cs.LGOct 6, 2026

Evolutionary One-Step Generators: Fast and Diverse Sampling for Discrete Design

Authors: Marcus Vukojevic, Erik Nielsen, Veronica Lachi, Andrea Passerini, Giovanni Iacca

Organizations: University of Trento, Italy · UiT The Arctic University of Norway, Tromsø, Norway

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Abstract

Several discrete design tasks, such as molecular discovery, require diverse collections of useful candidates at low computational cost. High validity alone does not guarantee a useful candidate library: repeatedly generating the same valid structures leaves few distinct alternatives. Training for both feasibility and diversity is challenging because many relevant criteria can only be evaluated after hard decoding. To address this challenge, we propose EGO (Evolutionary Generators with One-step inference), a framework for training compact generators directly on discrete outputs. The method combines distribution matching with structural constraints and optional diversity or history-dependent rewards, using antithetic low-rank evolution strategies without requiring criterion-specific differentiable surrogates. Once trained, the generator produces the entire graph in a single neural-network evaluation. On molecular generation benchmarks, our compact generator achieves over 50×50\times the valid-and-unique yield per estimated dense operation compared to recent one-step flow-map baselines while retaining high chemical validity. In scaffold completion, EGO achieves an observed 44.3×44.3\times speedup over MoLeR in generation to SMILES and produces approximately 10×10\times as many filter-passing proposals within matched time budgets for generation and screening. Beyond chemistry, EGO produces 1.54×1.54\times as many distinct held-out elite architectures as relaxed gradient training on NAS-Bench-101. The low generation cost may enable real-time candidate generation across discrete design tasks, supporting interactive exploration of constrained design spaces and rapid construction of candidate sets for downstream evaluation.

Figures & tables

Figure 2: Distinct valid yield versus estimated inference efficiency. Axes are valid-and-unique yield (VU) and VU per 101210^{12} estimated dense operations ( EVUE_{\mathrm{VU}} , log scale). The GuacaMol CFM-ECLD marker is a local 3/84-epoch feasibility checkpoint, not a converged baseline, and is excluded from ratio and ranking claims. Source protocols and sampling budgets differ, and EVUE_{\mathrm{VU}} is not wall-clock throughput. Full accounting and methods references appear in Appendix D .
Figure 3: Training efficiency and resource context. (a) Strict VUN under a shared active 30-minute ZINC250K wall-clock budget; RLOO versus EGO on QM9. (b) Training time (log scale). EGO reports optimizer-update time; DeFoG, Cometh, and CFM report elapsed training time on QM9/ZINC250K. ‡ Cometh GuacaMol used two A100s; † CFM GuacaMol: 84-epoch constant-throughput projection from a local three-epoch pilot. See Appendix D.4 .
Table 3
MetricEGO (ES)STESoft
Validity (%) ↑\uparrow72.3572.35 ±{}\pm{}1.761.7677.4577.45 ±{}\pm{}6.276.2789.56\mathbf{89.56} ±{}\pm{}0.82\mathbf{0.82}
Valid-and-unique yield (%) ↑\uparrow55.51\mathbf{55.51} ±{}\pm{}0.29\mathbf{0.29}33.2533.25 ±{}\pm{}6.026.0225.1225.12 ±{}\pm{}4.004.00
Held-out elite yield (%) ↑\uparrow2.22\mathbf{2.22} ±{}\pm{}0.08\mathbf{0.08}1.381.38 ±{}\pm{}0.290.291.441.44 ±{}\pm{}0.190.19
Mean test accuracy (%) ↑\uparrow91.8591.85 ±{}\pm{}0.060.0692.3892.38 ±{}\pm{}0.290.2992.70\mathbf{92.70} ±{}\pm{}0.10\mathbf{0.10}
Feature MMD ( ×10−3\times 10^{-3} ) ↓\downarrow27.7027.70 ±{}\pm{}2.702.7010.9010.90 ±{}\pm{}5.205.203.10\mathbf{3.10} ±{}\pm{}0.70\mathbf{0.70}
Table 2: NAS-Bench-101 results with SD from 10,000 proposals per selected model . The elite cutoff is the 90th percentile of benchmark validation accuracy in the architecture training partition. Mean test accuracy is averaged over valid proposals, including repetitions. Feature MMD compares the feature distributions of generated architectures and the elite target; lower values indicate a closer match. Protocol details appear in Appendix F.1 .
Appendix figures & tables37 assets

Supplementary material from the paper’s appendix.

Appendix

ExperimentCanvas sizeHidden layers
QM9 benchmark and matched-reward control93×7683\times 768
ZINC250K, primary experiment383×10243\times 1024
Full GuacaMol883×7683\times 768
Controlled molecular size sweep323×7683\times 768
ZINC250K active-time control383×7683\times 768
Appendix
Table 3: Molecular generator configurations. Canvas size is the maximum number of atom positions in the output. Hidden layers are shown as depth ×\times width. The ZINC250K active-time control uses the full 38-atom canvas with a smaller hidden width than the benchmark model.
