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Learning Where to Steer: Noise-Space Geometry for Efficient Offline Multi-Objective Optimization with Generative Models

Yuan Lu, Esha Singh, Yi-An Ma, Yusu Wang

arXiv:2609.38920v1cs.LG

TL;DR

Offline multi-objective optimization with generative models must steer samples toward Pareto trade-offs using a fixed dataset, since generated samples are not inherently better than the data. This paper steers initial noise along a few objective-relevant directions and reports the best average hypervolume rank among generative methods across 47 tasks.

  • Problem

    Generative models can sample within the observed data distribution, but their samples are not inherently biased toward better objective values or the Pareto front.

    Source: §1 · p. 1

  • Method

    The method learns a few objective-relevant noise directions and displaces initial noise along them, leaving the sampler unchanged.

    Source: §1 · pp. 2–3

  • Results

    Across 47 tasks and 14 strong baselines, steering with the proposed guidance operators attains the best average hypervolume rank among generative methods.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The results support noise-space steering as an efficient approach for generative MOO within the evaluated benchmark scope.

    Source: §1 · p. 3

    The reference’s distance to the true Pareto front depends on flow generalization beyond the data and cannot be bounded without querying the objectives.

    Source: §2.1 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

Offline multi-objective optimization seeks Pareto-front solutions from a fixed dataset without querying new objective values. Because generative models can reproduce the data distribution without favoring better solutions, they must be steered toward high-quality regions.

Noise-space geometry

  • The scalarized objective in noise space is often sensitive to only a few directions, with its average pullback metric approximately low-rank across nearly all considered datasets.The paper motivates steering along these objective-relevant directions to move generated samples toward the Pareto front.Source: §1 · p. 2

Noise-space steering

  • ASNS steers only initial noise along RFM-estimated leading AGOP eigenvectors, shares cached directions across trade-offs, and leaves the sampler unchanged.The directions are estimated once per task, and the original flow generates the candidate after the noise displacement.Source: §1 · p. 3

Theory

  • The theory shows that the selected directions capture the most squared gradient for their subspace dimension, increase the learned scalarized objective in expectation, and recover the flow's best attainable front up to errors.The stated errors arise from estimation and steering.Source: §1 · p. 3

Guidance operators

  • Two guidance-operator families complement noise steering: data-adaptive operators restrict gradients to learned geometry and the local data manifold, while Pareto-aware operators combine objective gradients into a first-order nondecreasing direction.The operators use information prior guidance discards and do not differentiate through the ODE solve.Source: §1 · p. 3

Results

  • Across 47 tasks and 14 strong baselines, ASNS with guidance attains the best average hypervolume rank among generative methods, while steering alone outranks the best prior generative method at up to an order of magnitude lower sampling cost.Ablations isolate the chosen directions, rank, weighting, and steering strength.Source: §1 · p. 3

2 Preliminaries

The setup defines offline multi-objective optimization from a fixed dataset and frames trade-offs through Pareto quality and scalarization. The problem setting then searches in noise space through a deterministic sampler, with the sampler’s attainable front serving as the reference rather than the unknown true front.

2 Preliminaries

  • Offline MOO maximizes m ≥ 2 objectives over a compact design space using a fixed dataset, with no objective queries and learned proxies used for optimization.The true objectives are reserved for final evaluation.Source: §2 · p. 4
  • A solution set should approximate and spread across the Pareto front, measured jointly by hypervolume; weighted sums over simplex preferences sweep trade-offs.The development is agnostic to the choice of scalarization.Source: §2 · p. 4

2.1 Problem setting

  • Composing proxies with a deterministic noise-to-solution sampler defines each objective on noise, and the method seeks a high scalarized value for each preference weight.The sampler may be a flow-matching model or a diffusion model sampled through its probability flow ODE.Source: §2.1 · p. 4, §2 · p. 4
  • Sweeping finite preference weights, drawing multiple noises per weight, and retaining proxy-nondominated candidates forms S, whose reference is the best front attainable by the sampler.The gap between this reference and the true front depends on flow generalization beyond the data and cannot be bounded without querying the true objectives.Source: §2.1 · pp. 4–5

3 Noise-Space Steering

Noise-space steering exploits a few objective-sensitive directions learned from function values, displacing the initial noise while leaving the sampler unchanged. The paper shows that this steering can improve the scalarized objective, while its benefit beyond the data support depends on the task.

