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

Marcus Vukojevic; Erik Nielsen; Veronica Lachi; Andrea Passerini; Giovanni Iacca

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

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 $50\times$ 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 $44.3\times$ speedup over MoLeR in generation to SMILES and produces approximately $10\times$ as many filter-passing proposals within matched time budgets for generation and screening. Beyond chemistry, EGO produces $1.54\times$ 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

10^{12}

estimated dense operations (

E_{\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

E_{\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 .

Metric	EGO (ES)		STE		Soft
Validity (%) $\uparrow$	$72.35$ ${}\pm{}$	$1.76$	$77.45$ ${}\pm{}$	$6.27$	$\mathbf{89.56}$ ${}\pm{}$	$\mathbf{0.82}$
Valid-and-unique yield (%) $\uparrow$	$\mathbf{55.51}$ ${}\pm{}$	$\mathbf{0.29}$	$33.25$ ${}\pm{}$	$6.02$	$25.12$ ${}\pm{}$	$4.00$
Held-out elite yield (%) $\uparrow$	$\mathbf{2.22}$ ${}\pm{}$	$\mathbf{0.08}$	$1.38$ ${}\pm{}$	$0.29$	$1.44$ ${}\pm{}$	$0.19$
Mean test accuracy (%) $\uparrow$	$91.85$ ${}\pm{}$	$0.06$	$92.38$ ${}\pm{}$	$0.29$	$\mathbf{92.70}$ ${}\pm{}$	$\mathbf{0.10}$
Feature MMD ( $\times 10^{-3}$ ) $\downarrow$	$27.70$ ${}\pm{}$	$2.70$	$10.90$ ${}\pm{}$	$5.20$	$\mathbf{3.10}$ ${}\pm{}$	$\mathbf{0.70}$

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

Experiment	Canvas size	Hidden layers
QM9 benchmark and matched-reward control	9	$3\times 768$
ZINC250K, primary experiment	38	$3\times 1024$
Full GuacaMol	88	$3\times 768$
Controlled molecular size sweep	32	$3\times 768$
ZINC250K active-time control	38	$3\times 768$

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 family	Information matched to training molecules
Atom states	Composition, including charge and hydrogen states.
Valence	Distribution of bond-order sums at atoms, in bins 0–8.
Bond types	Relative counts of single, double, and triple bonds.
WL1	Local atom neighborhoods after one refinement round; 64 bins.
WL2	Larger neighborhoods after two refinement rounds; 64 bins.
Spectrum	Global 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.

Experiment	Seeds	Reported training budget
QM9 benchmark	3	120,000 updates
Full GuacaMol	3	240,000 updates
Molecular size sweep	5	30,000 updates
ZINC250K active-time control	3	30 active minutes
QM9 matched-reward control	5	11.52M 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.

Coefficient	Value used in the experiments
Core feature weight $\lambda_{s}$	$1$
Risk weights $(\alpha_{\max},\alpha_{\mathrm{fail}},\alpha_{\mathrm{local}})$	$(1,0.75,0.5)$
Diversity thresholds $(a,b)$	$(0.35,0.75)$
Rank weights $(w_{O},w_{\Pi},w_{v},w_{c},w_{U})$	$(1,0.5,0.25,0.75,0.2)$

Appendix

Table 6: Scaffold coefficients

Feature family	Dimension	Purpose
Pixels	784	Fine image detail.
$2\times 2$ pooled means	196	Local stroke patterns.
$4\times 4$ pooled means	49	Coarse digit shape.
Row/column means	56	Horizontal and vertical ink placement.
Neighbor differences	56	Stroke transitions along rows and columns.
Global statistics	8	Ink 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.