Feature familyInformation matched to training molecules
Atom statesComposition, including charge and hydrogen states.
ValenceDistribution of bond-order sums at atoms, in bins 0–8.
Bond typesRelative counts of single, double, and triple bonds.
WL1Local atom neighborhoods after one refinement round; 64 bins.
WL2Larger neighborhoods after two refinement rounds; 64 bins.
SpectrumGlobal connectivity, using 10 bins of normalized-Laplacian eigenvalues.
Appendix
Table 4: Molecular features and their role in the fitness. Counts and histograms are normalized by the active atom count. Local-neighborhood histograms use bond-aware Weisfeiler–Lehman (WL) label refinement.
ExperimentSeedsReported training budget
QM9 benchmark3120,000 updates
Full GuacaMol3240,000 updates
Molecular size sweep530,000 updates
ZINC250K active-time control330 active minutes
QM9 matched-reward control511.52M graph evaluations
Appendix
Table 5: Budgets for the molecular comparisons. Each evaluated model produces 10,000 raw proposals for the primary yield metrics. Three-seed experiments use seeds 42, 43, and 44; the QM9 reward control uses seeds 42–46. Its row reports the largest matched reward budget.
CoefficientValue used in the experiments
Core feature weight λs\lambda_{s}11
Risk weights (αmax⁡,αfail,αlocal)(\alpha_{\max},\alpha_{\mathrm{fail}},\alpha_{\mathrm{local}})(1,0.75,0.5)(1,0.75,0.5)
Diversity thresholds (a,b)(a,b)(0.35,0.75)(0.35,0.75)
Rank weights (wO,wΠ,wv,wc,wU)(w_{O},w_{\Pi},w_{v},w_{c},w_{U})(1,0.5,0.25,0.75,0.2)(1,0.5,0.25,0.75,0.2)
Appendix
Table 6: Scaffold coefficients
Feature familyDimensionPurpose
Pixels784Fine image detail.
2×22\times 2 pooled means196Local stroke patterns.
4×44\times 4 pooled means49Coarse digit shape.
Row/column means56Horizontal and vertical ink placement.
Neighbor differences56Stroke transitions along rows and columns.
Global statistics8Ink amount, position, spread, and mirror asymmetry.
Appendix
Table 7: MNIST features used to build the fitness. All are computed directly from generated images, without digit labels.
Figure 5: Strict VUN across cumulative molecular size caps. Points and bars show mean ±\pm sample standard deviation across five seeds, with 30,000 updates and 10,000 final proposals per run. EGO includes rarity; EGO 0 removes it. Increasing the cap changes the molecular size mixture while the output canvas remains fixed. Error bars are not truncated at zero.
MethodStepsValid (%) ↑\uparrowUnique (%) ↑\uparrowFCD ↓\downarrowVU/s ↑\uparrow
Autoregressive generation
GraphARM [ Kong et al., 2023 ]n≤9n\leq 990.2595.621.22—
G2PT-base [ Chen et al., 2025 ]LL tokens99.096.80.06—
Iterative diffusion / flow
GDSS a [ Jo et al., 2022 ]1,00095.7298.462.900≈86.4\approx 86.4
DiGress [ Vignac et al., 2023 ]50099.096.2——
Appendix
Table 8: QM9 generation: published baselines and training comparisons at 120,000 updates. EGO and REINFORCE use the same 3×7683\times 768 generator and three training seeds, 42, 43, and 44; their dispersion is the sample SD. Uniqueness is conditional on validity, with the identity and postprocessing rules detailed in the text. VU/s counts valid, unique outputs per generation second. External rows retain the protocols, hardware, and uncertainty conventions of their sources. Gray rows identify one-step generators; dashes denote unavailable compatible evidence.
ArchitectureParams. (M)Full validStrict validStrict VUNFCD ↓\downarrow
2×7682\times 7680.86694.05±1.3694.05\pm 1.3692.03±0.8192.03\pm 0.8168.70±0.9868.70\pm 0.983.414±0.3363.414\pm 0.336
3×7683\times 7681.45693.46±0.8293.46\pm 0.8289.49±0.2289.49\pm 0.2268.31±1.0368.31\pm 1.033.534±0.6383.534\pm 0.638
4×7684\times 7682.04793.50±0.3193.50\pm 0.3189.29±1.4989.29\pm 1.4967.10±1.9467.10\pm 1.943.966±0.7893.966\pm 0.789
3×3843\times 3840.43394.30±1.1394.30\pm 1.1391.25±1.7291.25\pm 1.7267.06±2.1667.06\pm 2.163.452±0.2003.452\pm 0.200
3×10243\times 10242.46693.06±0.7293.06\pm 0.7289.35±1.3389.35\pm 1.3366.50±3.4266.50\pm 3.423.679±0.2933.679\pm 0.293
Appendix
Table 9: Complete EGO network size sweep on QM9 at the 120,000 update budget. Metrics are mean ±\pm sample standard deviation over seeds 42, 43, and 44. All columns except parameters and FCD are percentages. Parameter counts are rounded to millions; selected checkpoints can precede the budget milestone.
