Offline multi-objective optimization (MOO) seeks solutions with better objective trade-offs using only a fixed dataset, without querying the objectives. Diffusion models trained on such data have emerged as a promising approach, but their samples are not inherently better than the data and must be steered toward the Pareto front. Existing methods guide or condition every sampling step. We instead act on the initial noise and leave the sampling process unchanged. Across Off-MOO-Bench, we observe that the objectives, as functions of the noise, are sensitive to only a few directions. We estimate these directions once per task via a Recursive Feature Machine using function values alone, and a small cache serves every trade-off, so each candidate costs one noise displacement and one ODE solve. We prove that this displacement increases the learned scalarized objective in expectation, and that sweeping trade-offs recovers the flow's attainable front up to proxy and steering errors. With additional guidance, for which we introduce novel data-adaptive and Pareto-aware operators, our method attains the best average hypervolume rank among generative methods on 47 tasks, at comparable or lower sampling cost. Steering alone outranks the best prior generative method at a fraction of its sampling cost.
Figures & tables
Figure 1 : The noise-space objective is low rank. (a) Spectrum of Mw on IN-1K/MOP8. A handful of directions carry nearly all the sensitivity. (b) Displacing along them raises the true score. A trailing or random direction of equal norm does not.
Figure 2 : Noise-space RFM steering. (Left, once per task.) Random noise is decoded by the flow Φθ and scored by the proxies f^ , and the RFM recovers from these pairs a metric whose few leading directions are the ones that move the objectives. (Right, per trade off w ) . A noise vector is displaced along the cached directions for w and decoded by the unmodified flow in one ODE solve, with optional late guidance (Section 4 ).
Methods
DTLZ (7)
ZDT (5)
C-10/MOP (9)
IN-1K/MOP (9)
MORL (2)
RE (15)
All Tasks (47)
D (best)
12.43 ± 0.32
13.00 ± 0.45
10.58 ± 0.51
6.38 ± 0.10
4.10 ± 0.89
16.80 ± 0.27
12.02 ± 0.13
E2E
3.34 ± 0.61
5.16 ± 0.62
12.41 ± 0.55
11.38 ± 1.06
17.40 ± 1.95
8.37 ± 1.20
9.01 ± 0.59
E2E + GradNorm
9.71 ± 0.76
13.32 ± 0.59
13.80 ± 1.24
17.60 ± 0.41
7.70 ± 3.78
13.04 ± 0.67
13.37 ± 0.46
E2E + PcGrad
4.97 ± 1.19
7.56 ± 1.82
11.94 ± 0.36
12.42 ± 0.75
14.90 ± 3.45
6.05 ± 0.22
8.78 ± 0.33
MH
6.00 ± 0.96
4.88 ± 1.23
12.26 ± 1.68
13.58 ± 0.38
11.20 ± 3.56
6.59 ± 0.30
8.94 ± 0.43
MH + GradNorm
9.91 ± 0.45
11.36 ± 0.55
15.56 ± 1.42
16.47 ± 0.56
8.10 ± 6.18
11.37 ± 0.69
12.79 ± 0.56
Table 1 : Average rank ( ↓ ) per task family (number of tasks in parentheses), based on 100th-percentile hypervolume (Section 5.1 ); mean ± std over 5 seeds. Best per column in bold, second best underlined. The last two rows count tasks on which RFM-Cone wins, ties, or loses (W/T/L) against PCD and against ParetoFlow ⋆ , the better of ParetoFlow and ParetoFlow + on each task, with ties within 5×10−4 in hypervolume.
Method
DTLZ2
ZDT4
C-10/MOP1
MO-Hopper
RE21
RE61
Mean
RFM precomputation (one-off, shared by all RFM variants)
7.6 ± 0.1
7.4 ± 0.0
7.5 ± 0.1
22.7 ± 0.1
7.4 ± 0.1
7.5 ± 0.0
10.0
ParetoFlow
7.4 ± 0.1
8.9 ± 2.0
7.9 ± 0.1
10.8 ± 0.2
8.6 ± 0.3
11.8 ± 0.2
9.2
ParetoFlow+
12.4 ± 0.2
12.2 ± 0.2
11.1 ± 0.2
16.2 ± 0.5
13.5 ± 0.1
19.3 ± 0.1
14.1
PCD
6.8 ± 1.5
7.6 ± 2.3
22.2 ± 5.3
149.4 ± 30.0
22.6 ± 4.9
25.3 ± 4.2
39.0
RFM-AN (ours)
1.3 ± 0.0
1.3 ± 0.1
2.0 ± 0.0
6.3 ± 0.2
1.2 ± 0.4
1.2 ± 0.0
2.2
RFM-APG (ours)
3.3 ± 0.2
4.2 ± 0.4
8.2 ± 0.2
31.0 ± 1.3
2.8 ± 0.1
3.7 ± 0.1
8.9
Table 2 : Sampling wall-clock time in seconds (mean ± std over 5 seeds) to produce the final 256 candidates, same hardware for all methods (see § A.4 ). The first row is the one-off per-task RFM precomputation, cached and shared within seeds and variants and excluded from the sampling times. Best per column in bold.
Appendix figures & tables39 assets
Supplementary material from the paper’s appendix.
