SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-6.34575 · 10^4	3600.07	19.14
3000	big-M20	-6.34575 · 10^4	3600.11	18.10
3000	big-M5	-6.34575 · 10^4	2452.71	0.00
3000	ours	-6.34575 · 10^4	3600.33	19.31
4000	big-M50	-8.64284 · 10^4	3600.23	15.33
4000	big-M20	-8.64284 · 10^4	3600.11	14.04
4000	big-M5	-8.64284 · 10^4	1675.19	0.00
4000	ours	-8.64284 · 10^4	20.81	0.00
5000	big-M50	-1.09575 · 10^5	3600.08	11.60
5000	big-M20	-1.09575 · 10^5	3600.07	9.94
5000	big-M5	-1.09575 · 10^5	1316.91	0.00
5000	ours	-1.09575 · 10^5	20.65	0.00
6000	big-M50	-1.27642 · 10^5	3601.02	9.92
6000	big-M20	-1.27642 · 10^5	3600.06	9.92
6000	big-M5	-1.27642 · 10^5	1040.84	0.00
6000	ours	-1.27642 · 10^5	1.70	0.00
7000	big-M50	-1.50613 · 10^5	3600.45	8.32
7000	big-M20	-1.50613 · 10^5	3600.10	5.74
7000	big-M5	-1.50613 · 10^5	901.46	0.00
7000	ours	-1.50613 · 10^5	1.67	0.00
Table 101: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-1.59508 · 10^5	3601.50	19.07
3000	big-M20	-1.59508 · 10^5	3600.41	17.35
3000	big-M5	-1.59508 · 10^5	3600.05	6.40
3000	ours	-1.59508 · 10^5	3600.27	19.40
4000	big-M50	-2.17573 · 10^5	3600.23	14.78
4000	big-M20	-2.17573 · 10^5	3600.11	14.34
4000	big-M5	-2.17573 · 10^5	3600.36	4.63
4000	ours	-2.17573 · 10^5	99.53	0.00
5000	big-M50	-2.71985 · 10^5	3600.09	11.22
5000	big-M20	-2.71985 · 10^5	3600.06	11.29
5000	big-M5	-2.71985 · 10^5	2986.61	0.00
5000	ours	-2.71985 · 10^5	21.01	0.00
6000	big-M50	-3.16760 · 10^5	3600.07	9.96
6000	big-M20	-3.16760 · 10^5	3601.06	9.95
6000	big-M5	-3.16760 · 10^5	2025.19	0.00
6000	ours	-3.16760 · 10^5	20.91	0.00
7000	big-M50	-3.72609 · 10^5	3600.80	8.33
7000	big-M20	-3.72609 · 10^5	3600.81	8.12
7000	big-M5	-3.72609 · 10^5	1452.72	0.00
7000	ours	-3.72609 · 10^5	1.73	0.00
Table 102: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-3.33603 · 10^5	3601.02	18.78
3000	big-M20	-3.33603 · 10^5	3600.70	16.11
3000	big-M5	-3.33603 · 10^5	3600.22	7.55
3000	ours	-3.33603 · 10^5	3600.68	19.44
4000	big-M50	-4.53260 · 10^5	3600.12	14.95
4000	big-M20	-4.53260 · 10^5	3600.25	13.47
4000	big-M5	-4.53260 · 10^5	3600.05	6.23
4000	ours	-4.53260 · 10^5	3600.41	0.62
5000	big-M50	-5.63006 · 10^5	3600.07	11.69
5000	big-M20	-5.63006 · 10^5	3601.43	11.26