Architecture30,000 updates60,000 updates120,000 updates
2×7682\times 76863.0466.2368.70
3×7683\times 76863.1966.1968.31
4×7684\times 76862.5464.3367.10
3×3843\times 38461.0263.4067.06
3×10243\times 102462.1664.6666.50
Appendix
Table 10: EGO Strict VUN (%) on QM9 across training budgets, averaged over seeds 42, 43, and 44. Each result uses the checkpoint selected by validation within its budget. The dispersion at 120,000 updates is reported in Table 9 .
EGO architectureDense MFLOP ↓\downarrowEVUE_{\mathrm{VU}} ↑\uparrowTime (s) ↓\downarrowVU/s ↑\uparrow
3×7683\times 768 (primary)2.9082.908268,980.3268{,}980.31.159±0.0081.159\pm 0.008≈6,746\approx 6{,}746
2×7682\times 7681.7281.728445,409.0445{,}409.01.144±0.0081.144\pm 0.008≈6,725\approx 6{,}725
3×3843\times 3840.8640.864880,169.8880{,}169.81.164±0.0101.164\pm 0.010≈6,536\approx 6{,}536
4×7684\times 7684.0874.087186,692.3186{,}692.31.161±0.0091.161\pm 0.009≈6,574\approx 6{,}574
3×10243\times 10244.9254.925157,535.8157{,}535.81.167±0.0131.167\pm 0.013≈6,647\approx 6{,}647
Appendix
Table 11: EGO inference cost across the QM9 network size sweep at 120,000 updates. Dense MFLOP is the estimated work per proposal, and EVUE_{\mathrm{VU}} is the number of distinct valid outputs per 101210^{12} estimated dense operations. Times are seconds per 10,000 proposals through SMILES/CSV on an A100 40GB, with mean ±\pm sample SD over seeds 42, 43, and 44. VU/s is the mean distinct valid count divided by the mean time.
MethodFull valid ↑\uparrowStrict valid ↑\uparrowStrict unique yield ↑\uparrowStrict VUN ↑\uparrowFCD ↓\downarrow
EGO93.46±0.8293.46\pm 0.8289.49±0.2289.49\pm 0.2274.53±1.17\mathbf{74.53\pm 1.17}68.31±1.03\mathbf{68.31\pm 1.03}3.534±0.6383.534\pm 0.638
RLOO88.26±3.1288.26\pm 3.1285.89±2.4385.89\pm 2.4342.58±7.0942.58\pm 7.0939.39±7.6139.39\pm 7.613.907±0.1623.907\pm 0.162
RLOO + rarity96.80±2.19\mathbf{96.80\pm 2.19}95.77±2.10\mathbf{95.77\pm 2.10}48.86±4.1548.86\pm 4.1542.07±2.3842.07\pm 2.382.710±0.535\mathbf{2.710\pm 0.535}
Appendix
Table 12: QM9 training comparison at 120,000 updates: mean ±\pm sample SD across seeds 42, 43, and 44. All columns except raw FCD are percentages. Strict validity requires chemical validity and connectedness. Strict unique yield divides distinct valid connected molecules by all attempts; Strict VUN additionally requires absence from training. Bold values identify the best mean within this comparison.
MethodFull valid ↑\uparrowStrict valid ↑\uparrowStrict unique yield ↑\uparrowStrict VUN ↑\uparrowFCD ↓\downarrow
EGO76.71±3.6776.71\pm 3.6770.70±2.5970.70\pm 2.5940.71±2.5340.71\pm 2.5338.39±2.8938.39\pm 2.895.250±0.6605.250\pm 0.660
RLOO98.64±0.8698.64\pm 0.8697.77±1.0497.77\pm 1.0414.00±1.0514.00\pm 1.0512.05±0.5312.05\pm 0.534.161±0.5084.161\pm 0.508
Appendix
Table 13: QM9 comparison at 11,520,000 hard reward graph evaluations per method and seed: mean ±\pm sample SD over seeds 42 to 46. The current checkpoints are at 625 updates for EGO and 120,000 for RLOO. All columns except held-out raw FCD are percentages; yield metrics use all 10,000 proposals as the denominator. Both methods use the same reward and fixed scales, without rarity or rank shaping.
