.3. Furthermore, since the local search algorithms rely on the L2U approximation to evaluate the MAP configurations, we restricted the domains of the multi-valued variables to the first two values in the domain while shrinking and re-normalizing the corresponding CPTs. For each network we selected uniformly at random  $Q = 50\%$  of the variables to act as MAP variable and generated 10 random instances. As before, we indicate next to the average running times the number of instances solved
Available at https://www.bnlearn.com/bnrepository/
(a) Brain Tumour diagnosis
(b) Attack Intelligence report
Figure 2: Applications of Credal Marginal MAP inference.
by the respective algorithms within the time and memory limits. We can see again that CMBE(2) is competitive only on the easiest instances (e.g., child, mildew) while SA yields the best performance in terms of both running time and solution quality on the majority of the problem instances. In summary, the relatively large potentials hinder CMBE's performance, while the computational overhead incurred during the evaluation of relatively large neighborhoods of the current state slows down significantly SHC and TS compared with SA.
6.3 Applications
Figure 2a shows the credal network for the brain tumour diagnosis use case derived from the Bayesian network described in [11]. The variables are: MC - metastatic cancer, PD - Paget disease, B - brain tumour, ISC - increased serum calcium, H - headaches, M - memory loss, CT - scan result.
Considering the query variables  $B$  and  $ISC$ , the exact solution for both maximax and maximin CMMAP is  $(B = 0, ISC = 0)$  (obtained by both the CVE and DFS algorithms). In this case, the maximax and maximin scores are 0.837 and 0.42, respectively. Algorithms SHC, TS and SA also find the optimal configuration  $(B = 0, ISC = 0)$  which is evaluated by L2U to 0.8316 for maximax CMMAP and to 0.37296 for maximin CMMAP, respectively.
Figure 2b shows the credal network for the intelligence report analysis described in [9]. The variables are: As - assassination, C - coup/revolt, R - regime change, D - decision to invade, At - attack, B - build-up, P - propaganda, I - invasion.
Considering the query variables  $D$ ,  $At$  and  $I$ , the exact solution for both maximax and maximin CMMAP is  $(D = 0, At = 0, I = 0)$  and is obtained by both algorithms CVE and DFS. The corresponding scores are in this case 0.765 and 0.458806, respectively. The approximation schemes SHC, TS and SA also find the same optimal CMMAP configuration  $(D = 0, At = 0, I = 0)$  which is evaluated by L2U to 0.69651 for maximax CMMAP and to 0.305486 for maximin CMMAP, respectively.
We note that in both cases, the constrained induced width is 2 and therefore CMBE(2) coincides with the exact CVE. Therefore, all our approximation schemes found the optimal solutions.
The paper explores the Marginal MAP inference task in credal networks. We formally define the Credal Marginal MAP task and present new exact algorithms based on variable elimination and depth-first search. Subsequently, we introduce approximate algorithms using the mini-bucket partitioning or a combination of stochastic local search and approximate credal marginal inference. Our experiments on random and real-world credal networks demonstrate the effectiveness of our CMMAP algorithms. A potential direction for future work is to investigate branch-and-bound and best-first search strategies guided by a credal version of weighted mini-backets [32].
Fabio Cozman was supported in part by C4AI funding (FAPESP and IBM Corporation, grant 2019/07665-4) and CNPq (grant 305753/2022-3).
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