Fusion is the New Mutation: Bandit-Guided Evolution on Workflow Graphs

Zhiwei Shang; Jiahang Sun; Mingrong Gong; Mingze Kong; Zikun Qu; Pingchen Lu; Junhao Dong; Zhipiao Liu

cs.AIOct 4, 2026

Fusion is the New Mutation: Bandit-Guided Evolution on Workflow Graphs

Authors: Zhiwei Shang, Jiahang Sun, Mingrong Gong, Mingze Kong, Zikun Qu, Pingchen Lu, Junhao Dong, Zhipiao Liu, +4 more

Organizations: School of Data Science, The Chinese University of Hong Kong, Shenzhen, China · College of Computing and Data Science, Nanyang Technological University, Singapore · Huawei Technologies Co., Ltd., China · Artificial Intelligence Thrust, The Hong Kong University of Science and Technology (Guangzhou), China

Abstract

Automated agentic workflow optimization relies on costly evaluations, making it essential to allocate a limited evaluation budget effectively. Multi-parent fusion can reuse designs from previously discovered workflows, but identifying promising parent combinations requires learning from limited fusion feedback. We introduce DAGO (Directed Acyclic Graph Optimization), a contextual-bandit-guided framework that learns which parent workflows to fuse under a limited evaluation budget. DAGO formulates each candidate parent combination as an arm, represented by pretrained embeddings of its constituent workflows' code and prompts. A diagonal LinUCB policy learns a shared reward model across arms and balances exploitation of arms with high predicted offspring quality against uncertainty-driven exploration. After an arm is selected, an LLM generates a child workflow through summary-guided fusion, and the child's validation score serves as the reward for updating the bandit. A shared directed acyclic graph maintains discovered workflows and their multi-parent lineage, providing an expanding pool of parents for subsequent arm proposals. Across six benchmarks covering mathematical reasoning, code generation, and question answering, DAGO achieves the highest macro-average score among the evaluated baselines. Under matched validation-evaluation budgets, it improves over AFlow from 80.3 to 81.7 while reducing aggregate search expenditure by 11.2%. Ablation studies show that LinUCB-guided arm selection outperforms both random selection and its exploration-free variant, supporting the value of feedback-driven selection and exploration-exploitation balance.

Figures & tables

Figure 1: Overview of the DAGO framework. (a) The DAG stores workflows, validation scores, and parent lineage. (b) Annealed score sampling proposes parent combinations. (c) Diagonal LinUCB selects a combination from code-and-prompt embeddings. (d) Summary-guided fusion generates a child. Validation feedback updates LinUCB and the DAG; the highest-scoring workflow is returned.

Panel of Figure 2 — Figure 2: Overall performance across benchmarks and average ranking. Selected methods from Table 1 , with test scores averaged over three executions. (a) Solve rate / pass@1 / F1 for math / code / QA; brackets show DAGO’s gains over AFlow in points. (b) Six-task macro averages.

	QA		Code		Math
Method	HotpotQA	DROP	HumanEval	MBPP	GSM8K	MATH	Avg.
IO	68.1	68.3	87.0	71.8	92.7	48.6	72.8
CoT Wei et al. (2022)	67.9	78.5	88.6	71.8	92.4	48.8	74.7
CoT-SC Wang et al. (2023)	68.9	78.8	91.6	73.6	92.7	50.4	76.0
MedPrompt Nori et al. (2023)	68.3	78.0	91.6	73.6	90.0	50.0	75.3
MultiPersona Wang et al. (2024)	69.2	74.4	89.3	73.6	92.8	50.8	75.1

Table 1: Main results on six benchmarks. Means over three test executions (%), grouped by task category. Avg. is the six-task macro average; best scores are bold .

Figure 3: Validation best-so-far trajectory during optimization. Running best validation scores on HotpotQA, MATH, and MBPP.