Appendix
Task
d
m
∣D∣
Space
Reference point r
DTLZ
DTLZ1
7
3
60,000
Cont.
(−2.2,−2.2,−2.2)
DTLZ2
10
3
60,000
Cont.
(−2.07826,−2.44443,−2.45812)
DTLZ3
10
3
60,000
Cont.
(−2.22594,−2.07323,−2.1453)
DTLZ4
10
3
60,000
Cont.
(−2.29461,−2.91967,−2.63431)
DTLZ5
10
3
60,000
Cont.
(−2.74136,−2.99806,−2.6272)
Appendix
Table 3 : Task metadata and hypervolume reference points for all 47 tasks. Reference points are in normalized maximization coordinates (negated from the evaluator’s minimization convention). d is the number of design variables (categorical variables for NAS, whose logits have dimension D ), ∣D∣ the offline dataset size, and Cont. / Cat. continuous/categorical.
Flow and Proxy Training
Noise-Space Steering and Sampling
Parameter
Flow vθ
Proxy f^a
Parameter
Value(s)
Architecture
Residual MLP
MLP
Kernel
Laplace (Mahalanobis)
Depth
4 residual blocks
2 hidden layers
Bandwidth ℓ
20
Width
512
2048
Ridge λ
10−3
Activation
SiLU
LeakyReLU
RFM samples n
10,000 (5,000)
Normalization
LayerNorm
—
AGOP iterations T
7 (4)
Appendix
Table 4 : Hyperparameters for flow and proxy training (left) and noise-space steering and sampling (right), shared by all tasks. MORL values are given in parentheses where they differ.
Component
RFM / ParetoFlow+
PCD
Generative model
Rectified flow
EDM diffusion
Backbone
Residual MLP
Residual MLP
Hidden width
512
512
Residual blocks
4
4
Normalization
LayerNorm
LayerNorm
Input projection layers
1
2
Appendix
Table 5: Architecture comparison with the released PCD implementation. Main-path layer counts exclude the separate time/noise embedding MLP.
Figure 3 : Effective rank Eq. ( 13 ) of Mw at w=1/m against noise dimension D for all 47 tasks, log-log axes. The dashed line is a least squares fit, reff∝D0.42 , and the dotted line is reff=D .
Task
D
rmax
r∗
γ∗
seed s.d. at anchor
RE36
4
4
3
20
0.01
ZDT4
10
10
1
2.5
<0.01
C-10/MOP1
31
31
1
10
0.02–0.03
IN-1K/MOP8
174
174
4
20
0.01
MO-Hopper
10,184
300
100
10
0.03–0.04
Appendix
Table 6 : Per-task anchor settings for the sensitivity studies. D is the noise dimension and rmax the number of available directions. γ∗ is the largest steering strength within one seed standard deviation of the task’s best HV in preliminary sweeps, and r∗ the rank with the highest mean HV under soft weighting, with ties resolved toward the smaller rank. On MO-Hopper both are the main-result defaults. The last column is the seed standard deviation of normalized HV at these settings.
Figure 4 : Steering strength sweep. Normalized HV against the steering strength γ (top) and against the per-coordinate displacement ∥Δz∥2/D (bottom), mean ± s.d. over five seeds. The dotted line is the offline data. On MO-Hopper, solid and dashed lines are the RFM fits with r=100 , n=10,000 and r=300 , n=40,000 .
Figure 5 : Retained rank sweep for RFM-AN (top) and RFM-Cone (bottom) , mean ± s.d. over five seeds. The vertical line marks r∗ , the anchor rank of Table 6 , and the horizontal dotted line the offline data. Indicator weights lose HV as r grows on every task with room to add directions, while soft and geometric weights stay flat.
Figure 6 : Step length versus shape. Paired difference in normalized HV, soft weighting minus each hard mode, for RFM-AN, mean ± s.d. over five seeds. Positive values favor soft weighting, so a higher curve means that hard mode loses more. Solid lines use the original step, dashed lines rescale the hard step to the length of the soft step.
Figure 7 : Soft exponent. Top, normalized HV against α at r=r∗ (solid) and r=rmax (dashed), mean ± s.d. over five seeds. Bottom, the number of directions the weighting uses, reff of Eq. ( 13 ), at r=rmax . It falls from nearly the full basis at α=0.1 to a single direction at α=3 .
Figure 8 : Estimating the noise-space metric. Normalized HV against the sample budget n , the AGOP iterations T , the bandwidth ℓ , and the ridge λstat (rows), mean ± s.d. over five seeds. The dotted line is the offline data. On MO-Hopper, solid and dashed lines are the fits with r=100 and r=300 .
Figure 9 : Steering ascends under every scalarization. True scalarized objective against step size on ZDT2 (mean ± s.d. over five seeds, same w and base noise). In every panel the leading arms climb, the controls stay flat, and the reversed arm descends, and all four reach the same HV. Only the soft arm under the non-smooth Tch and aug-Tch becomes unreliable at large steps.