3.1 The objective is low rank in noise space

  • Across 47 tasks, the effective number of steering directions scales as reff ∝ D^0.42, with spectra concentrated on a few directions.Within benchmark families, effective rank changes little as D grows; differences appear mainly between families.Source: §3.1 · p. 5

3.2 Noise-space geometry

  • RFM estimates objective geometry from noise–proxy value pairs without differentiating through the sampler, and cached cross-objective blocks let each new trade-off use one contraction and eigendecomposition.The resulting directions are shared across noise vectors and trajectories, so one precomputation serves every trade-off.Source: §3.2 · pp. 6–7

3.3 Noise-space Steering

  • Under smoothness and bounded curvature, steering has positive expected scalarized-objective gain for 0 < γ < 2γ⋆, with the gain maximized at γ⋆.The gain grows linearly with steering strength while the curvature penalty grows quadratically.Source: §3.3 · p. 7
  • Among r-dimensional subspaces, the active subspace captures the largest expected squared gradient, and truncation bounds what the steering guarantee loses.Displacing along a leading eigenvector raises the true objective in the reported example, unlike a trailing eigenvector or equal-norm random direction.Source: §3.3 · pp. 7–8
  • As displacement grows, hypervolume degrades on ZDT4, saturates on NAS tasks, and increases on RE36, reflecting a task-dependent trade-off between reaching more of the attainable front and proxy error farther from the data.The paper notes that proxies are reliable only near the data, so which effect dominates depends on the task.Source: §3.3 · p. 8

4 Guided Noise-Space Steering

Late-trajectory guidance augments noise-space steering with geometry-adapted proxy-gradient steps, using cached RFM geometry without differentiating through the ODE solve. Data-adaptive operators use objective and sample geometry, while Pareto-aware guidance seeks directions that do not decrease any objective to first order.

Guidance setup

  • Because noise-only steering can stall when its mean gradient leaves the active subspace or the optimum lies beyond reach R, guidance replaces the gradient term from tstart onward.The RFM geometry is computed once per task, independent of noise and trajectory; each guided step uses one proxy gradient and avoids differentiating through the ODE solve.Source: §4 · p. 8

Data adaptive guidance

  • RFM-APG projects guidance onto intermediate-state active directions; RFM-DAMG spectrum-reweights it and projects onto a local-PCA tangent space, skipping that projection when rank is uncertified.Without a certified local rank, RFM-DAMG reduces to spectrum-reweighted guidance.Source: §4 · p. 8

Pareto aware guidance

  • RFM-Cone adjusts the preference direction toward lagging objectives, projects it onto the cone of non-decreasing proxy directions, and uses the MGDA min-norm direction if projection vanishes.Its step size uses shared onset and strength parameters across operators.Source: §4 · p. 8

Pareto-aware guarantees

  • RFM-Cone steps have nonnegative first-order change for every proxy; with Lipschitz proxy gradients, positively aligned objectives increase for small steps while the rest change only at second order.A strictly ascending cone direction exists exactly away from Pareto stationarity; at stationary points, the fallback is zero and the flow is unmodified.Source: §4 · pp. 8–9

Guidance cost and comparison

  • Steering alone adds no gradient evaluation; guided steps add one proxy backpropagation, with RFM-DAMG below ParetoFlow’s cost and RFM-Cone trading cost for the best rank (Table 2).RFM-Cone also requires an active-set projection whose cost grows with the number of objectives.Source: §4 · p. 9

5 Experiments

Across 47 Off-MOO-Bench tasks, noise steering improves generative-method hypervolume rank, and added guidance—especially RFM-Cone—can improve it further. RFM-AN is fastest, while the guided variants span accuracy–cost trade-offs.

Benchmark tasks.

  • Evaluation covers 47 Off-MOO-Bench tasks across six families, with 2–10,184 design dimensions, 2–8 objectives, and continuous, categorical, and mixed-integer spaces.Categorical NAS designs are optimized as continuous logits, and combinatorial tasks are excluded.Source: Benchmark tasks · p. 9, §5 · p. 9

5.2 Results

  • RFM-AN improves overall rank from 13.56 (ParetoFlow+) to 8.93 and outranks PCD (9.64); RFM-Cone is best among generative methods and second overall, beating ParetoFlow⋆ on 36 of 47 tasks.Every RFM variant outranks D(best) overall. RFM-Cone matches or beats PCD at about a third of its sampling cost.Source: Table 1 · p. 10, §5.2 · p. 10

5.1 Experimental Setup

  • Methods are compared using full-set hypervolume and task-family average rank over five seeds, against surrogate and generative baselines.Each method returns 256 solutions; all four RFM variants share a rectified-flow backbone and one geometry precomputation per task, and RFM-AN steers noise before an unmodified ODE solve. Proxy-based methods share other settings, while PCD follows its published configuration.Source: §5.1 · pp. 9–10

