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The Golden Path Hypothesis: Reusable Schedules in Diffusion Caching

Dong Wang, Wenwu Tang, Francesco Corti, Yun Cheng, Lothar Thiele, Olga Saukh

arXiv:2609.39343v1cs.AIcs.LG

TL;DR

Diffusion caching reduces repeated transformer computation, but it is unclear how much a high-quality cache schedule must change across prompts. The paper tests whether prompt-independent schedules can approach prompt-specific quality, analyzing trajectories and caching errors and searching schedules by final-output quality. Shared schedules retain quality on unseen prompts and datasets, while the experiments leave several model and evaluation-scope questions open.

  • Problem

    It remains unclear how much high-quality cache schedules need to vary across prompts, despite offline methods reusing schedules and adaptive methods selecting steps from prompt-dependent features.

    Source: §1 · p. 1

  • Method

    The paper tests the Golden Path Hypothesis through schedule comparisons, trajectory and error-propagation analysis, and searches on a small example set scored by final-output quality.

    Source: §1 · pp. 1, 3

  • Results

    Shared schedules approach prompt-specific quality, and schedules searched on a small example set retain quality on unseen prompts and datasets.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    The experiments support reusing shared schedules and scoring complete schedules by final-output quality because local approximation errors omit effects of earlier state changes.

    Source: §8 · p. 14

    Experiments cover transformer-based image and video models, not non-transformer diffusion models or other modalities, and do not systematically study scoring-example number or diversity.

    Source: §8 · p. 14

Abstract

from arXiv · show

Diffusion caching accelerates generation by replacing transformer computation with cached or predicted features at selected denoising steps. We introduce the Golden Path Hypothesis (GPH): under fixed inference conditions, prompt-independent cache schedules can achieve final-output quality comparable to the best prompt-specific schedules across prompts. We investigate the GPH across ten caching methods, four image and video models, and three cache ratios. Prompt-adaptive methods repeatedly select a small number of schedules, and reusing their most frequent schedules on new prompts closely matches the quality of prompt-specific choices. Exhaustive evaluation of 1.4 million schedules on four examples further identifies prompt-independent schedules that remain competitive on unseen prompts. To explain this transfer, we analyze denoising trajectories and the accumulation of caching errors. Latent-state trajectories exhibit similar structures across datasets and seeds, while an exact error decomposition shows that accumulated effects of earlier errors predict final latent-state error better than local approximation errors. This motivates searching for end-to-end schedules using final-output quality. With only a small set of examples, the resulting golden paths transfer across prompts and datasets, and can be tuned to the desired quality objective, including reconstruction fidelity or perceptual similarity.

1 Introduction

The paper asks how much high-quality cache schedules must vary across prompts and proposes the Golden Path Hypothesis: under fixed inference conditions, prompt-independent schedules can approach prompt-specific quality. It supports this hypothesis with recurring adaptive schedules, transfer to unseen prompts, trajectory analysis, and searches scored by final-output quality.

1 Introduction

  • Diffusion caching replaces repeated full computation with reused or predicted features, while prompt-dependent features motivate adaptive schedule selection.The paper asks how much a high-quality schedule actually needs to change across prompts.Source: §1 · p. 1
  • Across tested settings, shared schedules approach prompt-specific quality; exhaustive search also finds schedules competitive on thousands of unseen prompt–seed runs.The paper reports that prompt-adaptive methods repeatedly select a small number of schedules and that reusing their most frequent schedules closely matches prompt-specific choices.Source: §1 · pp. 2–3
  • Common denoising-trajectory patterns across datasets and seeds within each model support schedule reuse, despite trajectories having different spatial directions.Direction changes are smaller in the middle of denoising than near its ends, and normalized step displacements vary little across prompts over much of denoising.Source: §1 · pp. 2–3
  • Accumulated effects of earlier state changes correlate more strongly with final latent-state error than individual approximation errors.This motivates scoring candidate schedules by final-output quality rather than accumulating local prediction-error costs.Source: §1 · p. 3
  • Searching a small example set with final-output quality objectives finds schedules that retain quality on unseen prompts and datasets.Optimizing perceptual similarity instead of pixel fidelity selects different schedules and improves perceptual similarity to full-compute outputs on new prompts.Source: §1 · p. 3