Figure 10 : One precomputation serves every scalarization at a fraction of the cost of refitting. (a) HV of cache-based steering minus that of refit-based steering on ZDT2 (mean and 95% CI over five runs, grey band ±0.02 ). The cache matches or beats the refit at the balanced weight and trails it by a few percent elsewhere, similarly for all four scalarizations. (b) Cost per weight on a log scale. The precomputation runs once, a refit takes 0.56 s per weight, and cache assembly takes about a millisecond. Neither path runs the flow to build Mw . (c) Difference between the cache and refit matrices. With features held fixed, the refit reproduces the weighted-sum assembly to about 10−4 , so Eq. ( 8 ) is exact. A free refit differs by 6% to 24% for every scalarization, because it learns new features.
Figure 11 : Every scalarization reaches the same front quality. Hypervolume relative to the best offline designs (top) and IGD to the analytic front (bottom) at two steering budgets, mean and 95% CI over five seeds. The four scalarizations agree within the 2% tolerance on every task. More steering helps on the ZDT tasks, whose optima lie on the design boundary, and hurts IGD on DTLZ2, whose optima lie in its interior.
Figure 12 : Steering beyond the offline front on ZDT1. Best offline points, RFM-AN samples under weighted-sum scalarization at steering budgets ∥Δz∥2/D=0.3 and 1 , and the true Pareto front. Both objectives are minimized. The larger step moves the generated frontier further beyond the offline points and toward the true front.
Figure 13 : Guidance adds to steering and grows with its strength. Change in normalized HV over steering alone against the guidance strength, relative to its default (dotted line), on the five tasks where steering leaves headroom, mean and 95% CI over five seeds. The remaining three tasks are in Figure 14 .
Figure 14 : Guidance strength on all eight tasks. As Figure 13 , including ZDT4, IN-1K/MOP8, and MO-Swimmer.
Figure 15 : Guidance strength at the 50 th percentile. As Figure 14 , for the HV of the better half of the returned set.
Figure 16 : Guidance improves most runs. Fraction of task and seed pairs in which guidance raises HV over steering alone, against the guidance strength. The dashed line is chance.
Methods
MO-Hopper
MO-Swimmer
D (best)
6.33
3.70
E2E
5.56 ± 0.46
2.99 ± 0.10
E2E + GradNorm
6.17 ± 0.15
3.55 ± 0.30
E2E + PcGrad
5.65 ± 0.53
3.31 ± 0.17
MH
6.03 ± 0.22
3.43 ± 0.14
MH + GradNorm
6.07 ± 0.57
3.61 ± 0.15
Appendix
Table 7 : 100th-percentile hypervolume results for MORL. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
C-10/MOP1
C-10/MOP2
C-10/MOP3
C-10/MOP4
C-10/MOP5
C-10/MOP6
C-10/MOP7
C-10/MOP8
C-10/MOP9
D (best)
1.55
1.41
9.58
19.74
45.09
103.73
446.07
5.68
15.02
E2E
0.00 ± 0.00
0.00 ± 0.00
10.50 ± 0.14
22.27 ± 0.24
40.90 ± 2.56
104.16 ± 1.87
408.71 ± 28.08
5.79 ± 0.04
15.47 ± 0.19
E2E + GradNorm
1.22 ± 0.11
1.16 ± 0.08
10.48 ± 0.15
21.93 ± 0.38
39.60 ± 2.35
101.06 ± 3.63
402.82 ± 23.99
5.70 ± 0.09
14.08 ± 1.66
E2E + PcGrad
0.69 ± 0.63
0.46 ± 0.64
10.66 ± 0.09
22.29 ± 0.46
43.90 ± 0.15
102.28 ± 2.54
412.49 ± 33.85
5.79 ± 0.03
14.85 ± 0.41
MH
1.15 ± 0.10
0.99 ± 0.20
10.43 ± 0.17
21.71 ± 0.67
38.84 ± 0.12
104.04 ± 2.22
440.62 ± 13.07
5.82 ± 0.02
15.24 ± 0.41
MH + GradNorm
0.83 ± 0.56
0.94 ± 0.53
10.29 ± 0.21
21.64 ± 0.56
40.19 ± 2.21