5.2 Results

  • Each guidance operator improves RFM-AN’s overall rank, with RFM-Cone gaining most on DTLZ and ZDT; on NAS and MORL, our methods outrank or match every generative baseline in mean HV.The synthetic DTLZ and ZDT fronts lie far outside the data, where the best solutions can lie beyond one steering step and late proxy correction recovers only part of the gap.Source: §5.2 · pp. 10–11
  • Ablations show that leading-eigenvector displacement raises the true objective while trailing-eigenvector or equal-norm random displacement does not, and the gain saturates at a task-dependent step.Soft weighting keeps retained rank insensitive across a wide range, the RFM fit is robust, and one precomputation serves Tchebycheff scalarizations with the same front quality as weighted sums.Source: §5.2 · p. 11
  • RFM-AN is fastest on every timed task, about four times faster than ParetoFlow and 5× to 24× faster than PCD; RFM-Cone trades more cost for the best generative-method rank.RFM-DAMG is faster on average than ParetoFlow with a better HV rank, while both data-adaptive variants dominate generative baselines in rank and mean cost.Source: Table 2 · p. 11, §5.2 · p. 11

6 Conclusion … A.6.1 Validity of the Benchmarks

Noise-space steering learns objective geometry from function values, displaces initial noise while leaving the sampler unchanged, and uses one precomputation across trade-offs. Guidance extends its reach, while guarantees apply to proxies and surrogate search remains stronger on box-supported tasks whose fronts lie far outside the data.

6 Conclusion

  • The method attains the best average rank among generative methods, while steering alone samples fastest of all generative approaches.An RFM learns a few cached steering directions; each candidate requires one noise displacement and one ODE solve. The displacement provably ascends the scalarized proxy in expectation, but guarantees are on proxies; surrogate search remains stronger on some box-supported tasks, and the method requires a deterministic sampler and continuous design representation.Source: §6 · p. 11

A.1 Benchmark Tasks

  • Off-MOO-Bench covers synthetic, neural-architecture-search, multi-objective control, and engineering tasks, while excluding combinatorial tasks requiring discrete generative models.Synthetic tasks include DTLZ1–7 and ZDT1–4, ZDT6; NAS designs are continuous logits decoded by argmax, and MORL designs are full policy vectors. Some engineering tasks include integer variables, and none has a closed-form front.Source: §A.1 · p. 17

A.2 Evaluation Methodology

  • Evaluation uses hypervolume alone and returns 256 solutions per method; the main text reports the full set, while Appendix D.3 reports its better half.The dataset baseline uses the top 256 non-dominated offline designs and marks the level an offline method must exceed to show generalization beyond its data.Source: §A.2 · pp. 17–18

A.3 Hyperparameter and Tuning used for reporting results

  • RFM reporting uses defaults fixed before sensitivity studies rather than task-specific tuning, with MORL adjustments for its high-dimensional tasks.At dimension approximately 10^4, precomputation is compute-bound, so MORL uses fewer samples and AGOP iterations, retains 100 directions, and trains the flow for more epochs.Source: §A.3 · p. 18

A.4 Hardware and Compute

  • Sampling times are measured on shared hardware for 256 solutions, excluding training and oracle evaluation, with RFM precomputation reported separately.Precomputation takes 7.4–22.7 seconds on the six timed tasks and is reused across preference weights.Source: §A.4 · p. 19

A.5 Training Details

  • RFM variants and ParetoFlow+ share task-trained rectified-flow models and objective proxies, while ParetoFlow+ retains ParetoFlow’s sampling scheme to isolate method contributions.Inputs are normalized consistently between training and inference, outputs are mapped back to the original space, and PCD designs are clipped to task bounds before evaluation.Source: §A.5 · pp. 19–20

A.6 Baseline Algorithm Details

  • The flow backbone uses four residual blocks of width 512, and PGD-MOO is excluded because its released code could not run under the experimental setup.Objective proxies are independent two-hidden-layer MLPs of width 2048; the backbone follows PCD’s residual-MLP design with adaptations for rectified-flow velocity prediction. PGD-MOO’s code covers only RE, ZDT, and DTLZ, a limitation also noted in.Source: §A.6 · p. 20

A.6.1 Validity of the Benchmarks

  • Because hypervolume depends on normalization and reference points, the authors rerun methods under common task-specific settings to address benchmark-version differences.Their NSGA-II reruns obtain positive hypervolume on some tasks reported as zero in the PCD paper; GP-based baselines are omitted as costly at this data scale and weaker than DNN surrogates.Source: §A.6.1 · pp. 20–21

A.7 Limitations · B Ablation Studies

The guarantees apply to learned proxies and recovery is relative to the flow’s attainable front, not the true Pareto front. Steering has task-dependent limitations, including a need for deterministic sampling and continuous designs.