Autoregressive generation
Method	Steps	Valid (%) $\uparrow$	Unique (%) $\uparrow$	FCD $\downarrow$	VU/s $\uparrow$
GraphARM [ Kong et al., 2023 ]	$n\leq 9$	90.25	95.62	1.22	—
G2PT-base [ Chen et al., 2025 ]	$L$ tokens	99.0	96.8	0.06	—
Iterative diffusion / flow
GDSS a [ Jo et al., 2022 ]	1,000	95.72	98.46	2.900	$\approx 86.4$
DiGress [ Vignac et al., 2023 ]	500	99.0	96.2	—	—

Appendix

Table 8: QM9 generation: published baselines and training comparisons at 120,000 updates. EGO and REINFORCE use the same

3\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.

Architecture	Params. (M)	Full valid	Strict valid	Strict VUN	FCD $\downarrow$
$2\times 768$	0.866	$94.05\pm 1.36$	$92.03\pm 0.81$	$68.70\pm 0.98$	$3.414\pm 0.336$
$3\times 768$	1.456	$93.46\pm 0.82$	$89.49\pm 0.22$	$68.31\pm 1.03$	$3.534\pm 0.638$
$4\times 768$	2.047	$93.50\pm 0.31$	$89.29\pm 1.49$	$67.10\pm 1.94$	$3.966\pm 0.789$
$3\times 384$	0.433	$94.30\pm 1.13$	$91.25\pm 1.72$	$67.06\pm 2.16$	$3.452\pm 0.200$
$3\times 1024$	2.466	$93.06\pm 0.72$	$89.35\pm 1.33$	$66.50\pm 3.42$	$3.679\pm 0.293$

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.

Architecture	30,000 updates	60,000 updates	120,000 updates
$2\times 768$	63.04	66.23	68.70
$3\times 768$	63.19	66.19	68.31
$4\times 768$	62.54	64.33	67.10
$3\times 384$	61.02	63.40	67.06
$3\times 1024$	62.16	64.66	66.50

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 architecture	Dense MFLOP $\downarrow$	$E_{\mathrm{VU}}$ $\uparrow$	Time (s) $\downarrow$	VU/s $\uparrow$
$3\times 768$ (primary)	$2.908$	$268{,}980.3$	$1.159\pm 0.008$	$\approx 6{,}746$
$2\times 768$	$1.728$	$445{,}409.0$	$1.144\pm 0.008$	$\approx 6{,}725$
$3\times 384$	$0.864$	$880{,}169.8$	$1.164\pm 0.010$	$\approx 6{,}536$
$4\times 768$	$4.087$	$186{,}692.3$	$1.161\pm 0.009$	$\approx 6{,}574$
$3\times 1024$	$4.925$	$157{,}535.8$	$1.167\pm 0.013$	$\approx 6{,}647$

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

E_{\mathrm{VU}}

is the number of distinct valid outputs per

10^{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.

Method	Full valid $\uparrow$	Strict valid $\uparrow$	Strict unique yield $\uparrow$	Strict VUN $\uparrow$	FCD $\downarrow$
EGO	$93.46\pm 0.82$	$89.49\pm 0.22$	$\mathbf{74.53\pm 1.17}$	$\mathbf{68.31\pm 1.03}$	$3.534\pm 0.638$
RLOO	$88.26\pm 3.12$	$85.89\pm 2.43$	$42.58\pm 7.09$	$39.39\pm 7.61$	$3.907\pm 0.162$
RLOO + rarity	$\mathbf{96.80\pm 2.19}$	$\mathbf{95.77\pm 2.10}$	$48.86\pm 4.15$	$42.07\pm 2.38$	$\mathbf{2.710\pm 0.535}$

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.

Method	Full valid $\uparrow$	Strict valid $\uparrow$	Strict unique yield $\uparrow$	Strict VUN $\uparrow$	FCD $\downarrow$
EGO	$76.71\pm 3.67$	$70.70\pm 2.59$	$40.71\pm 2.53$	$38.39\pm 2.89$	$5.250\pm 0.660$
RLOO	$98.64\pm 0.86$	$97.77\pm 1.04$	$14.00\pm 1.05$	$12.05\pm 0.53$	$4.161\pm 0.508$

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.