2 The Golden Path Hypothesis

The Golden Path Hypothesis says that, under fixed inference conditions, a prompt-independent schedule can match the quality of the best prompt-specific schedule across prompts. The paper defines the conditions and comparison used to test that claim.

2 The Golden Path Hypothesis

  • Inference conditions comprise the model, sampler, noise-level sequence, cache ratio, approximation policy, and quality metric.The metric used to rank schedules is the objective; the cache ratio is K/N for K cached steps among N total steps.Source: §2 · p. 3
  • A golden path is a prompt-independent schedule whose final-output quality is comparable to each prompt’s best schedule under the same inference conditions.On new prompts, the comparison is with each prompt’s best evaluated quality.Source: §2 · p. 3
  • The main schedule comparisons use 50 denoising steps and cache ratios 0.58, 0.74, and 0.82, retaining each method’s approximation policy for baseline and reuse comparisons.Search experiments and comparisons against each prompt’s best evaluated schedule use residual reuse.Source: §2 · p. 3
  • Evaluations cover FLUX.1-dev and Qwen-Image on image datasets and HunyuanVideo and Wan2.1 on video datasets.PSNR and SSIM increase as cached outputs approach full-compute references, while LPIPS decreases.Source: §2 · pp. 3–4
  • Coverage measures the fraction of prompts whose schedule quality falls within an allowed metric-unit margin of each prompt’s best evaluated schedule.The paper reports PSNR coverage at margins of 0.25, 0.5, and 1.0 dB.Source: §2 · p. 4

3 Shared Schedules Recur and Transfer

Adaptive methods repeatedly choose a small set of schedules, and reusing frequent schedules on new prompts retains quality. Exhaustive search also finds a schedule that stays close to the best evaluated schedules across many unseen prompts.

3 Shared Schedules Recur and Transfer

  • Across image evaluations, the three most frequent schedules account for a median 89.8% of runs, with a 40.7–100% range across 96 method–model–dataset–ratio combinations.The four adaptive methods repeatedly make the same schedule choices across prompts despite selecting independently for each run.Source: §3 · p. 4

3.1 Shared schedules retain quality on new prompts

  • On new image prompts, shared schedules’ mean PSNR differs from adaptive methods by a median of 0.10 dB across 24 model–method–ratio combinations.The shared schedule’s mean PSNR ranges from 0.81 dB lower to 1.56 dB higher than the adaptive method’s.Source: §3.1 · p. 4
  • For video prompts, shared schedules are at most 0.25 dB below adaptive mean PSNR in 17 of 19 model–method–ratio combinations.Most frequent schedules remain unchanged when selected using prompts outside the evaluation set.Source: §3.1 · p. 4

3.2 A shared schedule approaches each prompt’s best evaluated quality

  • The schedule maximizing mean PSNR across four selection runs is within 0.269 dB of each run’s best schedule in three runs, and 1.249 dB in the fourth.The comparison exhaustively evaluates 1,370,754 schedules for FLUX.1-dev at cache ratio 0.82 with residual reuse.Source: Table 2 · p. 5, §3.2 · p. 5
  • Across four models and three cache ratios, schedules achieve majority coverage at a 1.0 dB margin, with all 12 lower confidence bounds above 50%.These are additional coverage tests reported in the appendix.Source: §3.2 · p. 5
  • The selected shared schedule stays within 0.5 dB of the best evaluated PSNR on 57% of new prompts and within 1.0 dB on 79%.The evaluation uses 2,381 new prompts and compares against the best mean PSNR among 337 candidate schedules.Source: §3.2 · p. 5, Figure 2 · p. 6

4 Common Patterns in Full-Compute Denoising

The section tests whether full-compute denoising trajectories share step-by-step patterns across prompts, datasets, and seeds within each model. It finds shared patterns in normalized displacement and direction changes, even though trajectories can occupy differently oriented latent-space planes.