97.93 ± 2.09
438.70 ± 16.65
5.36 ± 0.30
14.07 ± 0.75
Appendix
Table 8 : 100th-percentile hypervolume results for MO-NAS (Part 1, C-10/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
IN-1K/MOP1
IN-1K/MOP2
IN-1K/MOP3
IN-1K/MOP4
IN-1K/MOP5
IN-1K/MOP6
IN-1K/MOP7
IN-1K/MOP8
IN-1K/MOP9
D (best)
6.60
7.44
17.59
4.23
4.57
11.36
5.73
14.04
19.66
E2E
5.54 ± 0.22
5.67 ± 0.32
13.96 ± 0.20
4.51 ± 0.20
4.85 ± 0.05
12.49 ± 0.10
4.86 ± 0.09
9.37 ± 0.46
11.40 ± 0.92
E2E + GradNorm
5.33 ± 0.05
6.18 ± 0.08
13.30 ± 0.43
4.12 ± 0.13
4.36 ± 0.05
9.94 ± 0.20
4.75 ± 0.31
8.57 ± 0.23
10.37 ± 0.37
E2E + PcGrad
5.70 ± 0.13
6.27 ± 0.02
14.10 ± 0.29
4.48 ± 0.05
4.65 ± 0.06
11.86 ± 0.14
4.95 ± 0.10
9.54 ± 0.17
11.63 ± 0.14
MH
5.59 ± 0.04
5.96 ± 0.12
13.67 ± 0.08
4.52 ± 0.04
4.76 ± 0.05
11.53 ± 0.12
4.87 ± 0.04
9.43 ± 0.07
11.96 ± 0.31
MH + GradNorm
5.44 ± 0.22
6.28 ± 0.03
13.72 ± 0.45
3.30 ± 0.38
4.45 ± 0.49
8.68 ± 0.34
4.06 ± 0.15
8.71 ± 0.97
13.56 ± 1.91
Appendix
Table 9 : 100th-percentile hypervolume results for MO-NAS (Part 2, IN-1K/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE21
RE22
RE23
RE24
RE25
RE31
RE32
RE33
D (best)
4.18
4.81
4.75
4.60
4.79
10.61
10.61
10.53
E2E
4.26 ± 0.32
4.84 ± 0.00
4.84 ± 0.00
4.28 ± 0.02
4.35 ± 0.00
10.65 ± 0.00
10.63 ± 0.01
10.62 ± 0.00
E2E + GradNorm
4.47 ± 0.07
4.84 ± 0.00
4.81 ± 0.03
4.69 ± 0.15
4.84 ± 0.00
10.61 ± 0.06
10.62 ± 0.01
10.57 ± 0.04
E2E + PcGrad
4.60 ± 0.00
4.84 ± 0.00
4.84 ± 0.00
4.74 ± 0.16
4.84 ± 0.00
10.65 ± 0.00
10.64 ± 0.00
10.60 ± 0.01
MH
4.60 ± 0.00
4.84 ± 0.00
4.84 ± 0.00
4.63 ± 0.25
4.84 ± 0.00
10.65 ± 0.00
10.63 ± 0.00
10.62 ± 0.00
MH + GradNorm
4.41 ± 0.25
4.84 ± 0.00
4.80 ± 0.03
4.49 ± 0.76
4.84 ± 0.00
10.65 ± 0.01
7.88 ± 1.23
10.60 ± 0.02
Appendix
Table 10 : 100th-percentile hypervolume results for RE (Part 1). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
D (best)
10.61
12.43
9.90
17.50
20.98
15.35
8.99
E2E
10.65 ± 0.00
12.45 ± 0.00
9.86 ± 0.07
17.66 ± 0.03
21.06 ± 0.00
15.74 ± 0.01
10.80 ± 0.00
E2E + GradNorm
10.65 ± 0.00
12.44 ± 0.01
9.89 ± 0.00
17.53 ± 0.07
21.05 ± 0.01
15.32 ± 0.14
10.69 ± 0.02
E2E + PcGrad
10.65 ± 0.00
12.44 ± 0.00
9.89 ± 0.01
17.62 ± 0.04
21.06 ± 0.00
15.67 ± 0.01
10.77 ± 0.00
MH
10.64 ± 0.00
12.44 ± 0.00
9.89 ± 0.00
17.61 ± 0.11
21.06 ± 0.00
15.66 ± 0.01
10.78 ± 0.01
MH + GradNorm
10.64 ± 0.00
12.44 ± 0.00
9.88 ± 0.00
17.56 ± 0.03
20.94 ± 0.05
15.55 ± 0.04
10.77 ± 0.01
Appendix
Table 11 : 100th-percentile hypervolume results for DTLZ. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE34
RE35
RE36
RE37
RE41
RE42
RE61
D (best)
9.65
10.22
7.76
5.92
18.99
15.28
105.13
E2E
10.12 ± 0.00
10.52 ± 0.02
10.02 ± 0.12
6.75 ± 0.00
20.85 ± 0.05
22.38 ± 0.36
108.89 ± 0.03
E2E + GradNorm
9.85 ± 0.22
10.38 ± 0.02
9.83 ± 0.24
6.65 ± 0.02
17.52 ± 0.57
21.53 ± 0.26
102.69 ± 2.91
E2E + PcGrad
10.12 ± 0.00
10.58 ± 0.01
10.17 ± 0.22
6.75 ± 0.00
20.89 ± 0.03
22.71 ± 0.09
104.63 ± 0.85
MH
10.11 ± 0.00
10.56 ± 0.02
10.07 ± 0.02
6.75 ± 0.01
20.76 ± 0.04
22.51 ± 0.14
108.77 ± 0.16
MH + GradNorm
9.79 ± 0.07
10.41 ± 0.07
7.39 ± 1.88
6.68 ± 0.03
20.75 ± 0.06
22.24 ± 0.14
105.61 ± 2.98
Appendix
Table 12 : 100th-percentile hypervolume results for RE (Part 2). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D (best)
4.17
4.68
5.15
5.46
4.61
E2E
4.86 ± 0.00
5.71 ± 0.00
5.71 ± 0.03
4.68 ± 0.24
4.80 ± 0.00
E2E + GradNorm
4.16 ± 0.07
5.50 ± 0.14