5000	big-M5	-5.63006 · 10^5	3600.31	3.51
5000	ours	-5.63006 · 10^5	37.26	0.00
6000	big-M50	-6.57798 · 10^5	3600.53	9.98
6000	big-M20	-6.57798 · 10^5	3600.62	9.36
6000	big-M5	-6.57798 · 10^5	3600.05	2.82
6000	ours	-6.57798 · 10^5	20.64	0.00
7000	big-M50	-7.72266 · 10^5	3600.96	8.33
7000	big-M20	-7.72266 · 10^5	3600.57	7.80
7000	big-M5	-7.72266 · 10^5	3600.14	2.34
7000	ours	-7.72266 · 10^5	20.91	0.00
Table 103: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-7.41186 · 10^5	3600.75	18.15
3000	big-M20	-7.41186 · 10^5	3600.78	14.15
3000	big-M5	-7.41186 · 10^5	3600.34	5.92
3000	ours	-7.41186 · 10^5	3600.59	19.46
4000	big-M50	-1.00287 · 10^6	3600.07	14.98
4000	big-M20	-1.00287 · 10^6	3600.06	12.06
4000	big-M5	-1.00287 · 10^6	3600.36	5.06
4000	ours	-1.00287 · 10^6	3600.08	1.88
5000	big-M50	-1.24027 · 10^6	3600.07	11.88
5000	big-M20	-1.24027 · 10^6	3600.08	10.26
5000	big-M5	-1.24027 · 10^6	3600.22	4.34
5000	ours	-1.24027 · 10^6	469.43	0.00
6000	big-M50	-1.45416 · 10^6	3601.17	9.78
6000	big-M20	-1.45416 · 10^6	3600.60	9.11
6000	big-M5	-1.45416 · 10^6	3600.20	3.94
6000	ours	-1.45416 · 10^6	59.16	0.00
7000	big-M50	-1.70483 · 10^6	3602.06	8.34
7000	big-M20	-1.70483 · 10^6	3600.61	7.77
7000	big-M5	-1.70483 · 10^6	3600.22	3.39
7000	ours	-1.70483 · 10^6	21.12	0.00
Table 104: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-2.78468 · 10^6	3600.31	15.93
3000	big-M20	-2.78521 · 10^6	3600.05	10.59
3000	big-M5	-2.78521 · 10^6	3600.09	3.69
3000	ours	-2.78521 · 10^6	3600.23	19.46
4000	big-M50	-3.75203 · 10^6	3600.08	13.35
4000	big-M20	-3.75203 · 10^6	3600.04	8.60
4000	big-M5	-3.75203 · 10^6	3600.14	3.16
4000	ours	-3.75203 · 10^6	3601.10	3.50
5000	big-M50	-4.61877 · 10^6	3600.08	11.40
5000	big-M20	-4.61877 · 10^6	3600.05	7.62
5000	big-M5	-4.61877 · 10^6	3600.16	2.72
5000	ours	-4.61877 · 10^6	3600.40	0.93
6000	big-M50	-5.43826 · 10^6	3600.23	9.79
6000	big-M20	-5.43826 · 10^6	3600.11	6.92
6000	big-M5	-5.43826 · 10^6	3600.17	2.50
6000	ours	-5.43826 · 10^6	3600.26	0.33
7000	big-M50	-6.36767 · 10^6	3600.07	8.22
7000	big-M20	-6.36767 · 10^6	3600.24	6.06
7000	big-M5	-6.36767 · 10^6	3600.05	2.25
7000	ours	-6.36767 · 10^6	3600.58	0.11
Table 105: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.1).