A.7 Limitations

  • The guarantees concern learned proxies, and the recovery bound is relative to the best front the flow can generate rather than the true Pareto front.Source: §A.7 · p. 21
  • The best steering strength depends on the task, and the method lags surrogate-based methods when fronts lie far outside the data.Source: §A.7 · p. 21
  • The method requires a deterministic sampler and continuous design representation, excluding combinatorial tasks; planned extensions include adaptive steering strength and discrete generative models.The authors also plan to combine noise-space steering with surrogate-based search for fronts far from the data.Source: §A.7 · p. 21

B.1 Effective rank across tasks · B.2 Controlled Parameter studies · B.2.1 Steering strength and the reach limit

Across 47 tasks, the noise-space effective rank grows sublinearly with noise dimension and stays nearly fixed within task families. Controlled steering-strength sweeps reveal task-dependent reach limits, with a safe low-strength recommendation and a per-task tuning rule for sensitivity studies.

B.1 Effective rank across tasks

  • reff ∝ D^0.42 across 47 tasks spanning nearly four orders of magnitude in D, so the fraction of noise space affecting objectives falls to about one percent on MORL.RE uses 1.4 to 5.5 directions at D ≤ 7, NAS uses 9 to 27 at D ≤ 174, and MORL uses 65 and 100 directions at D ≈ 104, with its RFM capped at 100.Source: §B.1 · p. 22, Figure 3 · p. 22
  • Within families, effective rank remains stable as D grows: C-10/MOP stays between 10 and 15, IN-1K/MOP8 at D = 174 remains within its lower-D range, and ZDT stays at 6 to 7.The passage attributes this pattern to each family’s fixed intrinsic dimension of the data manifold learned by the flow.Source: §B.1 · p. 22

B.2 Controlled Parameter studies

  • The controlled studies vary seven noise-space steering hyperparameters, using five fixed-in-advance tasks and comparing steering-only RFM-AN with RFM-Cone guidance.Each setting generates 512 candidates, keeps 256 by proxy-prediction non-dominated sorting, and reports normalized HV over five seeds; the studies anchor per-task γ∗ and r∗ from preliminary sweeps.Source: §B.2 · pp. 22–23
  • A parameter is robust only if every swept value’s seed-paired difference from the anchor has a 95% confidence interval inside ±0.02 normalized HV.The study expects precomputation settings to matter mainly through surrogate error, while steering direction and task-dependent strength are theory-predicted sensitivities.Source: §B.2 · pp. 22–23

B.2.1 Steering strength and the reach limit

  • 15%: ZDT4’s normalized HV falls at γ = 20, while RE36 reaches 1.22× the offline data at γ = 20; NAS-task gains saturate or peak at moderate strengths.IN-1K/MOP8 saturates from γ = 2.5, and C-10/MOP1 peaks near γ = 5; MO-Hopper is insensitive to γ within seed noise under both RFM fits.Source: §B.2.1 · p. 24, Figure 4 · p. 24
  • γ ≤ 2.5 never loses HV on any task; γ = 2.5 is recommended when extra headroom is not expected, while task-specific γ∗ is within 0.016 of the sweep best on every task.The main-results default γ = 10 lies within 0.02 of each task’s best except ZDT4; MO-Hopper uses γ∗ = 10 because its HV does not depend on γ.Source: §B.2.1 · p. 24

B.2.2 Retained rank and hard versus soft weighting · B.2.3 Step length versus shape · B.2.4 Soft exponent

Across retained-rank, step-rescaling, and soft-exponent sweeps, soft weighting preserves hypervolume as weak directions are added and can outperform hard weighting even at equal step length. The experiments support α = 0.3 as a robust soft-steering choice.

B.2.2 Retained rank and hard versus soft weighting

  • Soft weights make HV less sensitive to retained rank because weak directions add little step length or curvature penalty, whereas indicator weights add a full penalty with little gain.The paper recommends soft weighting with the full basis, while reserving indicator weights for a small, well-chosen rank.Source: §B.2.2 · pp. 25–26
  • Past r*, indicator weights lose HV by about 23% on IN-1K/MOP8 and 14% on ZDT4, while soft weights stay within 0.02 of their r* value through each full basis.RFM-Cone reproduces the reported patterns; geometric weights are also flat, while soft weighting beats the geometric control by about 0.16 on RE36 and lies within 0.03 of it on the NAS tasks.Source: §B.2.2 · p. 25, Figure 5 · p. 25