4.1 Step-by-step changes across prompts

  • Median CV of normalized step displacement is 1.94%–2.35% across FLUX.1-dev datasets and 0.59%–0.63% across HunyuanVideo datasets; profiles have similar trends within each model.Distance from the endpoint line rises gradually before falling near the end; step displacements generally increase, while state norms first decrease and then increase.Source: §4.1 · pp. 6–7, Figure 3 · p. 7

4.2 Changes in direction

  • Turning angles change less in the middle of denoising than at the ends, with FLUX.1-dev profiles also sharing a local peak near step 18.The turning profiles use a five-step window and show similar direction-change patterns across datasets within each model.Source: §4.2 · p. 8, Figure 4 · p. 8

4.3 The shape and direction of individual trajectories

  • The first two principal components explain 89.9%–96.3% of centered off-chord variance, but when prompt and noise differ, median plane angles are about 88° in both image models.Pairs sharing initial noise have smaller median first principal angles than pairs sharing a prompt.Source: §4.3 · p. 8

5 Evaluating Schedules by Final-Output Quality

The section evaluates schedules by the final latent-state error they produce and examines how caching errors propagate. It finds that accumulated state effects track output error better than local approximation errors, and that schedule quality also depends on the approximation policy.

5.1 Current approximations and earlier errors

  • A cached-step velocity error has current and state contributions: the former compares approximate and full predictions at the cached state, while the latter compares full predictions at cached and full-compute states.At a full-computation step, the current contribution is zero.Source: §5.1 · p. 9
  • Summed state-contribution magnitudes correlate 0.988–0.996 with output-error magnitude, versus 0.579–0.748 for summed current-contribution magnitudes.Measurements used 100 PartiPrompts with FLUX.1-dev, SeaCache, and TeaCache at cache ratios near 0.58 and 0.82.Source: §5.1 · p. 9

5.2 Propagation from one cached step

  • A first-order analysis relates final output error to feature error propagated through the model, sampler, and remaining steps; independently measured single-step scores omit dependence on earlier choices.The single-step experiment replaces one feature on a full-compute trajectory, whereas scoring final outputs includes the effects of earlier choices.Source: §5.2 · pp. 10–11
  • Scores including propagation factors estimated from other prompts rank final output errors more accurately than local feature error alone; earlier cached steps rank higher under the step-only score.The table reports Spearman correlations with the L2 distance between final latent states, over 4,800 isolated cached steps per approximation policy.Source: §5.2 · p. 10, Table 3 · p. 10

5.3 The approximation policy changes schedule quality

  • At cache ratio 0.82, replacing residual reuse with second-order Hermite prediction lowers mean PSNR by 0.38 dB on BudCache and 6.68 dB on DiCache’s most frequent schedule.Schedule–policy interaction accounts for 13–27% of variation among 20 mean PSNR values at cache ratios 0.74 and 0.82; the gain or loss depends on the schedule.Source: §5.3 · p. 11, Figure 7 · p. 11

6 Searching for Golden Paths

The paper searches for fixed schedules by scoring complete generations on a small set of examples, then tests their quality on separate prompts. Schedules selected with perceptual similarity as the objective improve that metric on new prompts, while schedule choice trades off against PSNR.

