Geometric Generalisation of Surrogate Model-Based Optimisation to Combinatorial and Program Spaces

Table 7

Results for unimodal, parity, and symbolic regression problems obtained by SMBO, random search (RS), and genetic programming (GP).

(a) ⟨Unimodal⟩

md	SMBO			RS			GP
	Best	Average	SD	Best	Average	SD	Best	Average	SD
3	0.11	0.47	0.19	0.11	0.43	0.16	0.22	0.50	0.18
4	0.07	0.14	0.05	0.13	0.37	0.13	0.11	0.49	0.25
5	0.04	0.07	0.03	0.08	0.24	0.08	0.05	0.48	0.21
6	0.01	0.04	0.04	0.08	0.14	0.08	0.14	0.46	0.21
7	0.01	0.02	0.04	0.04	0.18	0.04	0.06	0.32	0.20

(b) ⟨4-OddParity⟩

md	SMBO			RS			GP
	Best	Average	SD	Best	Average	SD	Best	Average	SD
3	37.50	45.00	6.45	37.50	45.00	6.45	37.50	48.75	3.95
4	37.50	40.00	5.27	37.50	41.25	6.04	37.50	42.50	6.45
5	37.50	37.50	0.00	37.50	37.50	0.00	37.50	47.50	5.27
6	37.50	37.50	0.00	37.50	37.50	0.00	37.50	41.25	6.04
7	25.00	33.75	6.04	37.50	37.50	0.00	37.50	37.50	0.00

(c) ⟨SymbolicRegression⟩

md	SMBO			RS			GP
	Best	Average	SD	Best	Average	SD	Best	Average	SD
3	3.44	4.88	0.82	3.44	4.88	0.78	2.64	5.17	1.35
4	4.46	6.35	1.17	4.27	5.78	1.41	4.46	6.39	1.58
5	3.84	5.58	1.21	3.51	5.18	1.21	4.05	5.39	1.27
6	2.95	3.74	0.73	2.99	3.52	3.48	3.48	4.39	0.57
7	3.45	4.50	0.77	3.81	4.96	0.54	3.71	4.62	0.67

∗The best (minimum) and average fitness values for the best solution found by each algorithm, for md = 3, 4, 5, 6, 7, over 50 runs.