5.02 ± 0.06
5.09 ± 0.09
4.41 ± 0.08
E2E + PcGrad
4.86 ± 0.00
5.71 ± 0.00
5.74 ± 0.09
4.11 ± 0.12
3.90 ± 1.07
MH
4.85 ± 0.00
5.63 ± 0.01
5.80 ± 0.04
5.14 ± 0.18
4.78 ± 0.00
MH + GradNorm
4.82 ± 0.04
5.63 ± 0.08
5.42 ± 0.15
3.82 ± 0.08
3.32 ± 1.05
Appendix
Table 13 : 100th-percentile hypervolume results for ZDT. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
D (best)
–
–
–
–
–
–
–
E2E
10.65 ± 0.00
12.44 ± 0.00
9.83 ± 0.09
17.65 ± 0.03
21.06 ± 0.00
15.72 ± 0.01
10.80 ± 0.00
E2E + GradNorm
10.64 ± 0.00
12.43 ± 0.02
9.87 ± 0.00
17.35 ± 0.22
21.04 ± 0.02
15.31 ± 0.14
10.67 ± 0.04
E2E + PcGrad
10.64 ± 0.00
12.44 ± 0.00
9.88 ± 0.01
17.47 ± 0.08
21.06 ± 0.00
15.66 ± 0.01
10.77 ± 0.01
MH
10.64 ± 0.00
12.44 ± 0.00
9.87 ± 0.01
17.57 ± 0.13
21.05 ± 0.02
15.01 ± 0.78
10.77 ± 0.01
MH + GradNorm
10.64 ± 0.00
12.43 ± 0.00
9.86 ± 0.01
17.47 ± 0.03
20.89 ± 0.14
15.54 ± 0.04
10.76 ± 0.01
Appendix
Table 14 : 75th-percentile hypervolume results for DTLZ. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D (best)
–
–
–
–
–
E2E
4.86 ± 0.00
5.71 ± 0.00
5.65 ± 0.04
4.52 ± 0.26
4.79 ± 0.00
E2E + GradNorm
3.29 ± 0.03
5.21 ± 0.53
4.34 ± 0.06
4.80 ± 0.05
3.33 ± 1.08
E2E + PcGrad
4.86 ± 0.00
5.71 ± 0.00
5.55 ± 0.36
3.92 ± 0.19
1.97 ± 0.29
MH
4.85 ± 0.00
5.62 ± 0.01
5.77 ± 0.05
4.88 ± 0.11
4.76 ± 0.00
MH + GradNorm
4.82 ± 0.05
5.62 ± 0.08
5.29 ± 0.17
3.48 ± 0.05
3.28 ± 1.04
Appendix
Table 15 : 75th-percentile hypervolume results for ZDT. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
C-10/MOP1
C-10/MOP2
C-10/MOP3
C-10/MOP4
C-10/MOP5
C-10/MOP6
C-10/MOP7
C-10/MOP8
C-10/MOP9
D (best)
–
–
–
–
–
–
–
–
–
E2E
0.00 ± 0.00
0.00 ± 0.00
10.35 ± 0.21
22.10 ± 0.22
40.22 ± 2.23
103.01 ± 2.57
405.35 ± 29.34
5.75 ± 0.05
15.38 ± 0.18
E2E + GradNorm
1.20 ± 0.10
1.16 ± 0.08
10.02 ± 0.12
21.87 ± 0.35
38.07 ± 2.14
98.91 ± 3.63
402.47 ± 23.92
5.51 ± 0.09
13.78 ± 1.85
E2E + PcGrad
0.68 ± 0.62
0.26 ± 0.58
10.48 ± 0.24
22.08 ± 0.28
42.50 ± 0.80
101.72 ± 2.49
411.63 ± 33.26
5.77 ± 0.02
14.58 ± 0.32
MH
0.91 ± 0.52
0.98 ± 0.20
10.22 ± 0.32
21.36 ± 1.02
38.41 ± 0.62
101.89 ± 1.48
435.49 ± 6.14
5.77 ± 0.05
15.02 ± 0.42
MH + GradNorm
0.82 ± 0.55
0.91 ± 0.51
9.94 ± 0.43
21.22 ± 0.65
37.69 ± 1.26
96.99 ± 1.79
408.37 ± 24.70
5.27 ± 0.26
12.19 ± 1.39
Appendix
Table 16 : 75th-percentile hypervolume results for MO-NAS (Part 1, C-10/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
IN-1K/MOP1
IN-1K/MOP2
IN-1K/MOP3
IN-1K/MOP4
IN-1K/MOP5
IN-1K/MOP6
IN-1K/MOP7
IN-1K/MOP8
IN-1K/MOP9
D (best)
–
–
–
–
–
–
–
–
–
E2E
5.52 ± 0.23
5.66 ± 0.33
13.91 ± 0.20
4.49 ± 0.21
4.84 ± 0.04
11.83 ± 0.37
4.75 ± 0.15
9.31 ± 0.48
11.33 ± 0.91
E2E + GradNorm
5.26 ± 0.02
6.16 ± 0.09
13.23 ± 0.36
4.09 ± 0.12
4.31 ± 0.08
9.66 ± 0.14
4.54 ± 0.36
8.56 ± 0.23
10.29 ± 0.40
E2E + PcGrad
5.68 ± 0.13
6.25 ± 0.02
14.06 ± 0.26
4.44 ± 0.09
4.61 ± 0.04
11.77 ± 0.24
4.85 ± 0.11
9.52 ± 0.18
11.58 ± 0.15
MH
5.56 ± 0.05
5.87 ± 0.25
13.58 ± 0.04
4.48 ± 0.03
4.72 ± 0.03
10.98 ± 0.90
4.76 ± 0.03
9.33 ± 0.09
11.81 ± 0.44
MH + GradNorm
5.43 ± 0.23
6.22 ± 0.06
13.70 ± 0.43
3.22 ± 0.39
4.41 ± 0.50
8.52 ± 0.34
3.83 ± 0.24
8.66 ± 0.96
13.52 ± 1.90
Appendix
Table 17 : 75th-percentile hypervolume results for MO-NAS (Part 2, IN-1K/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
MO-Hopper
MO-Swimmer
D (best)
–
–
E2E
4.98 ± 0.19
2.55 ± 0.29
E2E + GradNorm
5.20 ± 0.36
3.17 ± 0.30