MethodStepsValid (%) ↑\uparrowUnique (%) ↑\uparrowFCD ↓\downarrowVU/s ↑\uparrow
Autoregressive generation and autoregressive diffusion
GraphARM [ Kong et al., 2023 ]n≤38n\leq 3888.2399.4616.26—
LO-ARM-st-sep [ Wang et al., 2025 ]D(n)D(n)96.26100.003.229—
PARD [ Zhao et al., 2024 ]blockwise95.2399.991.98—
Iterative diffusion and flow
GDSS a [ Jo et al., 2022 ]1,00097.0199.6414.656≈4.93\approx 4.93
Appendix
Table 14: Full ZINC250K comparison with published generators. Validity and conditional uniqueness are percentages, FCD is a distance, and VU/s is a measured output rate. EGO uses 120,000 training updates and reports mean ±\pm sample SD where available. Its FCD is a single recovered evaluation. Gray rows identify one-step generators; dashes denote unavailable measurements.
ArchitectureFull validStrict validConnectedUnique given validFull VUStrict VUNFCD ↓\downarrow
3×3843\times 38484.4284.42 ±4.62\pm 4.6279.5379.53 ±1.86\pm 1.8695.0895.08 ±2.76\pm 2.7693.1693.16 ±1.03\pm 1.0378.6278.62 ±3.88\pm 3.8874.7974.79 ±2.01\pm 2.0129.5829.58 ±1.49\pm 1.49
2×7682\times 76890.7290.72 ±5.31\pm 5.3186.7186.71 ±7.76\pm 7.7695.9395.93 ±2.50\pm 2.5097.0697.06 ±0.68\pm 0.6888.0888.08 ±5.66\pm 5.66——
3×7683\times 76888.3588.35 ±2.75\pm 2.7585.5985.59 ±4.20\pm 4.2097.2097.20 ±2.88\pm 2.8896.6996.69 ±0.24\pm 0.2485.4385.43 ±2.86\pm 2.86——
4×7684\times 76891.2391.23 ±0.29\pm 0.2986.3986.39 ±1.83\pm 1.8395.1695.16 ±1.59\pm 1.5994.8994.89 ±1.00\pm 1.0086.5686.56 ±0.73\pm 0.7381.70∗81.70^{*}28.89∗28.89^{*}
3×1024\boldsymbol{3\times 1024}94.0294.02 ±3.94\pm 3.9490.8490.84 ±6.97\pm 6.9796.8296.82 ±3.03\pm 3.0396.4296.42 ±0.21\pm 0.2190.6690.66 ±4.00\pm 4.0088.2288.22 ±6.86\pm 6.8630.36∗30.36^{*}
Appendix
Table 15: EGO architecture comparison on ZINC250K at 120,000 updates. Values are the mean ±\pm sample SD over the available runs per metric. All columns except FCD are percentages. Unique given valid is computed within each run among full valid molecules before aggregation; reconstruction from rounded logs is approximate. Stars mark single available evaluations, with no sample SD; dashes denote unavailable metrics. The highlighted architecture is the profile used in the main comparison.
Architecture30,000 updates60,000 updates120,000 updates
Full validity (%)
3×3843\times 38482.53±4.5382.53\pm 4.5383.62±4.2783.62\pm 4.2784.42±4.6284.42\pm 4.62
2×7682\times 76885.62±6.9885.62\pm 6.9888.73±5.7888.73\pm 5.7890.72±5.3190.72\pm 5.31
3×7683\times 76885.20±3.0085.20\pm 3.0086.23±2.3686.23\pm 2.3688.35±2.7588.35\pm 2.75
4×7684\times 76887.82±2.0587.82\pm 2.0590.68±0.1390.68\pm 0.1391.23±0.2991.23\pm 0.29
3×10243\times 102491.13±1.1791.13\pm 1.1792.54±2.2992.54\pm 2.2994.02±3.9494.02\pm 3.94
Appendix
Table 16: EGO validity on ZINC250K across training budgets. Each cell reports mean ±\pm sample SD over the available runs. Each milestone evaluates a checkpoint selected on validation data within the corresponding budget. Improvements describe aggregate means, rather than verified improvements in every individual trajectory.
ArchitectureVU/s (graph generation only) ↑\uparrow
3×3843\times 384—
2×7682\times 768722,733±257,183722{,}733\pm 257{,}183
3×7683\times 768841,619±28,323841{,}619\pm 28{,}323
4×7684\times 768(841,365±25,413)a(841{,}365\pm 25{,}413)^{a}
3×1024\boldsymbol{3\times 1024}862,142±46,109862{,}142\pm 46{,}109
Appendix
Table 17: Full VU per second of graph generation for the EGO architectures on ZINC250K at 120,000 updates: mean ±\pm sample SD of rates computed within each run. Molecular conversion and evaluation are excluded from the timing denominator. a The 4×7684\times 768 aggregate uses two available runs. Dashes denote unavailable rates; the highlighted profile is the configuration used in the main table.