G.3.10 Synthetic 2 Benchmark with  $\lambda_{2} = 0.1$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-6.34575 · 10^4	3600.12	19.14
3000	big-M20 + warm start	-6.34575 · 10^4	3600.22	18.10
3000	big-M5 + warm start	-6.34575 · 10^4	2424.12	0.00
3000	ours	-6.34575 · 10^4	3600.33	19.31
4000	big-M50 + warm start	-8.64284 · 10^4	3600.05	15.33
4000	big-M20 + warm start	-8.64284 · 10^4	3600.18	14.04
4000	big-M5 + warm start	-8.64284 · 10^4	1702.95	0.00
4000	ours	-8.64284 · 10^4	20.81	0.00
5000	big-M50 + warm start	-1.09575 · 10^5	3601.88	11.60
5000	big-M20 + warm start	-1.09575 · 10^5	3600.31	9.94
5000	big-M5 + warm start	-1.09575 · 10^5	1284.19	0.00
5000	ours	-1.09575 · 10^5	20.65	0.00
6000	big-M50 + warm start	-1.27642 · 10^5	3600.62	9.93
6000	big-M20 + warm start	-1.27642 · 10^5	3600.15	9.92
6000	big-M5 + warm start	-1.27642 · 10^5	1004.13	0.00
6000	ours	-1.27642 · 10^5	1.70	0.00
7000	big-M50 + warm start	-1.50613 · 10^5	3600.07	8.32
7000	big-M20 + warm start	-1.50613 · 10^5	3600.06	5.74
7000	big-M5 + warm start	-1.50613 · 10^5	794.91	0.00
7000	ours	-1.50613 · 10^5	1.67	0.00
Table 106: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-1.59508 · 10^5	3600.22	19.07
3000	big-M20 + warm start	-1.59508 · 10^5	3600.22	17.30
3000	big-M5 + warm start	-1.59508 · 10^5	3600.05	6.35
3000	ours	-1.59508 · 10^5	3600.27	19.40
4000	big-M50 + warm start	-2.17573 · 10^5	3600.10	14.78
4000	big-M20 + warm start	-2.17573 · 10^5	3600.06	14.34
4000	big-M5 + warm start	-2.17573 · 10^5	3600.04	4.63
4000	ours	-2.17573 · 10^5	99.53	0.00
5000	big-M50 + warm start	-2.71985 · 10^5	3600.07	11.22
5000	big-M20 + warm start	-2.71985 · 10^5	3600.10	11.29
5000	big-M5 + warm start	-2.71985 · 10^5	2980.53	0.00
5000	ours	-2.71985 · 10^5	21.01	0.00
6000	big-M50 + warm start	-3.16760 · 10^5	3600.07	9.96
6000	big-M20 + warm start	-3.16760 · 10^5	3600.74	9.95
6000	big-M5 + warm start	-3.16760 · 10^5	2034.39	0.00
6000	ours	-3.16760 · 10^5	20.91	0.00
7000	big-M50 + warm start	-3.72609 · 10^5	3601.58	8.33
7000	big-M20 + warm start	-3.72609 · 10^5	3600.38	8.12
7000	big-M5 + warm start	-3.72609 · 10^5	1513.64	0.00
7000	ours	-3.72609 · 10^5	1.73	0.00
Table 107: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-3.33603 · 10^5	3601.53	18.78
3000	big-M20 + warm start	-3.33603 · 10^5	3600.81	16.11
3000	big-M5 + warm start	-3.33603 · 10^5	3600.22	7.55
3000	ours	-3.33603 · 10^5	3600.68	19.44
4000	big-M50 + warm start	-4.53260 · 10^5	3600.07	15.04
4000	big-M20 + warm start	-4.53260 · 10^5	3600.26	13.47
4000	big-M5 + warm start	-4.53260 · 10^5	3600.21	6.23
4000	ours	-4.53260 · 10^5	3600.41	0.62
5000	big-M50 + warm start	-5.63006 · 10^5	3601.70	11.69
5000	big-M20 + warm start	-5.63006 · 10^5	3600.25	11.26
5000	big-M5 + warm start	-5.63006 · 10^5	3600.05	3.43
5000	ours	-5.63006 · 10^5	37.26	0.00
6000	big-M50 + warm start	-6.57798 · 10^5	3600.18	9.98