Autoregressive generation and autoregressive diffusion
Method	Steps	Valid (%) $\uparrow$	Unique (%) $\uparrow$	FCD $\downarrow$	VU/s $\uparrow$
GraphARM [ Kong et al., 2023 ]	$n\leq 38$	88.23	99.46	16.26	—
LO-ARM-st-sep [ Wang et al., 2025 ]	$D(n)$	96.26	100.00	3.229	—
PARD [ Zhao et al., 2024 ]	blockwise	95.23	99.99	1.98	—
Iterative diffusion and flow
GDSS a [ Jo et al., 2022 ]	1,000	97.01	99.64	14.656	$\approx 4.93$

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.

Architecture	Full valid	Strict valid	Connected	Unique given valid	Full VU	Strict VUN	FCD $\downarrow$
$3\times 384$	$84.42$ $\pm 4.62$	$79.53$ $\pm 1.86$	$95.08$ $\pm 2.76$	$93.16$ $\pm 1.03$	$78.62$ $\pm 3.88$	$74.79$ $\pm 2.01$	$29.58$ $\pm 1.49$
$2\times 768$	$90.72$ $\pm 5.31$	$86.71$ $\pm 7.76$	$95.93$ $\pm 2.50$	$97.06$ $\pm 0.68$	$88.08$ $\pm 5.66$	—	—
$3\times 768$	$88.35$ $\pm 2.75$	$85.59$ $\pm 4.20$	$97.20$ $\pm 2.88$	$96.69$ $\pm 0.24$	$85.43$ $\pm 2.86$	—	—
$4\times 768$	$91.23$ $\pm 0.29$	$86.39$ $\pm 1.83$	$95.16$ $\pm 1.59$	$94.89$ $\pm 1.00$	$86.56$ $\pm 0.73$	$81.70^{*}$	$28.89^{*}$
$\boldsymbol{3\times 1024}$	$94.02$ $\pm 3.94$	$90.84$ $\pm 6.97$	$96.82$ $\pm 3.03$	$96.42$ $\pm 0.21$	$90.66$ $\pm 4.00$	$88.22$ $\pm 6.86$	$30.36^{*}$

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.

Full validity (%)
Architecture	30,000 updates	60,000 updates	120,000 updates
$3\times 384$	$82.53\pm 4.53$	$83.62\pm 4.27$	$84.42\pm 4.62$
$2\times 768$	$85.62\pm 6.98$	$88.73\pm 5.78$	$90.72\pm 5.31$
$3\times 768$	$85.20\pm 3.00$	$86.23\pm 2.36$	$88.35\pm 2.75$
$4\times 768$	$87.82\pm 2.05$	$90.68\pm 0.13$	$91.23\pm 0.29$
$3\times 1024$	$91.13\pm 1.17$	$92.54\pm 2.29$	$94.02\pm 3.94$

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.

Architecture	VU/s (graph generation only) $\uparrow$
$3\times 384$	—
$2\times 768$	$722{,}733\pm 257{,}183$
$3\times 768$	$841{,}619\pm 28{,}323$
$4\times 768$	$(841{,}365\pm 25{,}413)^{a}$
$\boldsymbol{3\times 1024}$	$862{,}142\pm 46{,}109$

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\times 768

aggregate uses two available runs. Dashes denote unavailable rates; the highlighted profile is the configuration used in the main table.

Timing boundary	Time (s)	Full VU/s $\uparrow$
Graph generation only	$\approx 0.011$	$862{,}142\pm 46{,}109$
Generation through SMILES/CSV	$\approx 3.59$	$2{,}525\pm 160$

Appendix

Table 18: Measured timing boundaries for EGO

3\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.