Timing boundaryTime (s)Full VU/s ↑\uparrow
Graph generation only≈0.011\approx 0.011862,142±46,109862{,}142\pm 46{,}109
Generation through SMILES/CSV≈3.59\approx 3.592,525±1602{,}525\pm 160
Appendix
Table 18: Measured timing boundaries for EGO 3×10243\times 1024 on ZINC250K at 120,000 updates. Time refers to 10,000 raw proposals and is reported approximately. VU/s uses fully valid distinct outputs, and reports mean ±\pm sample SD of rates computed within each run. The broader boundary includes molecular conversion and CSV export; subsequent distributional evaluation is excluded.
kkMethodFull ↑\uparrowStrict ↑\uparrowStrict VUN ↑\uparrowConn. ↑\uparrowMMDcell↓\mathrm{MMD}_{\rm cell}\downarrow
16EGO97.33 ±\pm 1.1194.91 ±\pm 1.5790.84 ±\pm 0.7797.55 ±\pm 0.940.0084 ±\pm 0.0007
16EGO 086.60 ±\pm 2.8183.82 ±\pm 4.1877.39 ±\pm 3.7597.07 ±\pm 1.850.0052 ±\pm 0.0009
16STE (feature-ST)99.21 ±\pm 0.9196.52 ±\pm 1.757.51 ±\pm 4.8397.31 ±\pm 2.290.0538 ±\pm 0.0351
16Soft95.01 ±\pm 1.7489.33 ±\pm 7.1910.81 ±\pm 4.2194.29 ±\pm 6.820.0430 ±\pm 0.0313
24EGO95.48 ±\pm 2.9891.69 ±\pm 3.1989.45 ±\pm 2.6796.18 ±\pm 2.120.0099 ±\pm 0.0015
24EGO 075.58 ±\pm 11.0069.38 ±\pm 14.0163.30 ±\pm 10.9693.17 ±\pm 4.790.0065 ±\pm 0.0012
Appendix
Table 19: Complete cumulative ZINC250K sweep. Values are mean ±\pm sample standard deviation across five seeds, with 30,000 updates per run. Full, Strict, Strict VUN, and Conn. are percentages; all use the complete attempt budget as the denominator. Lower MMDcell\mathrm{MMD}_{\rm cell} is better, with comparisons restricted to a fixed cap.
kkContrastΔ\Delta ValidityΔ\Delta VUN103ΔMMDcell10^{3}\Delta\mathrm{MMD}_{\rm cell}
16EGO −- EGO 011.08  [6.72, 15.44]11.08\;[6.72,\,15.44]13.45  [8.62, 18.28]13.45\;[8.62,\,18.28]3.23  [2.35, 4.12]3.23\;[2.35,\,4.12]
16EGO 0 −- Soft−5.51  [−13.45, 2.44]-5.51\;[-13.45,\,2.44]66.58  [59.67, 73.48]66.58\;[59.67,\,73.48]−37.80  [−76.99, 1.39]-37.80\;[-76.99,\,1.39]
16EGO 0 −- STE−12.70  [−19.28, −6.12]-12.70\;[-19.28,\,-6.12]69.88  [61.27, 78.49]69.88\;[61.27,\,78.49]−48.64  [−91.73, −5.55]-48.64\;[-91.73,\,-5.55]
16STE −- Soft7.19  [−1.20, 15.58]7.19\;[-1.20,\,15.58]−3.30  [−13.92, 7.32]-3.30\;[-13.92,\,7.32]10.84  [−66.83, 88.50]10.84\;[-66.83,\,88.50]
24EGO −- EGO 022.31  [6.40, 38.21]22.31\;[6.40,\,38.21]26.15  [13.23, 39.08]26.15\;[13.23,\,39.08]3.40  [0.85, 5.95]3.40\;[0.85,\,5.95]
24EGO 0 −- Soft−5.82  [−37.61, 25.97]-5.82\;[-37.61,\,25.97]49.17  [40.16, 58.17]49.17\;[40.16,\,58.17]−54.89  [−104.17, −5.60]-54.89\;[-104.17,\,-5.60]
Appendix
Table 20: ZINC250K paired contrasts: mean difference with its unadjusted paired 95% confidence interval in brackets, over five seeds. Each difference is the first method minus the second. Validity and VUN use the strict definition, with differences reported in percentage points; MMD differences are multiplied by 10310^{3} . Positive validity or yield differences and negative MMD differences favor the first method.