6000	big-M20 + warm start	-6.57798 · 10^5	3600.55	9.36
6000	big-M5 + warm start	-6.57798 · 10^5	3600.04	2.89
6000	ours	-6.57798 · 10^5	20.64	0.00
7000	big-M50 + warm start	-7.72266 · 10^5	3600.63	8.33
7000	big-M20 + warm start	-7.72266 · 10^5	3600.06	7.80
7000	big-M5 + warm start	-7.72266 · 10^5	3600.15	2.46
7000	ours	-7.72266 · 10^5	20.91	0.00
Table 108: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-7.41186 · 10^5	3601.56	18.15
3000	big-M20 + warm start	-7.41186 · 10^5	3600.35	14.15
3000	big-M5 + warm start	-7.41186 · 10^5	3600.27	5.92
3000	ours	-7.41186 · 10^5	3600.59	19.46
4000	big-M50 + warm start	-1.00287 · 10^6	3600.09	14.98
4000	big-M20 + warm start	-1.00287 · 10^6	3600.12	12.06
4000	big-M5 + warm start	-1.00287 · 10^6	3600.07	5.06
4000	ours	-1.00287 · 10^6	3600.08	1.88
5000	big-M50 + warm start	-1.24027 · 10^6	3600.40	11.88
5000	big-M20 + warm start	-1.24027 · 10^6	3600.98	10.26
5000	big-M5 + warm start	-1.24027 · 10^6	3600.22	4.34
5000	ours	-1.24027 · 10^6	469.43	0.00
6000	big-M50 + warm start	-1.45416 · 10^6	3600.05	9.78
6000	big-M20 + warm start	-1.45416 · 10^6	3600.06	9.06
6000	big-M5 + warm start	-1.45416 · 10^6	3600.19	3.94
6000	ours	-1.45416 · 10^6	59.16	0.00
7000	big-M50 + warm start	-1.70483 · 10^6	3600.07	8.34
7000	big-M20 + warm start	-1.70483 · 10^6	3600.28	7.77
7000	big-M5 + warm start	-1.70483 · 10^6	3600.24	3.39
7000	ours	-1.70483 · 10^6	21.12	0.00
Table 109: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-2.78521 · 10^6	3600.07	15.91
3000	big-M20 + warm start	-2.78521 · 10^6	3600.71	10.37
3000	big-M5 + warm start	-2.78521 · 10^6	3600.08	3.69
3000	ours	-2.78521 · 10^6	3600.23	19.46
4000	big-M50 + warm start	-3.75203 · 10^6	3600.27	13.35
4000	big-M20 + warm start	-3.75203 · 10^6	3600.05	8.60
4000	big-M5 + warm start	-3.75203 · 10^6	3600.09	3.16
4000	ours	-3.75203 · 10^6	3601.10	3.50
5000	big-M50 + warm start	-4.61877 · 10^6	3600.12	11.40
5000	big-M20 + warm start	-4.61877 · 10^6	3600.06	7.62
5000	big-M5 + warm start	-4.61877 · 10^6	3600.09	2.72
5000	ours	-4.61877 · 10^6	3600.40	0.93
6000	big-M50 + warm start	-5.43826 · 10^6	3600.07	9.79
6000	big-M20 + warm start	-5.43826 · 10^6	3600.33	6.97
6000	big-M5 + warm start	-5.43826 · 10^6	3600.24	2.50
6000	ours	-5.43826 · 10^6	3600.26	0.33
7000	big-M50 + warm start	-6.36767 · 10^6	3600.07	8.22
7000	big-M20 + warm start	-6.36767 · 10^6	3600.67	6.06
7000	big-M5 + warm start	-6.36767 · 10^6	3600.07	2.25
7000	ours	-6.36767 · 10^6	3600.58	0.11
Table 110: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.1). All baselines use our beam search solution as a warm start.
G.3.11 Synthetic 2 Benchmark with  $\lambda_{2} = 10.0$  with No Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-6.33576 · 104	3600.31	18.47
3000	big-M20	-6.33576 · 104	3600.43	17.82
3000	big-M5	-6.33576 · 104	2349.33	0.00
3000	ours	-6.33576 · 104	3601.41	4.54
4000	big-M50	-8.63287 · 104	3600.85	15.21