6 Searching for Golden Paths

  • The search scores complete schedules by mean PSNR on eight prompt–seed runs, then selects among procedures using 50 separate COCO validation prompts.Hill climbing, simulated annealing, and greedy coordinate ascent are compared with random search under residual reuse.Source: §6 · p. 12
  • With residual reuse, searched schedules have higher mean PSNR than BudCache and random search in every Table 4 comparison, and beat MeanCache in 19 of 24; the other five are 0.02–0.09 dB lower.Source: §6 · p. 12, Table 4 · p. 12
  • DPCache and MeanCache score local prediction errors on full-compute trajectories, whereas BudCache scores complete cached generations using final latent MSE.Latent MSE differs from pixel fidelity and perceptual similarity.Source: §6 · p. 12
  • Across all 48 comparisons, LPIPS-selected schedules have lower mean LPIPS and higher mean SSIM than PSNR-selected schedules, while PSNR is higher in 15 and lower in 33.The comparison keeps the approximation policy fixed and spans two procedures, two models, three cache ratios, and four datasets.Source: §6 · p. 12, Table 5 · p. 13

7 Related Work

Prior work addresses feature reuse and prediction, adaptive or fixed schedule selection, and trajectory geometry. This paper extends schedule-selection work by testing shared schedules against prompt-specific choices and examining final-output quality as the selection objective.

Feature reuse and prediction

Schedule selection

  • DPCache and MeanCache select fixed schedules from errors measured on full-compute trajectories, while BudCache uses final latent MSE; the paper tests whether shared schedules approach prompt-specific quality.The authors' study also examines how output-quality objectives affect selected schedules and their quality on new prompts.Source: §7 · p. 13

Trajectory geometry

  • Prior geometry work uses trajectory geometry for sampling times or cache-specific curvature, while this study examines full-compute trajectories across image and video inputs.Within each model, relative displacements and direction changes follow similar patterns despite differing spatial directions.Source: §7 · pp. 13–14

8 Discussion and Conclusion

The experiments support reusable schedules and motivate evaluating complete schedules by final-output quality; the authors also identify limits in the models and questions studied. The appendices specify the evaluation setup, methods, measures, and separation of schedule selection from evaluation.

8 Discussion and Conclusion

  • Experiments support the GPH: shared schedules can match the best evaluated prompt-specific quality, and searches on a few examples transfer to new prompts.The effect of an approximation policy depends on the schedule, and changing the objective can improve the chosen metric on new prompts.Source: §8 · p. 14
  • The study covers transformer-based image and video models, but not non-transformer diffusion models or other data modalities.The authors also have not systematically studied scoring-example number and diversity or the distribution and distances of golden paths.Source: §8 · p. 14

AI Disclosure

  • The authors report using AI tools for implementation, data work, figures, drafting, and review, while retaining responsibility for the final scientific decisions and reported results.They state that they formulated the GPH, designed the methodology, and interpreted the results.Source: AI Disclosure · p. 14

Reproducibility Statement

  • The repository provides model-specific implementations, generation and evaluation scripts, search and analysis code, and setup and data-preparation instructions.The paper identifies the repository as the source for reproducing experiments.Source: Reproducibility Statement · p. 14

A.1 Sampling conditions

  • The image trajectory analyses use four datasets, with turning profiles and plane-orientation comparisons based on 1,440 trajectories per image model.The video trajectory analysis uses 4,629 trajectories across two video models and two datasets.Source: §A.1 · pp. 17–18

A.2 Methods

  • The methods distinguish adaptive schedules chosen during each prompt–seed run from offline searches that select a fixed schedule before generation.The method table summarizes both schedule selection and approximation policy, including the search experiments.Source: §A.2 · p. 18, Table 6 · p. 18

A.3 Aggregation and evaluation measures

  • Quality comparisons pair each accelerated output with a full-compute output using the same prompt and seed; PSNR, SSIM, and LPIPS respectively assess reconstruction, structure, and deep visual-feature similarity.Results are averaged over prompts within each seed and then over three seeds; bold indicates the best mean within each dataset, ratio, and metric.Source: §A.3 · p. 18