MethodFull validStrict validStrict VUNUpdates ( 10310^{3} )
EGO86.94±5.2686.94\pm 5.2679.69±8.4779.69\pm 8.4777.91±7.9277.91\pm 7.9225.84±0.2925.84\pm 0.29
EGO 071.53±4.9571.53\pm 4.9561.71±3.5861.71\pm 3.5853.73±3.7053.73\pm 3.7025.95±0.2625.95\pm 0.26
STE79.12±4.1579.12\pm 4.1557.26±11.2757.26\pm 11.2729.50±1.7629.50\pm 1.76145.73±2.39145.73\pm 2.39
Soft92.82±4.7092.82\pm 4.7066.81±25.4666.81\pm 25.465.78±9.865.78\pm 9.86156.58±3.30156.58\pm 3.30
Appendix
Table 21: Full ZINC250K after 30 minutes of active training. Values are mean ±\pm sample SD across three seeds, with 10,000 proposals per seed. Quality metrics are percentages; updates are in thousands.
MethodStepsValid (%) ↑\uparrowVU (%) ↑\uparrowFCD ↓\downarrowVU/s ↑\uparrow
Autoregressive generation
SMILES LSTM [ Brown et al., 2019 ]LL tokens95.995.9a95.9^{a}≈0.46\approx 0.46—
G2PT-base [ Chen et al., 2025 ]LL tokens94.694.6a94.6^{a}≈0.34\approx 0.34—
G2PT-large [ Chen et al., 2025 ]LL tokens95.395.3a95.3^{a}≈0.38\approx 0.38—
Iterative diffusion and flow
DiGress [ Vignac et al., 2023 ]50085.285.2a85.2^{a}≈1.93\approx 1.93—
Appendix
Table 22: GuacaMol generation, including the local CFM-ECLD pilot after three epochs and 14,919 updates, using seed 42. The pilot is an early checkpoint, not a converged baseline, and is excluded from efficiency ratio and ranking claims. EGO reports mean ±\pm sample SD over three training seeds. VU is the percentage of raw attempts that yield distinct valid molecules, except for the proxies marked a . FCD is a distance, with lower values preferred. Gray rows identify generators that use one network evaluation. Published results retain their source protocols; EGO and the local CFM-ECLD pilot use our offline evaluation protocol. The final block reports the separate EGO experiment on n≤32n\leq 32 .
kkMethodFull ↑\uparrowStrict ↑\uparrowStrict VUN ↑\uparrowConn. ↑\uparrowMMDcell↓\mathrm{MMD}_{\rm cell}\downarrow
16EGO94.57 ±\pm 2.6690.27 ±\pm 2.3482.12 ±\pm 2.1995.68 ±\pm 1.430.0091 ±\pm 0.0024
16EGO 086.71 ±\pm 5.4982.60 ±\pm 6.8266.01 ±\pm 4.0895.79 ±\pm 1.590.0064 ±\pm 0.0016
16STE (feature-ST)99.72 ±\pm 0.3897.88 ±\pm 1.956.39 ±\pm 1.3898.16 ±\pm 1.630.0364 ±\pm 0.0189
16Soft97.75 ±\pm 2.0992.52 ±\pm 4.417.74 ±\pm 7.5394.77 ±\pm 2.620.0387 ±\pm 0.0162
24EGO92.80 ±\pm 5.4286.27 ±\pm 6.4781.53 ±\pm 6.8193.44 ±\pm 1.540.0109 ±\pm 0.0037
24EGO 074.42 ±\pm 7.9168.65 ±\pm 7.9358.68 ±\pm 8.2293.99 ±\pm 3.170.0088 ±\pm 0.0024
Appendix
Table 23: Complete cumulative GuacaMol sweep. Values are mean ±\pm sample standard deviation across five seeds, with 30,000 updates per run. Full, Strict, Strict VUN, and Conn. are percentages; all use the complete attempt budget as the denominator. Lower MMDcell\mathrm{MMD}_{\rm cell} is better, with comparisons restricted to a fixed cap.
kkContrastΔ\Delta ValidityΔ\Delta VUN103ΔMMDcell10^{3}\Delta\mathrm{MMD}_{\rm cell}
16EGO −- EGO 07.687.68 [−0.68, 16.03][-0.68,\,16.03]16.1216.12 [10.91, 21.32][10.91,\,21.32]2.632.63 [0.49, 4.76][0.49,\,4.76]
16EGO 0 −- Soft−9.92-9.92 [−15.07, −4.78][-15.07,\,-4.78]58.2758.27 [50.18, 66.35][50.18,\,66.35]−32.20-32.20 [−52.40, −12.01][-52.40,\,-12.01]
16EGO 0 −- STE−15.28-15.28 [−22.69, −7.88][-22.69,\,-7.88]59.6259.62 [55.76, 63.48][55.76,\,63.48]−29.99-29.99 [−52.55, −7.43][-52.55,\,-7.43]
16STE −- Soft5.365.36 [0.12, 10.61][0.12,\,10.61]−1.35-1.35 [−9.21, 6.50][-9.21,\,6.50]−2.21-2.21 [−25.74, 21.31][-25.74,\,21.31]
24EGO −- EGO 017.6217.62 [5.49, 29.75][5.49,\,29.75]22.8422.84 [12.98, 32.70][12.98,\,32.70]2.032.03 [−2.51, 6.58][-2.51,\,6.58]
24EGO 0 −- Soft−22.08-22.08 [−35.91, −8.26][-35.91,\,-8.26]50.7350.73 [33.29, 68.17][33.29,\,68.17]−89.57-89.57 [−152.35, −26.79][-152.35,\,-26.79]
Appendix
Table 24: GuacaMol paired contrasts: mean difference with its unadjusted paired 95% confidence interval in brackets, over five seeds. Each difference is the first method minus the second. Validity and VUN use the strict definition, with differences reported in percentage points; MMD differences are multiplied by 10310^{3} . Positive validity or yield differences and negative MMD differences favor the first method.