$k$	Method	Full $\uparrow$	Strict $\uparrow$	Strict VUN $\uparrow$	Conn. $\uparrow$	$\mathrm{MMD}_{\rm cell}\downarrow$
16	EGO	97.33 $\pm$ 1.11	94.91 $\pm$ 1.57	90.84 $\pm$ 0.77	97.55 $\pm$ 0.94	0.0084 $\pm$ 0.0007
16	EGO 0	86.60 $\pm$ 2.81	83.82 $\pm$ 4.18	77.39 $\pm$ 3.75	97.07 $\pm$ 1.85	0.0052 $\pm$ 0.0009
16	STE (feature-ST)	99.21 $\pm$ 0.91	96.52 $\pm$ 1.75	7.51 $\pm$ 4.83	97.31 $\pm$ 2.29	0.0538 $\pm$ 0.0351
16	Soft	95.01 $\pm$ 1.74	89.33 $\pm$ 7.19	10.81 $\pm$ 4.21	94.29 $\pm$ 6.82	0.0430 $\pm$ 0.0313
24	EGO	95.48 $\pm$ 2.98	91.69 $\pm$ 3.19	89.45 $\pm$ 2.67	96.18 $\pm$ 2.12	0.0099 $\pm$ 0.0015
24	EGO 0	75.58 $\pm$ 11.00	69.38 $\pm$ 14.01	63.30 $\pm$ 10.96	93.17 $\pm$ 4.79	0.0065 $\pm$ 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

\mathrm{MMD}_{\rm cell}

is better, with comparisons restricted to a fixed cap.

$k$	Contrast	$\Delta$ Validity	$\Delta$ VUN	$10^{3}\Delta\mathrm{MMD}_{\rm cell}$
16	EGO $-$ EGO 0	$11.08\;[6.72,\,15.44]$	$13.45\;[8.62,\,18.28]$	$3.23\;[2.35,\,4.12]$
16	EGO 0 $-$ Soft	$-5.51\;[-13.45,\,2.44]$	$66.58\;[59.67,\,73.48]$	$-37.80\;[-76.99,\,1.39]$
16	EGO 0 $-$ STE	$-12.70\;[-19.28,\,-6.12]$	$69.88\;[61.27,\,78.49]$	$-48.64\;[-91.73,\,-5.55]$
16	STE $-$ Soft	$7.19\;[-1.20,\,15.58]$	$-3.30\;[-13.92,\,7.32]$	$10.84\;[-66.83,\,88.50]$
24	EGO $-$ EGO 0	$22.31\;[6.40,\,38.21]$	$26.15\;[13.23,\,39.08]$	$3.40\;[0.85,\,5.95]$
24	EGO 0 $-$ Soft	$-5.82\;[-37.61,\,25.97]$	$49.17\;[40.16,\,58.17]$	$-54.89\;[-104.17,\,-5.60]$

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

10^{3}

. Positive validity or yield differences and negative MMD differences favor the first method.

Method	Full valid	Strict valid	Strict VUN	Updates ( $10^{3}$ )
EGO	$86.94\pm 5.26$	$79.69\pm 8.47$	$77.91\pm 7.92$	$25.84\pm 0.29$
EGO 0	$71.53\pm 4.95$	$61.71\pm 3.58$	$53.73\pm 3.70$	$25.95\pm 0.26$
STE	$79.12\pm 4.15$	$57.26\pm 11.27$	$29.50\pm 1.76$	$145.73\pm 2.39$
Soft	$92.82\pm 4.70$	$66.81\pm 25.46$	$5.78\pm 9.86$	$156.58\pm 3.30$

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.