4000	big-M20	-8.63287 · 104	3600.06	13.93
4000	big-M5	-8.63287 · 104	1577.17	0.00
4000	ours	-8.63287 · 104	20.33	0.00
5000	big-M50	-1.09474 · 105	3600.28	11.85
5000	big-M20	-1.09474 · 105	3600.55	10.17
5000	big-M5	-1.09474 · 105	1627.54	0.00
5000	ours	-1.09474 · 105	20.71	0.00
6000	big-M50	-1.27542 · 105	3600.37	9.90
6000	big-M20	-1.27542 · 105	3600.09	9.81
6000	big-M5	-1.27542 · 105	1071.17	0.00
6000	ours	-1.27542 · 105	1.69	0.00
7000	big-M50	-1.50512 · 105	3600.56	8.31
7000	big-M20	-1.50512 · 105	3600.07	8.30
7000	big-M5	-1.50512 · 105	856.04	0.00
7000	ours	-1.50512 · 105	1.70	0.00
Table 111: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-1.59409 · 10^5	3600.72	18.52
3000	big-M20	-1.59409 · 10^5	3600.07	17.23
3000	big-M5	-1.59409 · 10^5	3600.44	5.92
3000	ours	-1.59409 · 10^5	3600.62	4.68
4000	big-M50	-2.17472 · 10^5	3600.07	14.64
4000	big-M20	-2.17472 · 10^5	3600.82	14.15
4000	big-M5	-2.17472 · 10^5	3600.05	4.05
4000	ours	-2.17472 · 10^5	86.90	0.00
5000	big-M50	-2.71885 · 10^5	3603.74	11.16
5000	big-M20	-2.71885 · 10^5	3601.16	11.14
5000	big-M5	-2.71885 · 10^5	2522.90	0.00
5000	ours	-2.71885 · 10^5	20.75	0.00
6000	big-M50	-3.16660 · 10^5	3600.07	9.93
6000	big-M20	-3.16660 · 10^5	3600.26	9.92
6000	big-M5	-3.16660 · 10^5	1893.58	0.00
6000	ours	-3.16660 · 10^5	21.05	0.00
7000	big-M50	-3.72509 · 10^5	3601.17	8.31
7000	big-M20	-3.72509 · 10^5	3600.31	8.29
7000	big-M5	-3.72509 · 10^5	1472.84	0.00
7000	ours	-3.72509 · 10^5	1.71	0.00
Table 112: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-3.33503 · 10^5	3600.07	18.43
3000	big-M20	-3.33503 · 10^5	3600.06	16.05
3000	big-M5	-3.33503 · 10^5	3600.22	7.54
3000	ours	-3.33503 · 10^5	3600.25	6.21
4000	big-M50	-4.53159 · 10^5	3600.16	14.84
4000	big-M20	-4.53159 · 10^5	3600.90	13.44
4000	big-M5	-4.53159 · 10^5	3600.21	6.24
4000	ours	-4.53159 · 10^5	3600.95	0.35
5000	big-M50	-5.62906 · 10^5	3600.16	11.59
5000	big-M20	-5.62906 · 10^5	3600.07	11.13
5000	big-M5	-5.62906 · 10^5	3600.36	3.65
5000	ours	-5.62906 · 10^5	31.47	0.00
6000	big-M50	-6.57698 · 10^5	3600.16	9.95
6000	big-M20	-6.57698 · 10^5	3600.52	9.33
6000	big-M5	-6.57698 · 10^5	3600.37	2.90
6000	ours	-6.57698 · 10^5	21.24	0.00
7000	big-M50	-7.72166 · 10^5	3600.12	8.31
7000	big-M20	-7.72166 · 10^5	3600.05	7.81
7000	big-M5	-7.72166 · 10^5	3600.05	2.33
7000	ours	-7.72166 · 10^5	20.37	0.00
Table 113: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-7.41086 · 10^5	3600.65	17.88
3000	big-M20	-7.41086 · 10^5	3600.14	14.12
3000	big-M5	-7.41086 · 10^5	3600.35	5.94
3000	ours	-7.41086 · 10^5	3600.48	8.32
4000	big-M50	-1.00277 · 10^6	3600.09	14.85
4000	big-M20	-1.00277 · 10^6	3600.47	11.77
4000	big-M5	-1.00277 · 10^6	3600.05	5.04
4000	ours	-1.00277 · 10^6	3600.62	1.66
5000	big-M50	-1.24017 · 10^6	3600.25	11.82
5000	big-M20	-1.24017 · 10^6	3600.06	10.25
5000	big-M5	-1.24017 · 10^6	3600.22	4.34
5000	ours	-1.24017 · 10^6	386.72	0.00