E2E + PcGrad
4.95 ± 0.21
2.79 ± 0.06
MH
5.35 ± 0.20
2.95 ± 0.14
MH + GradNorm
5.57 ± 0.46
2.96 ± 0.09
Appendix
Table 18 : 75th-percentile hypervolume results for MORL. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE21
RE22
RE23
RE24
RE25
RE31
RE32
RE33
D (best)
–
–
–
–
–
–
–
–
E2E
4.25 ± 0.31
4.75 ± 0.21
4.72 ± 0.27
4.27 ± 0.03
4.23 ± 0.14
10.61 ± 0.06
10.63 ± 0.01
10.61 ± 0.01
E2E + GradNorm
4.46 ± 0.07
4.84 ± 0.00
4.16 ± 0.03
4.31 ± 0.02
4.55 ± 0.02
9.93 ± 0.03
10.18 ± 0.33
9.30 ± 0.74
E2E + PcGrad
4.59 ± 0.00
4.84 ± 0.00
4.84 ± 0.00
3.97 ± 0.06
4.55 ± 0.06
10.63 ± 0.03
10.64 ± 0.00
9.70 ± 0.85
MH
4.59 ± 0.00
4.84 ± 0.00
4.84 ± 0.00
4.52 ± 0.39
4.54 ± 0.67
10.65 ± 0.00
10.63 ± 0.00
10.60 ± 0.01
MH + GradNorm
4.35 ± 0.33
4.40 ± 0.60
4.75 ± 0.04
4.21 ± 0.96
4.82 ± 0.02
10.49 ± 0.35
7.66 ± 1.09
10.58 ± 0.02
Appendix
Table 19 : 75th-percentile hypervolume results for RE (Part 1). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE34
RE35
RE36
RE37
RE41
RE42
RE61
D (best)
–
–
–
–
–
–
–
E2E
10.09 ± 0.01
10.52 ± 0.02
9.87 ± 0.12
6.71 ± 0.03
20.74 ± 0.10
22.27 ± 0.39
108.63 ± 0.10
E2E + GradNorm
9.57 ± 0.73
10.38 ± 0.02
6.06 ± 0.68
6.62 ± 0.02
17.41 ± 0.57
21.43 ± 0.20
100.45 ± 3.18
E2E + PcGrad
10.08 ± 0.02
10.58 ± 0.00
10.03 ± 0.22
6.72 ± 0.01
20.79 ± 0.04
22.66 ± 0.10
103.26 ± 1.44
MH
10.08 ± 0.01
10.55 ± 0.02
9.93 ± 0.04
6.70 ± 0.02
20.62 ± 0.05
22.44 ± 0.08
108.31 ± 0.47
MH + GradNorm
9.73 ± 0.03
10.35 ± 0.03
6.54 ± 1.27
6.65 ± 0.02
20.63 ± 0.06
22.13 ± 0.16
105.13 ± 3.20
Appendix
Table 20 : 75th-percentile hypervolume results for RE (Part 2). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
D (best)
–
–
–
–
–
–
–
E2E
10.64 ± 0.00
12.43 ± 0.01
9.73 ± 0.12
17.61 ± 0.03
21.05 ± 0.01
15.70 ± 0.02
10.79 ± 0.00
E2E + GradNorm
10.62 ± 0.01
12.40 ± 0.08
9.83 ± 0.02
17.16 ± 0.30
20.83 ± 0.40
15.16 ± 0.22
10.59 ± 0.08
E2E + PcGrad
10.62 ± 0.01
12.44 ± 0.00
9.85 ± 0.02
17.25 ± 0.17
21.06 ± 0.00
15.62 ± 0.02
10.61 ± 0.08
MH
10.62 ± 0.01
12.44 ± 0.01
9.84 ± 0.03
17.38 ± 0.42
21.03 ± 0.04
14.31 ± 0.49
10.77 ± 0.02
MH + GradNorm
10.63 ± 0.01
12.42 ± 0.01
9.80 ± 0.06
17.25 ± 0.14
20.83 ± 0.16
15.46 ± 0.06
10.69 ± 0.03
Appendix
Table 21 : 50th-percentile hypervolume results for DTLZ. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D (best)
–
–
–
–
–
E2E
4.86 ± 0.00
5.70 ± 0.00
5.18 ± 0.64
4.40 ± 0.32
4.78 ± 0.01
E2E + GradNorm
3.08 ± 0.10
5.07 ± 0.61
4.21 ± 0.07
4.65 ± 0.02
3.02 ± 0.90
E2E + PcGrad
4.86 ± 0.00
5.70 ± 0.00
5.06 ± 0.50
3.80 ± 0.20
1.82 ± 0.22
MH
4.84 ± 0.01
5.52 ± 0.11
5.68 ± 0.07
4.72 ± 0.11
4.75 ± 0.00
MH + GradNorm
4.49 ± 0.79
5.56 ± 0.10
4.94 ± 0.26
3.22 ± 0.13
3.21 ± 0.99
Appendix
Table 22 : 50th-percentile hypervolume results for ZDT. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
C-10/MOP1
C-10/MOP2
C-10/MOP3
C-10/MOP4
C-10/MOP5
C-10/MOP6
C-10/MOP7
C-10/MOP8
C-10/MOP9
D (best)
–
–
–
–
–
–
–
–
–
E2E
0.00 ± 0.00
0.00 ± 0.00
10.10 ± 0.25
21.02 ± 0.32
38.49 ± 2.38
99.40 ± 2.67
400.60 ± 24.57
5.72 ± 0.05
15.22 ± 0.22
E2E + GradNorm
1.20 ± 0.10
1.12 ± 0.08
9.66 ± 0.14
20.43 ± 0.69
37.02 ± 2.45
94.45 ± 4.46
377.97 ± 19.27
4.88 ± 0.39
13.50 ± 1.79
E2E + PcGrad
0.66 ± 0.61
0.26 ± 0.57
9.99 ± 0.17
20.50 ± 0.49
40.11 ± 2.06
97.21 ± 2.04
396.72 ± 30.39
5.72 ± 0.06
14.41 ± 0.32
MH
1.10 ± 0.04
0.95 ± 0.16
9.74 ± 0.31
19.56 ± 1.18
37.19 ± 0.65
99.48 ± 2.29
409.60 ± 20.37
5.74 ± 0.07