MetricCFM-ECLD pilot
Full validity (%)19.67
Strict validity (%)4.64
Uniqueness among full valid molecules (%)100.00
Full VU (%)19.67
Strict VU (%)4.64
Full VUN (%)19.66
Appendix
Table 25: CFM-ECLD pilot on full GuacaMol, seed 42, after three epochs and 14,919 updates. This is a single pilot checkpoint without an estimate of variation across training seeds. Full validity allows disconnected molecules; strict validity also requires connectivity.
DatasetMethodVU (%)Dense MFLOPEVUE_{\mathrm{VU}}
QM9GraphARM≈86.30\approx 86.30——
 G2PT-base≈95.83\approx 95.83—67.1u67.1^{u}
 CatFlow (100)≈99.76\approx 99.76—169u169^{u}
 DeFoG (500)≈95.72\approx 95.7286,234.24011.1
 MELD≈96.49\approx 96.49—1.72u1.72^{u}
 MolGAN≈10.20\approx 10.200.714142,945
Appendix
Table 26: Dense inference work and efficiency for the molecular comparison. VU is a percentage of raw proposals, and MFLOP denotes 10610^{6} counted operations per proposal, including all network evaluations. EGO counts use local architectures; published CFM counts use reconstructed architectures. The GuacaMol CFM count uses the local three-epoch pilot. Estimates marked u retain the values used in the main figure, but lack a complete recovered layer count. Dashes indicate unavailable counts. The pilot is excluded from ranking and ratio claims.
MethodQM9ZINC250KGuacaMol
EGO0.2999±0.00120.2999\pm 0.00122.098314.8468±0.069714.8468\pm 0.0697
DeFoG [ Qin et al., 2025 ]6.514141
Cometh [ Siraudin et al., 2025 ]6—≈96\approx 96
CFM [ Roos et al., 2026 ]≈48\approx 48≈144\approx 144≈2,200\approx 2{,}200 to 2,8002{,}800
Appendix
Table 27: Training time in hours for the methods discussed in the main text. EGO measures synchronized optimizer updates; external values use the reported elapsed training durations. The GuacaMol CFM entry is a constant-throughput projection to 84 epochs from the local three-epoch pilot. It is not a measured full training duration. A dash denotes unavailable evidence.
Metric42 / 4143 / 4244 / 43
Mean deadline per query (s)6.1766.1055.974
Mean proposals screened, EGO1,822.941,947.701,907.35
Mean proposals screened, MoLeR256.0256.0256.0
Filter-passing proposals, EGO55,15252,97556,064
Filter-passing proposals, MoLeR5,4345,4855,417
Passing fraction (%), EGO30.2527.2029.39
Appendix
Table 28: Generation and screening at matched time. Each column reports one replicate on the same 100 scaffold queries; headers identify the EGO training seed and MoLeR sampling seed. Screened counts are means per query; passing counts are totals over the panel. Passing fractions use all screened proposals. The outcome block partitions all 100 queries. Higher-QED counts require passing candidates from both methods; exclusive successes and failures of both methods remain explicit.
Figure 6: Highest QED completions that pass screening for the prespecified scaffold 23 at matched time. Each row shows one replicate, using EGO training seeds 42, 43, and 44 and sampling seeds 41, 42, and 43 from the same MoLeR checkpoint trained on ZINC. The common budgets are 5.64, 5.55, and 5.13 seconds. The supplied scaffold is highlighted. Table 28 reports results over all 100 queries.