B.2.3 Step length versus shape

  • Soft weighting wins through both shorter steps as weak directions are added and better allocation to strong directions even at equal step length.Rescaling removes the longer-step effect, so remaining gaps reflect how the step is distributed across directions.Source: §B.2.3 · p. 26
  • 14% indicator gap on ZDT4 falls to zero after length matching, while IN-1K/MOP8 retains a 2%–6% gap for r ≥ 8 and RE36 retains about 10% at r = 4.The MO-Hopper pattern also matches ZDT4: its gap closes after the hard step is rescaled to the soft step's length.Source: §B.2.3 · p. 26, Figure 6 · p. 26
  • On RE36, soft weighting leads the geometric control by about 17% at r = 3 with or without rescaling; on C-10/MOP1, its 3% lead after rescaling reverses the control’s up-to-2.6% original-step edge.The results show that the geometric control's edge on C-10/MOP1 comes from step length rather than better weighting; RFM-Cone shows the same pattern.Source: §B.2.3 · p. 26

B.2.4 Soft exponent

  • At α = 0.1, full-basis HV drops 12% on IN-1K/MOP8, 4% on ZDT4, and 2%–3% on C-10/MOP1; at α = 3, RE36 loses 34%–39%.These losses match the predicted failures from spreading the step over nearly the full basis or collapsing it onto one direction.Source: §B.2.4 · p. 27, Figure 7 · p. 27
  • 0.1% is the maximum gap from the best on every task for α = 0.3 at r*, and its gap stays within 3% at rmax.At r*, ZDT4 and C-10/MOP1 are flat because r* = 1 leaves nothing to reweight, and MO-Hopper is flat within seed noise; RFM-Cone follows RFM-AN throughout.Source: §B.2.4 · p. 27
  • Any α in [0.3, 1] is within 3% of the best on every task except RE36 at r*, where α = 1 loses 5%; α = 0.3 has worst-case loss no greater than 3%.At α = 0.3, the step uses about 17 of 174 directions on IN-1K/MOP8 and 64 of 300 on MO-Hopper.Source: §B.2.4 · p. 27

B.2.5 Estimating the noise-space metric

The estimated noise-space metric produces stable hypervolume across tested estimation settings, while the smallest tested budget is cheaper and remains close to the anchor. The method estimates the metric from decoded noise samples using an RFM surrogate and AGOP updates.

B.2.5 Estimating the noise-space metric

  • The RFM estimates c Mw from decoded noise samples by alternating kernel-ridge surrogate fits with AGOP updates; n, T, ℓ, and λ affect estimation error, not the steering step.The converged Mahalanobis metrics are the diagonal blocks of c Mw, which converges to the surrogate metric as n grows (Proposition 2).Source: §B.2.5 · p. 28
  • HV changes by at most 1.6% across tested parameter ranges, and MO-Hopper is flat within seed noise under both fits.The ranges are 4× for n, 3× for T, 4× for ℓ, and 100× for λstat; RFM-Cone follows RFM-AN throughout.Source: Figure 8 · p. 28, §B.2.5 · pp. 28–29
  • At n = 5,000 and T = 4, HV stays within 1% of the anchor on every task, with precomputation taking 0.4 s versus 0.95 s.This smallest tested setting supports cheaper precomputation on the MORL tasks.Source: §B.2.5 · p. 29
  • Parameter robustness does not mean the directions are unimportant: changing how the step is spread over them changes HV by up to 23%.Because objectives depend on only a few directions, the RFM recovers them reliably under the tested settings.Source: §B.2.5 · p. 29
  • The recommended defaults are n = 10,000, T = 7, ℓ = 20, and λ = 10−3, with a smaller budget usable when precomputation is expensive.Source: §B.2.5 · p. 29

B.2.6 Robustness to the choice of scalarization · B.2.7 Steering beyond the offline Support on ZDT1

Across scalarizations, cached-noise steering reaches equivalent sweep-level front quality; on ZDT1, it extends generated trade-offs beyond the best offline points toward the true Pareto front.