14.87 ± 0.50
MH + GradNorm
0.77 ± 0.55
0.90 ± 0.50
9.64 ± 0.53
20.41 ± 0.55
35.25 ± 3.28
94.56 ± 1.45
382.77 ± 13.02
5.17 ± 0.27
10.98 ± 1.23
Appendix
Table 23 : 50th-percentile hypervolume results for MO-NAS (Part 1, C-10/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
IN-1K/MOP1
IN-1K/MOP2
IN-1K/MOP3
IN-1K/MOP4
IN-1K/MOP5
IN-1K/MOP6
IN-1K/MOP7
IN-1K/MOP8
IN-1K/MOP9
D (best)
–
–
–
–
–
–
–
–
–
E2E
5.49 ± 0.24
5.63 ± 0.30
13.84 ± 0.20
4.44 ± 0.21
4.64 ± 0.19
9.89 ± 0.71
4.58 ± 0.14
9.03 ± 0.45
10.89 ± 0.83
E2E + GradNorm
5.22 ± 0.03
6.13 ± 0.08
12.91 ± 0.30
3.96 ± 0.11
4.28 ± 0.10
9.42 ± 0.32
4.39 ± 0.36
8.24 ± 0.23
10.06 ± 0.74
E2E + PcGrad
5.55 ± 0.24
6.17 ± 0.05
13.94 ± 0.34
4.27 ± 0.14
4.43 ± 0.25
10.64 ± 1.21
4.75 ± 0.12
9.42 ± 0.22
11.49 ± 0.09
MH
5.51 ± 0.07
5.79 ± 0.22
13.37 ± 0.10
4.30 ± 0.19
4.46 ± 0.21
9.44 ± 0.30
4.63 ± 0.10
9.21 ± 0.09
11.74 ± 0.42
MH + GradNorm
5.37 ± 0.25
6.15 ± 0.05
13.40 ± 0.48
3.19 ± 0.37
4.38 ± 0.49
8.28 ± 0.19
3.65 ± 0.33
8.28 ± 0.88
13.28 ± 2.17
Appendix
Table 24 : 50th-percentile hypervolume results for MO-NAS (Part 2, IN-1K/MOP). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
MO-Hopper
MO-Swimmer
D (best)
–
–
E2E
4.87 ± 0.13
2.20 ± 0.43
E2E + GradNorm
4.93 ± 0.27
2.89 ± 0.14
E2E + PcGrad
4.85 ± 0.21
2.58 ± 0.06
MH
5.00 ± 0.15
2.68 ± 0.11
MH + GradNorm
5.41 ± 0.39
2.66 ± 0.18
Appendix
Table 25 : 50th-percentile hypervolume results for MORL. For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE21
RE22
RE23
RE24
RE25
RE31
RE32
RE33
D (best)
–
–
–
–
–
–
–
–
E2E
4.25 ± 0.31
4.46 ± 0.21
4.71 ± 0.27
4.27 ± 0.03
4.14 ± 0.28
10.56 ± 0.12
10.63 ± 0.01
10.60 ± 0.01
E2E + GradNorm
4.33 ± 0.28
4.82 ± 0.02
3.74 ± 0.05
4.20 ± 0.03
4.36 ± 0.05
9.78 ± 0.07
9.84 ± 0.39
8.19 ± 0.84
E2E + PcGrad
4.59 ± 0.00
4.84 ± 0.00
4.84 ± 0.00
3.53 ± 0.05
4.27 ± 0.21
10.39 ± 0.55
10.64 ± 0.00
7.85 ± 0.42
MH
4.59 ± 0.00
4.77 ± 0.15
4.84 ± 0.00
4.46 ± 0.36
4.29 ± 0.78
10.65 ± 0.00
10.63 ± 0.00
10.58 ± 0.03
MH + GradNorm
4.01 ± 0.92
4.02 ± 0.87
4.09 ± 0.14
3.39 ± 0.53
4.58 ± 0.24
10.39 ± 0.57
7.41 ± 0.94
9.50 ± 1.95
Appendix
Table 26 : 50th-percentile hypervolume results for RE (Part 1). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Methods
RE34
RE35
RE36
RE37
RE41
RE42
RE61
D (best)
–
–
–
–
–
–
–
E2E
10.07 ± 0.02
10.49 ± 0.03
9.66 ± 0.17
6.68 ± 0.04
20.58 ± 0.11
21.83 ± 0.58
108.30 ± 0.18
E2E + GradNorm
7.21 ± 1.27
10.37 ± 0.01
5.42 ± 0.03
6.59 ± 0.02
16.90 ± 1.03
19.65 ± 2.26
96.02 ± 3.08
E2E + PcGrad
10.07 ± 0.02
10.57 ± 0.00
9.65 ± 0.26
6.69 ± 0.01
20.69 ± 0.05
22.55 ± 0.15
101.10 ± 2.70
MH
10.06 ± 0.01
10.55 ± 0.02
9.74 ± 0.09
6.67 ± 0.03
20.48 ± 0.07
22.35 ± 0.12
107.94 ± 0.65
MH + GradNorm
8.69 ± 0.80
10.06 ± 0.32
4.96 ± 1.15
6.62 ± 0.03
20.48 ± 0.10
21.47 ± 0.14
104.36 ± 4.19
Appendix
Table 27 : 50th-percentile hypervolume results for RE (Part 2). For each task, methods within one standard deviation (or 5×10−4 ) of the best are bolded and the best of the remaining methods is underlined; the shaded row marks the RFM variant with the most task-best results overall.