A.4 Calibration, schedule search, and evaluation separation

  • Predictors and fixed schedules are prepared separately from evaluation data, and the most frequent adaptive schedules are selected on 544 PartiPrompts before reuse on the other 1,088.The schedule-selection procedure combines counts across three base seeds for each model, method, and target ratio.Source: §A.4 · p. 19

A.5 Complete per-model results

The appendix reports complete model-level quality tables alongside schedule reuse, frequency, cache-ratio, and runtime analyses. Reusing selected schedules generally remains close to adaptive choices on held-out image and video prompts, while random schedules provide a weaker comparison in the reported tests.

A.5 Complete per-model results

  • The complete per-model tables report quality by method, dataset, target cache ratio, and metric; the Wan2.1 table includes PSNR, SSIM, LPIPS, temporal-LPIPS change, and VBench.The table captions specify averaging over three base seeds and define the metric abbreviations.Source: §A.5 · p. 19, Table 10 · p. 21

A.6 Cache-ratio alignment

  • Adaptive methods’ realized cache counts remain close to target ratios, but can differ from them; the largest image deviation is 0.543 cached steps for FLUX SenCache at target 29.The appendix reports realized mean counts separately by dataset.Source: §A.6 · p. 22

A.7 Online Runtime and Offline Preparation Cost

  • Offline preparation varies substantially: adaptive ratio calibration costs 67.5 H100 hours for HunyuanVideo and 68.0 for Wan2.1, while exhaustive search is estimated at about 3,600 H100 hours.The exhaustive-search estimate uses 5,483,016 generated cached images and a measured 2.37 seconds per image.Source: §A.7 · p. 25

B.1 Schedule-distribution statistics

  • Across 96 image combinations, the three most frequent schedules account for a median 89.8% of runs, and the same schedule is most frequent under all three seeds in 85 combinations.The mean effective schedule count is 2.23 for SeaCache, 3.99 for SenCache, 5.56 for TeaCache, and 5.55 for DiCache.Source: §B.1 · pp. 25–26, Table 19 · p. 26

B.2 Reusing adaptive schedules on images

  • Most frequent adaptive schedules selected on 544 prompts are reused on 1,088 held-out image prompts; random schedules are also tested as a control.The image random control samples cached steps uniformly without replacement while retaining full computation at steps 0, 1, 2, and 49.Source: §B.2 · pp. 26–27

B.3 Reusing adaptive schedules on videos

  • On video prompts, fixed schedules lose at most 0.25 dB in 17 of 19 comparisons; random-schedule controls have median PSNR differences from −5.9 to −0.3 dB, favoring adaptive methods.Two comparisons are exceptions to the fixed-schedule result; DiCache is excluded from the random control because its adaptive run uses a different approximation policy.Source: §B.3 · pp. 27, 29, Table 22 · p. 30

B.2 Reusing adaptive schedules on images

  • On held-out image prompts, fixed schedules change mean PSNR by −0.81 to +1.56 dB, with median absolute mean difference of 0.10 dB; 22 of 24 confidence intervals exclude zero.Schedules are selected on 544 prompts and evaluated on the other 1,088; positive differences favor the fixed schedule.Source: §B.2 · pp. 26–27, Table 20 · p. 28

B.4.1 Candidate sets and schedule selection

  • Coverage tests compare a fixed schedule with the best schedule in candidate sets of 9–17 or 13–32 schedules, using coverage margins and a corrected lower confidence bound.The enlarged sets include every available distinct schedule at the same cache ratio under residual reuse.Source: §B.4.1 · pp. 29–30

B.4.2 PSNR coverage

Shared schedules cover many prompts near their candidate-set best across PSNR, SSIM, and LPIPS, while exhaustive search finds schedules that perform consistently across four calibration examples.