B.2.6 Robustness to the choice of scalarization

  • Steering raises hypervolume 3.6%–4.6% under weighted sum, Tchebycheff, augmented Tchebycheff, and softmin-Tchebycheff, using the same cached precomputation without refitting.The top-1 arm climbs, trailing and random directions show no systematic gain, and the reversed direction worsens the objective; the four scalarizations reach hypervolume within one percentage point of one another.Source: §B.2.6 · p. 30, Figure 9 · p. 30
  • Under nonsmooth Tchebycheff variants, the soft arm stalls near step 0.5 with standard deviation 0.24, while softmin-Tchebycheff keeps climbing with standard deviation 0.05.Theorem 1’s ascent guarantee assumes smoothness; the top-1 arm remains consistent even under the nonsmooth scalarizations.Source: §B.2.6 · p. 30
  • Cache assembly requires no refit or flow evaluation and costs less than 0.2% of refitting—about 1 ms versus 0.56 s per weight—while weighted-sum assembly matches fixed-feature refitting within 0.02%.The cache recombines per-objective gradients for nonlinear scalarizations as well as weighted sums.Source: Figure 10 · p. 31, §B.2.6 · p. 31
  • A full refit leads in average hypervolume by 3.1% for weighted sum and 4.8%–6.6% for the Tchebycheff family; cache quality matches or beats refitting at balanced weights but trails near simplex ends.Shared cached features fit mixtures of objectives, whereas a refit tailors features to one weight.Source: Figure 10 · p. 31, §B.2.6 · pp. 31–32
  • All 24 scalarization comparisons fall within 2% of weighted-sum hypervolume; 21 confidence intervals do too, while ZDT3 at the larger budget favors weighted sum by 1.6%–1.9%.On ZDT tasks, every scalarization finishes 1%–8% above the best offline designs; on DTLZ2, results reach 99.3%–99.6% of the offline best.Source: Figure 11 · p. 32, §B.2.6 · p. 32
  • On ZDT, larger steering budgets raise hypervolume 2%–3% and lower IGD 15%–37%; on DTLZ2, hypervolume rises 0.2% while IGD worsens about 34%.The larger budget pushes more points to the design boundary, where ZDT optima lie, but crowds points toward the edges of DTLZ2’s interior front.Source: §B.2.6 · pp. 32–33

B.2.7 Steering beyond the offline Support on ZDT1

  • On ZDT1, even steering budget 0.3 extends the generated frontier below the best offline points; budget 1 moves it further toward the true Pareto front without trajectory guidance.The plotted objectives use the original minimization convention.Source: Figure 12 · p. 33, §B.2.7 · p. 33

B.3 Guidance Ablations … C.3.1 Assumptions

The appendix evaluates guidance over noise-space steering, describes how steering reuses cached noise-space geometry, and states the smoothness conditions and limitations of its expected-ascent guarantee.

B.3 Guidance Ablations

  • At default strength, RFM-Cone raises HV significantly on five of eight tasks by 0.3–1.5 pp, reaching 6.6 pp on ZDT2 at 4×.On the other three tasks it stays within 0.6 pp of steering alone up to 2×; significant losses appear only at 4×, on MO-Swimmer (3.3 pp) and IN-1K/MOP8. This follows the predicted second-order turnover.Source: Figure 13 · p. 34, §B.3 · p. 34
  • At default strength, RFM-DAMG gains 1.7 pp on ZDT2, while RFM-APG loses 3.0 pp on ZDT4 at 2×; RFM-Cone raises 50th-percentile HV on five tasks by 1.1–2.1 pp.At default strength, guidance improves over steering alone in 78% of task-seed pairs for RFM-Cone, 72% for RFM-APG, and 65% for RFM-DAMG; all remain above 60% at every strength. Doubling guidance raises HV further on five of eight tasks without significant loss on any, while the main results use the conservative default.Source: §B.3 · p. 34, Figure 15 · p. 36, Figure 16 · p. 37

C.1 Notation and setup

  • Because the flow sampler is deterministic, each objective can be treated as a function of its seed, and the method optimizes a scalarized proxy objective in noise space.The setting is offline: objectives are not queried, and proxies are trained on the fixed dataset.Source: §C.1 · p. 35

Noise-space geometry.

  • The active frame uses the fewest sensitivity directions that retain all but an εrank-fraction of objective sensitivity; when effective rank is small relative to D, only a few noise directions matter.The discarded spectral tail is the truncation error bounded in Proposition 3.Source: Noise-space geometry · p. 36

C.2 The steering algorithm

  • A one-time RFM precomputation estimates objective geometry from prior seeds, and each weight reuses its cache for a noise displacement followed by an unmodified ODE solve.For each new weight, the algorithm assembles the metric, computes its eigendecomposition, orients directions using cached Pearson correlations, then displaces and decodes the noise.Source: §C.2 · p. 37

C.3 First-order guarantee

  • Under smoothness of the noise-space objective, a steering step provably increases the scalarized objective in expectation, and the AGOP frame is the best rank-r frame for that step.The guarantee holds for any fixed sign pattern; Pearson-based sign selection targets the pattern that maximizes the bound, but is not an assumption.Source: §C.3 · pp. 37–38