Method	Steps	Valid (%) $\uparrow$	VU (%) $\uparrow$	FCD $\downarrow$	VU/s $\uparrow$
Autoregressive generation
SMILES LSTM [ Brown et al., 2019 ]	$L$ tokens	95.9	$95.9^{a}$	$\approx 0.46$	—
G2PT-base [ Chen et al., 2025 ]	$L$ tokens	94.6	$94.6^{a}$	$\approx 0.34$	—
G2PT-large [ Chen et al., 2025 ]	$L$ tokens	95.3	$95.3^{a}$	$\approx 0.38$	—
Iterative diffusion and flow
DiGress [ Vignac et al., 2023 ]	500	85.2	$85.2^{a}$	$\approx 1.93$	—

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\leq 32

$k$	Method	Full $\uparrow$	Strict $\uparrow$	Strict VUN $\uparrow$	Conn. $\uparrow$	$\mathrm{MMD}_{\rm cell}\downarrow$
16	EGO	94.57 $\pm$ 2.66	90.27 $\pm$ 2.34	82.12 $\pm$ 2.19	95.68 $\pm$ 1.43	0.0091 $\pm$ 0.0024
16	EGO 0	86.71 $\pm$ 5.49	82.60 $\pm$ 6.82	66.01 $\pm$ 4.08	95.79 $\pm$ 1.59	0.0064 $\pm$ 0.0016
16	STE (feature-ST)	99.72 $\pm$ 0.38	97.88 $\pm$ 1.95	6.39 $\pm$ 1.38	98.16 $\pm$ 1.63	0.0364 $\pm$ 0.0189
16	Soft	97.75 $\pm$ 2.09	92.52 $\pm$ 4.41	7.74 $\pm$ 7.53	94.77 $\pm$ 2.62	0.0387 $\pm$ 0.0162
24	EGO	92.80 $\pm$ 5.42	86.27 $\pm$ 6.47	81.53 $\pm$ 6.81	93.44 $\pm$ 1.54	0.0109 $\pm$ 0.0037
24	EGO 0	74.42 $\pm$ 7.91	68.65 $\pm$ 7.93	58.68 $\pm$ 8.22	93.99 $\pm$ 3.17	0.0088 $\pm$ 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

\mathrm{MMD}_{\rm cell}

is better, with comparisons restricted to a fixed cap.

$k$	Contrast	$\Delta$ Validity	$\Delta$ VUN	$10^{3}\Delta\mathrm{MMD}_{\rm cell}$
16	EGO $-$ EGO 0	$7.68$ $[-0.68,\,16.03]$	$16.12$ $[10.91,\,21.32]$	$2.63$ $[0.49,\,4.76]$
16	EGO 0 $-$ Soft	$-9.92$ $[-15.07,\,-4.78]$	$58.27$ $[50.18,\,66.35]$	$-32.20$ $[-52.40,\,-12.01]$
16	EGO 0 $-$ STE	$-15.28$ $[-22.69,\,-7.88]$	$59.62$ $[55.76,\,63.48]$	$-29.99$ $[-52.55,\,-7.43]$
16	STE $-$ Soft	$5.36$ $[0.12,\,10.61]$	$-1.35$ $[-9.21,\,6.50]$	$-2.21$ $[-25.74,\,21.31]$
24	EGO $-$ EGO 0	$17.62$ $[5.49,\,29.75]$	$22.84$ $[12.98,\,32.70]$	$2.03$ $[-2.51,\,6.58]$
24	EGO 0 $-$ Soft	$-22.08$ $[-35.91,\,-8.26]$	$50.73$ $[33.29,\,68.17]$	$-89.57$ $[-152.35,\,-26.79]$

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

10^{3}

. Positive validity or yield differences and negative MMD differences favor the first method.

Metric	CFM-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.

Dataset	Method	VU (%)	Dense MFLOP	$E_{\mathrm{VU}}$
QM9	GraphARM	$\approx 86.30$	—	—
	G2PT-base	$\approx 95.83$	—	$67.1^{u}$
	CatFlow (100)	$\approx 99.76$	—	$169^{u}$
	DeFoG (500)	$\approx 95.72$	86,234.240	11.1
	MELD	$\approx 96.49$	—	$1.72^{u}$
	MolGAN	$\approx 10.20$	0.714	142,945

Appendix

Table 26: Dense inference work and efficiency for the molecular comparison. VU is a percentage of raw proposals, and MFLOP denotes

10^{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.