6000	big-M50	-1.45406 · 10^6	3600.07	9.75
6000	big-M20	-1.44919 · 10^6	3600.53	9.46
6000	big-M5	-1.45406 · 10^6	3600.20	3.95
6000	ours	-1.45406 · 10^6	54.13	0.00
7000	big-M50	-1.70473 · 10^6	3600.81	8.32
7000	big-M20	-1.70473 · 10^6	3600.06	7.68
7000	big-M5	-1.70473 · 10^6	3600.20	3.41
7000	ours	-1.70473 · 10^6	21.30	0.00
Table 114: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 10.0$ ).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-2.78507 · 10^6	3600.17	15.95
3000	big-M20	-2.78507 · 10^6	3600.56	10.36
3000	big-M5	-2.78507 · 10^6	3600.09	3.69
3000	ours	-2.78507 · 10^6	3600.78	9.79
4000	big-M50	-3.75192 · 10^6	3600.06	13.19
4000	big-M20	-3.75192 · 10^6	3600.05	8.60
4000	big-M5	-3.75192 · 10^6	3600.41	3.16
4000	ours	-3.75192 · 10^6	3600.32	3.23
5000	big-M50	-4.61867 · 10^6	3600.17	11.32
5000	big-M20	-4.61867 · 10^6	3600.56	7.66
5000	big-M5	-4.61867 · 10^6	3600.22	2.72
5000	ours	-4.61867 · 10^6	3600.30	0.89
6000	big-M50	-5.43816 · 10^6	3600.12	9.75
6000	big-M20	-5.43816 · 10^6	3600.33	6.91
6000	big-M5	-5.43816 · 10^6	3600.20	2.50
6000	ours	-5.43816 · 10^6	3600.22	0.32
7000	big-M50	-6.36757 · 10^6	3600.32	8.20
7000	big-M20	-6.36757 · 10^6	3600.05	6.05
7000	big-M5	-6.36757 · 10^6	3600.37	2.25
7000	ours	-6.36757 · 10^6	3600.06	0.10
Table 115: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 10.0).
G.3.12 Synthetic 2 Benchmark with  $\lambda_{2} = 10.0$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-6.33576 · 104	3600.12	18.47
3000	big-M20 + warm start	-6.33576 · 104	3600.44	17.82
3000	big-M5 + warm start	-6.33576 · 104	2370.86	0.00
3000	ours	-6.33576 · 104	3601.41	4.54
4000	big-M50 + warm start	-8.63287 · 104	3600.29	15.21
4000	big-M20 + warm start	-8.63287 · 104	3600.09	13.93
4000	big-M5 + warm start	-8.63287 · 104	1870.35	0.00
4000	ours	-8.63287 · 104	20.33	0.00
5000	big-M50 + warm start	-1.09474 · 105	3600.15	11.85
5000	big-M20 + warm start	-1.09474 · 105	3600.43	10.17
5000	big-M5 + warm start	-1.09474 · 105	1146.28	0.00
5000	ours	-1.09474 · 105	20.71	0.00
6000	big-M50 + warm start	-1.27542 · 105	3601.39	9.89
6000	big-M20 + warm start	-1.27542 · 105	3600.08	9.81
6000	big-M5 + warm start	-1.27542 · 105	1038.32	0.00
6000	ours	-1.27542 · 105	1.69	0.00
7000	big-M50 + warm start	-1.50512 · 105	3601.02	8.31
7000	big-M20 + warm start	-1.50512 · 105	3600.99	7.98
7000	big-M5 + warm start	-1.50512 · 105	853.59	0.00
7000	ours	-1.50512 · 105	1.70	0.00
Table 116: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-1.59409 · 10^5	3600.67	18.52
3000	big-M20 + warm start	-1.59409 · 10^5	3601.34	17.23
3000	big-M5 + warm start	-1.59409 · 10^5	3600.06	5.92
3000	ours	-1.59409 · 10^5	3600.62	4.68
4000	big-M50 + warm start	-2.17472 · 10^5	3600.08	14.64
4000	big-M20 + warm start	-2.17472 · 10^5	3600.35	14.15
4000	big-M5 + warm start	-2.17472 · 10^5	3600.06	3.77
4000	ours	-2.17472 · 10^5	86.90	0.00