C.3.1 Assumptions

  • Theorem 1 assumes a twice continuously differentiable noise-space objective with square-integrable gradient and bounded curvature, but assumes neither proxy fidelity nor gradient signal in the active subspace at every seed.Bounded derivatives of the flow and proxies support the smoothness condition; a smooth proxy approximation can handle LeakyReLU. The expected gain is positive for a small enough step unless the mean gradient is orthogonal to the active subspace, and individual seeds can also have nearly orthogonal gradients.Source: §C.3.1 · p. 38

C.3.2 Steering ascends the scalarized objective · C.3.3 The active subspace is the optimal steering subspace

Steering along weighted active directions increases the scalarized objective in expectation when its step size is admissible, and the active subspace captures the most expected squared gradient among rank-r subspaces. These guarantees are first-order and global over the noise distribution, with caveats for sign errors, orthogonal mean gradients, and decoder reach.

C.3.2 Steering ascends the scalarized objective

  • Steering with the population-optimal signs increases the scalarized objective in expectation whenever the mean gradient has a component in the weighted active directions.For hard cutoff weights, the guarantee holds for 0 < γ < 2γ⋆; the bound is vacuous when the mean gradient is orthogonal to the active subspace.Source: §C.3.2 · pp. 39–40
  • The population Pearson sign maximizes expected gain, while its empirical cached estimate converges to it as sample size grows when the mean directional gradient is nonzero.At finite sample size, the empirical and population signs can disagree.Source: §C.3.2 · p. 40
  • The expected ascent gain depends on how much weighted mean gradient the active directions capture, while the sign selects the direction of movement and γ sets its size.The guarantee is first-order and local; because the curvature bound is worst-case, it explains ascent but does not prescribe γ, which is set empirically.Source: §C.3.2 · p. 40

C.3.3 The active subspace is the optimal steering subspace

  • Among all rank-r subspaces, the active subspace span(Uw) captures the maximum expected squared gradient, equal to the sum of the top-r eigenvalues of Mw.When the rth and (r+1)th eigenvalues differ, this maximizing span is unique.Source: §C.3.3 · p. 40
  • The ascent guarantee and subspace result together make the top-r eigenvectors of Mw the directions that both capture the most expected squared gradient and support ascent.The directions and sign are global averages from Mw and the cache, rather than directions or signs tailored to each noise vector.Source: §C.3.3 · p. 41
  • Sign errors degrade gracefully: steering still ascends if mis-signed directions carry less than half the weighted mean gradient.High-signal directions are where the Pearson sign is most reliable.Source: §C.3.3 · p. 41
  • The gain vanishes when the mean gradient is orthogonal to the active subspace, and steering displacement must remain within the range where the decoder works reliably.The degenerate case motivates complementary operators in Section C.6.Source: §C.3.3 · p. 41

C.4 Estimation and truncation of the metric

The empirical AGOP estimates the population metric of the surrogate gradients, while weighting and truncation determine how estimation error affects the steering directions. The total metric error also includes RFM approximation and proxy gaps, which are not removed by more noise samples.

Estimation and truncation of the metric

  • The empirical AGOP converges to the surrogate-gradient population metric at the Monte Carlo rate O(n^-1/2).The consistency result treats the surrogate as fixed and estimates its metric from n noise samples.Source: §C.4 · pp. 41, 43
  • Soft weighting yields steering-weight error O_p(n^-min(α,1)/2) without requiring an eigengap.For α ≥ 1 the rate is O_p(n^-1/2); for 0 < α < 1 it is O_p(n^-α/2).Source: §C.4 · p. 42
  • Hard top-r truncation has subspace error O_p(n^-1/2/∆r), so stability depends on a positive gap between the retained and discarded eigenvalues.Near-ties can make the retained subspace rotate substantially under small perturbations.Source: §C.4 · pp. 42–43
  • Total metric error separates Monte Carlo error, assumed RFM gradient approximation error, and an offline proxy gap that cannot be closed without objective queries.The Monte Carlo term vanishes as n grows and the RFM term as the surrogate improves; the proxy gap remains irreducible without additional assumptions.Source: §C.4 · p. 43

C.4.1 Truncation error of the active subspace … D Additional Results

This appendix analyzes active-subspace truncation and noise-space guidance, including conditions for recovering the attainable front by sweeping trade-off weights. It also examines preconditioned and Pareto-aware guidance operators, their effects on objectives, and when strict ascent is possible or guidance stops.