B.4.2 PSNR coverage

  • All 12 smaller candidate sets have corrected lower bounds above 0.5 at the 0.5 dB margin (lowest 0.529), while random schedules cover at most 6% at 0.25 dB.The sets contain 9–17 schedules; the reported shared schedule is selected by coverage at 0.25 dB.Source: §B.4.2 · p. 30, Table 25 · p. 32
  • Under the same correction, enlarged candidate sets have lower bounds above 0.5 in 7 of 12 combinations at 0.25 dB, 9 of 12 at 0.5 dB, and all 12 at 1.0 dB.Each candidate set uses the same 1,088 prompts at base seed 42 with residual reuse for every schedule; the schedule with highest coverage at 0.25 dB is reported at all three margins.Source: §B.4.2 · p. 30, Table 24 · p. 31

B.4.3 Quality margins and coverage under SSIM and LPIPS

  • The three PSNR margins—0.25, 0.5, and 1.0 dB—are 0.42, 0.84, and 1.69 times the median 0.592 dB quality change per additional cached step.For SSIM and LPIPS, the corresponding margins are about one half, one, and two times each metric’s median change per additional cached step.Source: §B.4.3 · p. 32
  • All 12 SSIM bounds and 10 of 12 LPIPS bounds exceed 0.5 at the middle margin; at the tightest margin, seven SSIM and eight LPIPS bounds do so.At the middle margin, HunyuanVideo bounds are 0.477 at K = 37 and 0.481 at K = 41 for LPIPS.Source: §B.4.3 · p. 32, Table 26 · p. 33

B.5.1 Search space and schedule selection

  • 76 schedules fall within 0.1 dB of the highest mean PSNR after all 1,370,754 FLUX.1-dev schedules are evaluated at cache ratio 0.82.The schedule selected by four-run mean PSNR is 0.412 dB below the four runs’ individual best schedules on average; 42 schedules are within 1.0 dB of all four individual maxima.Source: Figure 8 · p. 34, §B.5.1 · p. 34

B.5.2 Evaluation data and fixed-schedule results

A schedule selected by mean PSNR on four examples remains competitive beyond those examples, although metric and dataset comparisons differ. The evaluation uses a deduplicated set of 337 candidate schedules.

B.5.2 Evaluation data and fixed-schedule results

  • The larger evaluation tests 337 distinct schedules, selected from four-run scores, uniform samples, and existing or constructed controls.The two selected schedules and method comparisons also run on 30,000 DiffusionDB runs.Source: §B.5.2 · pp. 34–35
  • The mean-PSNR-selected schedule has the highest mean PSNR and SSIM and lowest mean LPIPS on DrawBench, GenEval-style, and PartiPrompts; on DiffusionDB, SeaCache leads in SSIM and LPIPS.On DiffusionDB, the selected schedule has higher PSNR than SeaCache.Source: §B.5.2 · p. 35, Table 29 · p. 35
  • Under residual reuse, the selected schedule exceeds MeanCache by 0.178 dB mean PSNR across the first three datasets.Against SeaCache, it wins on 72.3% of prompts for PSNR, 56.2% for SSIM, and 45.9% for LPIPS across those datasets.Source: §B.5.2 · p. 35

B.5.3 Schedule rankings across datasets

Schedule rankings from four calibration runs remain similar in larger evaluations, but quality also depends on the approximation policy and model-specific trajectory patterns. The appendix reports trajectory measurements across models and numerical precision checks for turning angles.