5000	big-M50 + warm start	-2.71885 · 10^5	3600.07	11.16
5000	big-M20 + warm start	-2.71885 · 10^5	3600.19	11.09
5000	big-M5 + warm start	-2.71885 · 10^5	2700.47	0.00
5000	ours	-2.71885 · 10^5	20.75	0.00
6000	big-M50 + warm start	-3.16660 · 10^5	3600.06	9.93
6000	big-M20 + warm start	-3.16660 · 10^5	3600.74	9.92
6000	big-M5 + warm start	-3.16660 · 10^5	1742.17	0.00
6000	ours	-3.16660 · 10^5	21.05	0.00
7000	big-M50 + warm start	-3.72509 · 10^5	3601.61	8.31
7000	big-M20 + warm start	-3.72509 · 10^5	3600.25	8.11
7000	big-M5 + warm start	-3.72509 · 10^5	1466.07	0.00
7000	ours	-3.72509 · 10^5	1.71	0.00
Table 117: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-3.33503 · 10^5	3600.08	18.43
3000	big-M20 + warm start	-3.33503 · 10^5	3600.16	16.05
3000	big-M5 + warm start	-3.33503 · 10^5	3600.22	7.54
3000	ours	-3.33503 · 10^5	3600.25	6.21
4000	big-M50 + warm start	-4.53159 · 10^5	3600.06	14.84
4000	big-M20 + warm start	-4.53159 · 10^5	3600.21	13.44
4000	big-M5 + warm start	-4.53159 · 10^5	3600.22	6.24
4000	ours	-4.53159 · 10^5	3600.95	0.35
5000	big-M50 + warm start	-5.62906 · 10^5	3600.08	11.59
5000	big-M20 + warm start	-5.62906 · 10^5	3601.71	11.22
5000	big-M5 + warm start	-5.62906 · 10^5	3600.13	3.66
5000	ours	-5.62906 · 10^5	31.47	0.00
6000	big-M50 + warm start	-6.57698 · 10^5	3600.14	9.95
6000	big-M20 + warm start	-6.57698 · 10^5	3600.97	9.33
6000	big-M5 + warm start	-6.57698 · 10^5	3600.14	2.91
6000	ours	-6.57698 · 10^5	21.24	0.00
7000	big-M50 + warm start	-7.72166 · 10^5	3601.52	8.31
7000	big-M20 + warm start	-7.72166 · 10^5	3600.30	7.81
7000	big-M5 + warm start	-7.72166 · 10^5	3600.46	2.33
7000	ours	-7.72166 · 10^5	20.37	0.00
Table 118: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-7.41086 · 10^5	3601.32	17.88
3000	big-M20 + warm start	-7.41086 · 10^5	3600.23	14.06
3000	big-M5 + warm start	-7.41086 · 10^5	3600.11	5.94
3000	ours	-7.41086 · 10^5	3600.48	8.32
4000	big-M50 + warm start	-1.00277 · 10^6	3600.44	14.85
4000	big-M20 + warm start	-1.00277 · 10^6	3600.09	11.77
4000	big-M5 + warm start	-1.00277 · 10^6	3600.26	5.04
4000	ours	-1.00277 · 10^6	3600.62	1.66
5000	big-M50 + warm start	-1.24017 · 10^6	3600.27	11.82
5000	big-M20 + warm start	-1.24017 · 10^6	3600.69	10.25
5000	big-M5 + warm start	-1.24017 · 10^6	3600.22	4.34
5000	ours	-1.24017 · 10^6	386.72	0.00
6000	big-M50 + warm start	-1.45406 · 10^6	3600.26	9.75
6000	big-M20 + warm start	-1.45406 · 10^6	3600.07	9.09
6000	big-M5 + warm start	-1.45406 · 10^6	3600.18	3.95
6000	ours	-1.45406 · 10^6	54.13	0.00
7000	big-M50 + warm start	-1.70473 · 10^6	3600.20	8.32
7000	big-M20 + warm start	-1.70473 · 10^6	3600.12	7.68
7000	big-M5 + warm start	-1.70473 · 10^6	3600.04	3.41
7000	ours	-1.70473 · 10^6	21.30	0.00
Table 119: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-2.78507 · 10^6	3600.07	15.95
3000	big-M20 + warm start	-2.78507 · 10^6	3600.04	10.36
3000	big-M5 + warm start	-2.78507 · 10^6	3600.09	3.69