C.4.1 Truncation error of the active subspace

  • The expected squared gradient invisible to the retained subspace equals the discarded spectral tail, which the effective-rank rule keeps below an εrank fraction.For soft weighting, the decomposition remains valid and is gap-free; weak directions add only a negligible amount to the curvature penalty.Source: §C.4.1 · p. 44

C.5 Pareto-front recovery

  • As steering strength grows, the flow may reconstruct less reliably because steered noise moves away from the prior region, and this enters recovery through reach and proxy errors.For the hard cutoff, the step size is γ√r; the passage characterizes reliability degradation as smooth rather than a hard limit.Source: §C.5 · p. 45

C.5.1 Pareto-front recovery as distance to an oracle

  • Under the stated front-curvature and smoothness conditions, weight sweeping yields an approximate Pareto front, with its error governed by truncation, reach, proxy, and grid errors.With an exact proxy, full-rank steering, no in-subspace reach shortfall, and a dense weight grid, the recovered front converges in Hausdorff distance to the target front.Source: §C.5.1 · pp. 45, 47
  • The recovery guarantee for linear weights covers the front reached by supporting hyperplanes, while steering also works with smooth scalarizations beyond linear weights.The passage reports that all four tested scalarizations ascend and reach the same front quality; extending the front-recovery theorem beyond linear scalarization is left to future work.Source: §C.5.1 · p. 48

C.6 Guided Noise-Space RFM Steering

  • Guided steering adds a proxy-gradient term to the ODE velocity to address gradients outside the active subspace and per-weight optima beyond the noise-space step’s reach.The stated operators are RFM-APG and RFM-DAMG, which precondition the scalarized gradient, and RFM-Cone, which combines per-objective gradients into a common ascent direction.Source: §C.6 · pp. 48–49

C.6.1 Preconditioned ascent: adaptive, projected, manifold

  • The preconditioners use AGOP spectra or local data covariance to shape guidance, and any positive-semidefinite preconditioner gives an ascent direction.RFM-APG rebuilds its active subspace at guidance onset, while RFM-DAMG applies the manifold preconditioner after the adaptive one.Source: §C.6.1 · pp. 49–50
  • Adaptive weighting concentrates ascent on data-learned directions with greater objective sensitivity, increasingly so as α grows; flat retained spectra yield ordinary projected ascent.The weighting is strongest when the AGOP spectrum is peaked, indicating a few dominant objective-sensitive directions.Source: §C.6.1 · p. 50
  • RFM-DAMG applies a locally estimated tangent-space projection after adaptive weighting, but the combined nonsymmetric operator ascends only when its gradient inner product is nonnegative.Without local-rank certification, it falls back to adaptive weighting alone, whose ascent guarantee applies directly.Source: §C.6.1 · p. 51
  • Cone projection, with an MGDA min-norm fallback, gives every proxy objective a nonnegative directional derivative; smooth objectives with positive derivative improve for sufficiently small steps.When a derivative is zero, the objective is flat to first order and can lose at most a second-order amount.Source: §C.6.1 · p. 51
  • A strictly ascending direction exists exactly when zero is outside the convex hull of the proxy gradients; the min-norm fallback is zero exactly at Pareto-stationary points.At such points, guidance switches off and decoding continues with the unmodified flow.Source: §C.6.1 · pp. 51–52

D.1 100th Percentile Results · D.2 75th Percentile Results · D.3 50th Percentile Results

Appendix D reports per-task hypervolume results at the 100th, 75th, and 50th percentiles across benchmark families. Its table captions describe how near-best methods and the RFM variant with the most task-best results are marked.

D.1 100th Percentile Results

  • The 100th-percentile hypervolume of the full returned set is reported, and its average ranks are used to compute Table 1.The tables cover MORL, C-10/MOP, IN-1K/MOP, RE, DTLZ, and ZDT.Source: §D.1 · p. 53
  • Table 7 bolds methods within one standard deviation or 5 × 10−4 of the best, underlines the best remaining method, and shades the RFM variant with the most task-best results.Source: Table 7 · p. 53

D.2 75th Percentile Results

  • The 75th-percentile hypervolume tables cover DTLZ, ZDT, C-10/MOP, IN-1K/MOP, MORL, and RE.Source: §D.2 · p. 57
  • Table 14 bolds methods within one standard deviation or 5 × 10−4 of the best, underlines the best remaining method, and shades the RFM variant with the most task-best results.Source: Table 14 · p. 57

D.3 50th Percentile Results

  • The 50th-percentile hypervolume tables cover DTLZ, ZDT, C-10/MOP, IN-1K/MOP, MORL, and RE.Source: §D.3 · p. 60
  • Table 21 bolds methods within one standard deviation or 5 × 10−4 of the best, underlines the best remaining method, and shades the RFM variant with the most task-best results.Source: Table 21 · p. 61
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