B.5.3 Schedule rankings across datasets

  • Calibration and held-out rankings correlate at 0.926 across 337 schedules, with 39 schedules in the top 64 on both axes; the four-run-selected schedule ranks 36th on held-out mean PSNR.Across nine dataset–seed combinations, rank correlations range from 0.896 to 0.946, and 34–42 schedules from the four-run top 64 remain in the larger-evaluation top 64 in each combination.Source: §B.5.3 · pp. 35–36, Figure 9 · p. 36

B.5.4 Five approximation policies

  • Across approximation policies, mean PSNR for the four-run-selected schedule varies by 1.262 dB on three smaller datasets and 0.899 dB on DiffusionDB.For the highest-residual-reuse-PSNR schedule, the corresponding ranges are 3.963 dB and 3.508 dB; the caption reports that two-anchor prediction improves all three quality metrics for the four-run-selected schedule.Source: §B.5.4 · pp. 36–37, Table 30 · p. 37

C.1 Step-by-step profiles for the other models

  • Normalized step-displacement CV across datasets is 0.67%–1.17% for Qwen-Image and 1.25%–1.28% for Wan2.1, with a start-of-trajectory value reaching 19.27% in Wan2.1.Qwen-Image also has a distinct last-step displacement drop, associated with a smaller final sampler noise-level change.Source: §C.1 · p. 37

C.2 Distance from the line joining the endpoints

  • The normalized off-chord distance peaks at 0.0925–0.1079 of chord length for FLUX and 0.0358–0.0529 for Qwen-Image, at states 40–41 and 37–39, respectively.Across each dataset’s median profile, the distance rises slowly and returns rapidly to zero at the final latent.Source: §C.2 · p. 37

C.3 Numerical precision of turning angles

  • With a five-step angle window, the signal-to-rounding-floor ratio is 3.42–25.5 for FLUX and at least 2.61 for Qwen-Image.For a 30-trajectory Qwen-Image subset, minimum ratios increase to 3.76, 5.13, and 6.54 for window sizes 7, 9, and 11.Source: §C.3 · pp. 37–38

C.4 Samples for the spatial-direction measurements

The trajectory analyses sample image and video runs, decompose error into state and current contributions, and compare complete cached trajectories with final quality. Accumulated state effects show stronger associations with output error and quality than local or timing measures.

C.4 Samples for the spatial-direction measurements

  • The spatial-direction measurements use 1,440 trajectories per image model, including prompt-matched and noise-matched pairs, plus 20,000 pairs sharing neither prompt nor noise.The three plotted examples use all 51 latent states and trajectory-specific principal components; their first two components explain 94.56%–95.17% of centered off-chord variance.Source: §C.4 · pp. 38–39

D.1 State and current contributions

  • Accumulated state contributions correlate more strongly with output-error magnitude than accumulated current contributions: 0.988–0.996 versus 0.579–0.748 across the four method–ratio combinations.The measurements use FLUX.1-dev, SeaCache and TeaCache, 100 PartiPrompts, and cache ratios near 0.58 and 0.82.Source: §D.1 · p. 39

D.2 Complete cached trajectories

  • Across six model–ratio combinations, median final latent-state gap and mean PSNR have Spearman correlations from −0.98 to −0.95; early gap alone explains more PSNR variation than first-cache-step position (R2 = 0.715 versus 0.268).Using prompts outside schedule selection, the pooled correlation between early gap and mean PSNR is −0.82.Source: §D.2 · p. 40
  • Geometry-based controls use derivatives of mean normalized off-chord distance and speed to assign cached-interval costs, then use dynamic programming to select schedules.The interval cost increases for longer noise-level gaps since the last full step.Source: §D.2 · p. 40

D.3 Derivation and measurement of the four factors

The analysis measures how feature errors propagate into final-state errors, then separates schedule and approximation-policy effects on quality. Both the propagation profile and the schedule–policy interaction matter for interpreting caching performance.