Offline multi-objective optimization (Offline MOO) seeks Pareto-optimal designs from static datasets without additional environment interactions. Existing generative methods typically guide sampling with external surrogate or preference models, which adds training complexity and may provide unreliable guidance. We propose ParetoPilot, a plug-and-play method that guides designs to Pareto front at inference time using a pre-trained conditional diffusion model without any surrogate. ParetoPilot introduces an Infer-Perturb-Guide (IPG) engine within the reverse diffusion process. IPG first infers the individual conditional target for each sample in the batch by aligning its conditional and unconditional predictions. It then perturbs these targets collectively across the batch, balancing convergence toward the Pareto front and diversity among samples. Finally, the engine guides the generative trajectory toward the Pareto front by injecting these perturbed targets via standard Classifier-Free Guidance (CFG). Experiments on 51 tasks demonstrate that ParetoPilot achieves the best overall ranking among 16 methods and competitive hypervolume improvement.
Ruiqing Sun, Sen Yang, Dawei Feng +3
College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China · Bioinformatics Center of AMMS, Beijing 100080, China
Offline multi-objective optimization requires not only moving the objective vectors of candidate designs toward the Pareto front, but also distributing them effectively along it. Generative methods have recently emerged as a natural approach because they learn a distribution over feasible designs while allowing generation to be steered toward promising designs. Existing methods, however, largely retain classical sample-wise guidance strategies, leaving the distribution-level modeling capability of generative methods underused. We propose ParetoTransport, a training-free guidance method for pre-trained flow-matching models that explicitly specifies and refines a population-level distribution in objective space. ParetoTransport guides a flow-matching sampler to iteratively transport the empirical offline distribution toward the Pareto front, with Wasserstein matching to intermediate proxy distributions. This directly controls distributional displacement and mass allocation along the front. We establish a convergence result and demonstrate state-of-the-art performance on standard offline MOO benchmarks, extending recent evaluations beyond hypervolume to generational distance, inverted generational distance, and Wasserstein distance.
Stephanie Holly, Sepp Hochreiter, Werner Zellinger
LIT AI Lab and Institute for Machine Learning JKU Linz, Austria
Existing reward alignment methods for diffusion and flow models rely on multi-step stochastic trajectories, making them difficult to extend to deterministic generators. A natural alternative is noise-space optimization, but existing approaches require backpropagation through the generator and reward pipeline, limiting applicability to differentiable settings. To address this, here we present ZeNO (Zeroth-order Noise Optimization), a gradient-free framework that formulates noise optimization as a path-integral control problem, estimable from zeroth-order reward evaluations alone. When instantiated with an Ornstein--Uhlenbeck reference process, the update connects to Langevin dynamics implicitly targeting a reward-tilted distribution. ZeNO enables effective inference-time scaling and demonstrates strong performance across diverse generators and reward functions, including a protein structure generation task where backpropagation is infeasible.