3000	ours	-2.78507 · 10^6	3600.78	9.79
4000	big-M50 + warm start	-3.75192 · 10^6	3600.06	13.19
4000	big-M20 + warm start	-3.75192 · 10^6	3600.11	8.61
4000	big-M5 + warm start	-3.75192 · 10^6	3600.15	3.16
4000	ours	-3.75192 · 10^6	3600.32	3.23
5000	big-M50 + warm start	-4.61867 · 10^6	3600.06	11.32
5000	big-M20 + warm start	-4.61867 · 10^6	3600.79	7.64
5000	big-M5 + warm start	-4.61867 · 10^6	3600.30	2.72
5000	ours	-4.61867 · 10^6	3600.30	0.89
6000	big-M50 + warm start	-5.43816 · 10^6	3600.39	9.75
6000	big-M20 + warm start	-5.43816 · 10^6	3600.05	6.91
6000	big-M5 + warm start	-5.43816 · 10^6	3600.05	2.50
6000	ours	-5.43816 · 10^6	3600.22	0.32
7000	big-M50 + warm start	-6.36757 · 10^6	3600.10	8.20
7000	big-M20 + warm start	-6.36757 · 10^6	3600.28	6.05
7000	big-M5 + warm start	-6.36757 · 10^6	3600.37	2.25
7000	ours	-6.36757 · 10^6	3600.06	0.10
Table 120: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ₂ = 10.0). All baselines use our beam search solution as a warm start.
G.4 Comparison with MOSEK Solver, SubsetSelectionCIO, and L0BNB
G.4.1 Synthetic 1 Benchmark with  $\lambda_{2} = 0.001$
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-4.79406 · 10^5	2	2.46	0.00
100	SubsetSelectCIO + warm start	-4.79406 · 10^5	2	2.72	0.00
100	L0BnB	-2.10878 · 10^6	10	1420.85	0.00
100	MSK persp.	-2.10878 · 10^6	10	932.06	0.00
100	MSK persp. + warm start	-2.10878 · 10^6	10	898.00	0.00
100	Convex MSK opt. persp.	-2.10878 · 10^6	10	0.10	0.00
100	ours	-2.10878 · 10^6	10	0.29	0.00
500	SubsetSelectCIO	-4.79201 · 10^5	2	3.91	0.00
500	SubsetSelectCIO + warm start	-4.79201 · 10^5	2	3.59	0.00
500	L0BnB	-2.12833 · 10^6	10	1754.22	0.00
500	MSK persp.	-2.12833 · 10^6	10	3606.30	0.09
500	MSK persp. + warm start	-2.12833 · 10^6	10	3600.23	0.09
500	Convex MSK opt. persp.	-2.12833 · 10^6	10	6.79	0.01
500	ours	-2.12833 · 10^6	10	0.37	0.00
1000	SubsetSelectCIO	-4.71433 · 10^5	2	4.36	0.00
1000	SubsetSelectCIO + warm start	-4.71433 · 10^5	2	4.39	0.00
1000	L0BnB	-2.11005 · 10^6	10	2114.74	0.00
1000	MSK persp.	-	-	3673.42	-
1000	MSK persp. + warm start	-2.11005 · 10^6	10	3608.90	0.00
1000	Convex MSK opt. persp.	-2.11005 · 10^6	10	54.02	0.01
1000	ours	-2.11005 · 10^6	10	0.58	0.00
3000	SubsetSelectCIO	-4.72136 · 10^5	2	5.51	0.00
3000	SubsetSelectCIO + warm start	-4.72136 · 10^5	2	9.19	0.00
3000	L0BnB	-2.10230 · 10^6	10	2302.34	1.20
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-	-	3600.00	-
3000	Convex MSK opt. persp.	-2.10230 · 10^6	10	1759.45	0.02
3000	ours	-2.10230 · 10^6	10	1.73	0.00
5000	SubsetSelectCIO	-4.62647 · 10^5	2	12.04	0.00
5000	SubsetSelectCIO + warm start	-4.62647 · 10^5	2	12.66	0.00
5000	L0BnB	-2.09635 · 10^6	10	2643.23	5.18
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.09635 · 10^6	10	3760.81	0.05
5000	ours	-2.09635 · 10^6	10	4.22	0.00
Table 121: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.1, λ2 = 0.001).