D.3 Derivation and measurement of the four factors

  • Larger median D10 is associated with lower mean final-image PSNR across fixed schedules and approximation policies.D10 is the normalized L2 gap between cached and full-compute states at state 10; the figure relates median D10 over 49 prompts to mean PSNR over 1,632 PartiPrompts.Source: Figure 11 · p. 41
  • For residual reuse, the complete error product’s Spearman correlation with output-error magnitude is 0.886, versus 0.528 for local feature error and 0.867 for step index.The complete product also has the highest reported correlation for second-order Hermite prediction (0.896) and first-order Taylor prediction (0.899).Source: §D.3 · p. 42
  • Propagation profiles transfer across prompt subsets: their Spearman correlations range from 0.9955 to 0.9981, with correlation changes below 0.003.The profiles estimated from one half of the prompts retained their predictive correlation when evaluated on the other half.Source: §D.3 · p. 42

D.4 Schedule and approximation policy comparisons

  • BudCache and SeaCache differ by up to 3.24 dB on images and 3.01 dB on video under residual reuse.The comparison averages equally across four image datasets or two video datasets at the same target ratio.Source: §D.4 · p. 43
  • Schedule–approximation interactions account for 13–27% of mean-PSNR variation at cache ratios 0.74 and 0.82.The estimates use an additive fit to 20 combinations of four schedules and five approximation policies.Source: Table 32 · p. 43, §D.4 · p. 43

E.1 Search space, scoring examples, and evaluation

The appendix evaluates schedule search, candidate selection, and quality-objective choices, and compares approximation policies on fixed schedules. Its results show that searched schedules improve PSNR over BudCache, while the objective and approximation policy affect quality tradeoffs.

E.1 Search space, scoring examples, and evaluation

  • At K=29/37/41, the search spaces contain 1,749,695,026,860/1,101,716,330/1,370,754 schedules, scored by mean PSNR against same-prompt, same-seed full-compute outputs on eight fixed examples.The eight prompts were fixed before the search and do not appear in the named evaluation prompt sets.Source: §E.1 · p. 43

E.2 Search parameters

  • Procedures share budgets of 1,400, 700, and 400 evaluations at K=29, 37, and 41; hill climbing uses one-swap moves, while coordinate ascent optimizes movable full-step positions individually.Hill climbing, annealing, and coordinate ascent use stopping rules based on improvements relative to standard error; random search stops at its evaluation limit.Source: §E.2 · pp. 44–45

E.3 Candidate validation and selection

  • Validation on 50 separate COCO prompts selects the highest-validation-mean candidate; it changes the eight-example choice in eight search runs.Each procedure retains three distinct candidates before validation, and identical schedules are merged across procedures.Source: §E.3 · p. 46, Table 37 · p. 46

E.4 Results on five quality measures

  • Searched schedules improve PSNR over BudCache in every reported model, cache-ratio, and dataset combination.The comparisons use residual reuse and matched prompts and seeds.Source: Table 38 · p. 47

E.5 Schedules selected by different objectives

  • The standardized PSNR–LPIPS objective yields higher SSIM and lower LPIPS in all 48 tests, and higher PSNR in 26, than PSNR-selected schedules.Changing the objective changes the quality tradeoff within the same search space.Source: §E.5 · p. 48

E.6 Changing the approximation policy

  • Two-anchor extrapolation gives the highest equal-weight mean PSNR in all six model–ratio combinations, gaining 1.672/0.989/0.071 dB for FLUX.1-dev and 1.433/0.454/0.157 dB for Qwen-Image at ratios 0.58/0.74/0.82.Each schedule was held fixed while the approximation policy changed.Source: §E.6 · p. 48, Table 39 · p. 48

E.7 Comparing search procedures using exhaustive scores

  • Ten search algorithms are compared using stored exhaustive scores for all 1,370,754 FLUX.1-dev schedules that cache 41 of 50 steps.The comparisons inform the choice of search procedures and hyperparameters for Section 6.Source: §E.7 · p. 48, Table 40 · p. 49
  • Three schedules fall within 0.01 dB of the exhaustive best, while scoring-example subsets incur mean losses of 0.315, 0.284, and 0.219 dB for one, two, and three runs.The individual subset losses range from 0.000 to 1.017 dB; one-swap climbs from 2,000 random starts converge to the exhaustive best in 54.6% of climbs.Source: §E.7 · p. 49

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