MetricEGOMoLeR
Usable completions (%)77.67±2.5777.67\pm 2.5797.58±0.0997.58\pm 0.09
Distinct usable yield (%)66.78±1.1566.78\pm 1.1585.80±0.1285.80\pm 0.12
Filter passing fraction (%)30.3±1.130.3\pm 1.119.4±0.119.4\pm 0.1
Distinct filtered completions20,245±75520{,}245\pm 75516,684±7916{,}684\pm 79
Scaffold coverage (/100)77.7±1.577.7\pm 1.586.3±0.686.3\pm 0.6
Generation to SMILES (s)41.82±8.3941.82\pm 8.391,851.82±8.051{,}851.82\pm 8.05
Appendix
Table 29: Scaffold completion at the same proposal budget. Each run uses 100 queries and 1,000 proposals per query. Values are the mean ±\pm sample SD over three EGO training seeds and three sampling seeds from one MoLeR checkpoint trained from scratch on ZINC. Distinct-library counts are deduplicated within each query. The filter passing fraction is conditional on distinct usable completions. Generation time excludes screening.
MethodSeed typeSeedDistinct filtered count
EGOTraining4219,52719{,}527
EGOTraining4320,17520{,}175
EGOTraining4421,03321{,}033
MoLeRSampling4116,76316{,}763
MoLeRSampling4216,68316{,}683
MoLeRSampling4316,60616{,}606
Appendix
Table 30: Distinct completions that pass screening in each run. Each count is the sum of distinct counts within 100 queries at 1,000 proposals per query. MoLeR sampling seeds share one checkpoint trained with seed 42.
Figure 7: Examples of scaffold completions from EGO with training seed 43 and externally pretrained MoLeR with sampling seed 42. The supplied core is highlighted. All displayed molecules pass the screening criteria. Three scaffolds were selected by a fixed hash ordering among queries with at least two distinct passing completions from both methods. The first two distinct passing completions in proposal order are shown for each method and scaffold.
Figure 8: Selected examples of larger scaffold extensions from EGO with training seed 44 and externally pretrained MoLeR with sampling seed 42. The supplied core is highlighted, and all displayed molecules pass screening. Scaffolds were ranked by the second-largest addition of heavy atoms from EGO among queries with at least two distinct passing completions from both methods. The three selected scaffolds show the two largest passing extensions from each method. These examples illustrate larger extensions rather than typical outputs.
MethodUpdate timeCompilationValidation
EGO46.12±0.5946.12\pm 0.5948.41±0.7548.41\pm 0.7539.58±1.9139.58\pm 1.91
STE34.06±0.3834.06\pm 0.3854.18±0.8854.18\pm 0.8830.84±1.0130.84\pm 1.01
Soft34.37±0.6534.37\pm 0.6554.51±1.1854.51\pm 1.1816.36±1.2816.36\pm 1.28
Appendix
Table 31: NAS training time in seconds, mean ±\pm sample standard deviation across seeds 42–44. Columns measure separate stages of the 15,000-update runs; none alone is total runtime.
MethodLeNet Fréchet ↓\downarrowPredicted classes / 10 ↑\uparrow
EGO599.8±47.1599.8\pm 47.16.33±0.586.33\pm 0.58
STE522.4±95.6522.4\pm 95.67.67±0.587.67\pm 0.58
Soft420.6±89.1\mathbf{420.6\pm 89.1}8.33±0.58\mathbf{8.33\pm 0.58}
Appendix
Table 32: Controlled Energy comparison on binarized MNIST, mean ±\pm sample standard deviation over seeds 42–44. Distances use 2,000 generated and 2,000 test images. Coverage uses all 10,000 generated images, with at least 50 predictions required to count a digit class.
MetricEGOSTESoft
LeNet KID ↓\downarrow0.4517±0.04900.4517\pm 0.04900.3868±0.08530.3868\pm 0.08530.3069±0.07180.3069\pm 0.0718
Pooled-image MMD 2 ↓\downarrow0.0414±0.00080.0414\pm 0.00080.0235±0.00940.0235\pm 0.00940.0145±0.00620.0145\pm 0.0062
Class-balance TV ↓\downarrow0.545±0.0200.545\pm 0.0200.463±0.0450.463\pm 0.0450.401±0.0990.401\pm 0.099
Normalized class entropy ↑\uparrow0.628±0.0350.628\pm 0.0350.738±0.0440.738\pm 0.0440.754±0.1080.754\pm 0.108
Appendix
Table 33: Additional MNIST diagnostics, mean ±\pm sample standard deviation across seeds 42–44. LeNet KID uses classifier embeddings; pooled-image MMD 2 uses raw block means. Class metrics use predicted labels on 10,000 images.
MethodSelected updatePredicted classes / 10
EGO29,000 / 30,000 / 30,0006 / 6 / 7
STE500 / 30,000 / 5008 / 8 / 7
Soft1,500 / 28,000 / 1,0008 / 9 / 8
Appendix
Table 34: MNIST checkpoint selection and predicted digit coverage. Entries follow seed order 42, 43, 44. Every run completes 30,000 updates; coverage is measured on 10,000 final images.
Figure 9: Binarized MNIST samples generated by EGO , STE, and Soft, from top to bottom.

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