Method	QM9	ZINC250K	GuacaMol
EGO	$0.2999\pm 0.0012$	2.0983	$14.8468\pm 0.0697$
DeFoG [ Qin et al., 2025 ]	6.5	14	141
Cometh [ Siraudin et al., 2025 ]	6	—	$\approx 96$
CFM [ Roos et al., 2026 ]	$\approx 48$	$\approx 144$	$\approx 2{,}200$ to $2{,}800$

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.

Metric	42 / 41	43 / 42	44 / 43
Mean deadline per query (s)	6.176	6.105	5.974
Mean proposals screened, EGO	1,822.94	1,947.70	1,907.35
Mean proposals screened, MoLeR	256.0	256.0	256.0
Filter-passing proposals, EGO	55,152	52,975	56,064
Filter-passing proposals, MoLeR	5,434	5,485	5,417
Passing fraction (%), EGO	30.25	27.20	29.39

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.

Metric	EGO	MoLeR
Usable completions (%)	$77.67\pm 2.57$	$97.58\pm 0.09$
Distinct usable yield (%)	$66.78\pm 1.15$	$85.80\pm 0.12$
Filter passing fraction (%)	$30.3\pm 1.1$	$19.4\pm 0.1$
Distinct filtered completions	$20{,}245\pm 755$	$16{,}684\pm 79$
Scaffold coverage (/100)	$77.7\pm 1.5$	$86.3\pm 0.6$
Generation to SMILES (s)	$41.82\pm 8.39$	$1{,}851.82\pm 8.05$

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.

Method	Seed type	Seed	Distinct filtered count
EGO	Training	42	$19{,}527$
EGO	Training	43	$20{,}175$
EGO	Training	44	$21{,}033$
MoLeR	Sampling	41	$16{,}763$
MoLeR	Sampling	42	$16{,}683$
MoLeR	Sampling	43	$16{,}606$

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.

Method	Update time	Compilation	Validation
EGO	$46.12\pm 0.59$	$48.41\pm 0.75$	$39.58\pm 1.91$
STE	$34.06\pm 0.38$	$54.18\pm 0.88$	$30.84\pm 1.01$
Soft	$34.37\pm 0.65$	$54.51\pm 1.18$	$16.36\pm 1.28$

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.

Method	LeNet Fréchet $\downarrow$	Predicted classes / 10 $\uparrow$
EGO	$599.8\pm 47.1$	$6.33\pm 0.58$
STE	$522.4\pm 95.6$	$7.67\pm 0.58$
Soft	$\mathbf{420.6\pm 89.1}$	$\mathbf{8.33\pm 0.58}$

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.

Metric	EGO	STE	Soft
LeNet KID $\downarrow$	$0.4517\pm 0.0490$	$0.3868\pm 0.0853$	$0.3069\pm 0.0718$
Pooled-image MMD 2 $\downarrow$	$0.0414\pm 0.0008$	$0.0235\pm 0.0094$	$0.0145\pm 0.0062$
Class-balance TV $\downarrow$	$0.545\pm 0.020$	$0.463\pm 0.045$	$0.401\pm 0.099$
Normalized class entropy $\uparrow$	$0.628\pm 0.035$	$0.738\pm 0.044$	$0.754\pm 0.108$

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.

Method	Selected update	Predicted classes / 10
EGO	29,000 / 30,000 / 30,000	6 / 6 / 7
STE	500 / 30,000 / 500	8 / 8 / 7
Soft	1,500 / 28,000 / 1,000	8 / 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.

Panel of Figure 9 — Figure 9: Binarized MNIST samples generated by EGO , STE, and Soft, from top to bottom.

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