Table 8

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Parameter	Value	Description / usage
Budget	Initial population size $N$	10	Number of diverse workflows generated at initialization (inserted into the DAG before bandit-guided fusion).
	Max iterations $T$	30	Maximum number of fusion iterations in Algorithm 1 .
	Stopping window $C_{\mathrm{patience}}$	5	Check for a lack of relative improvement over the most recent five fusion-round validation scores.
	Minimum iterations $T_{\min}$	20	Enable early stopping only after at least $T_{\min}$ iterations.
Arm proposal	Candidate-arm cap $M$	200	Propose $M_{t}=\min\{M,\binom{\|\mathcal{V}_{t-1}\|}{J}\}$ distinct parent tuples in round $t$ .
	Parents per arm $J$	5	Number of parent workflows fused to synthesize one child.

Appendix

Table 8: DAGO hyperparameters (default unless otherwise specified).

Dataset	Domain	Val Size	Test Size	Metric	Notes
GSM8K	Math Reasoning	264	1,055	Accuracy	Grade-school word problems
MATH	Math Reasoning	119	486	Accuracy	Level 5 (hardest) problems only
HumanEval	Code Generation	33	131	Pass@1	Python function synthesis
MBPP	Code Generation	86	341	Pass@1	Mostly Basic Python Problems
HotpotQA	Question Answering	200	800	F1	Multi-hop reasoning required
DROP	Question Answering	200	800	F1	Discrete reasoning over paragraphs

Appendix

Table 9: Benchmark datasets and evaluation protocol used in our experiments (aligned with AFlow). Val Size is the fixed validation subset used for reward estimation during search (held fixed across iterations), and Test Size is the disjoint evaluation subset used for reporting final results.

Component	Parameter	Reported value
Initialization	$N$ , $q_{\mathrm{init}}$	$10$ , $1$
Fusion budget	$T$ , $q$ , $B_{\max}$	$30$ , $3$ , $100$
Stopping	$T_{\min}$ , $C_{\mathrm{patience}}$ , $\delta_{\mathrm{stop}}$	$20$ , $5$ , $10^{-4}$
Candidate cap / arm size	$M$ , $J$	$200$ , $5$
Sampling	$\tau_{\mathrm{init}}\to\tau_{\mathrm{final}}$ , $p_{\min}$	$2.0\to 0.5$ , $0.01$
Bandit	$\lambda$ , $\nu$	$0.2$ , $0.1$

Appendix

Table 10: Reported settings used to specify the revised procedure. The table does not introduce new tuned parameters.

Metric	GSM8K	MATH	HumanEval	MBPP	HotpotQA	DROP
Total Iterations	28	27	25	25	30	30
Early Stopped	Yes	Yes	Yes	Yes	No	No
Total DAG Nodes	38	37	35	35	40	40
Total DAG Edges	140	135	125	125	150	150
Best Initial Score	95.8%	50.4%	87.9%	83.7%	77.2%	82.9%
Best Fusion Score	95.8%	55.7%	88.9%	86.0%	79.3%	84.0%

Appendix

Table 11: DAGO optimization statistics per benchmark. Initial scores use one execution of the fixed validation split; fusion scores average three executions. Relative improvement is measured against the best initial score. Fusion-reward standard deviations describe variation across evaluated fusion nodes, not variation across independent searches or test executions.

Pattern	GSM8K	MATH	HumanEval	MBPP	HotpotQA	DROP
Multi-Solution Ensemble	✓	✓	✓	✓	✓	–
Computational Verification	✓	✓	✓	✓	–	–
Conditional Retry	–	–	✓	✓	–	–
Explicit Problem Analysis	–	–	✓	✓	✓	✓
Type-Aware Processing	–	–	–	–	–	✓
Format-Aware Generation	–	✓	✓	✓	✓	✓

Appendix

Table 12: Emergent workflow patterns discovered by DAGO across task types. ✓ indicates pattern presence.

Dataset	Best Node	Direct Parents	Unique Ancestors	Max Depth
GSM8K	11	5	5	1
MATH	15	5	5	1
HumanEval	12	5	7	2
MBPP	23	5	13	3
HotpotQA	29	5	21	7
DROP	25	5	18	7

Appendix

Table 13: Ancestral analysis of best fusion workflows. Unique ancestors counts distinct nodes in the ancestral DAG.

Optimization Configuration
Statistic	GSM8K	MATH	HumanEval	MBPP	HotpotQA	DROP
Total Iterations	28	27	25	25	30	30
Early Stopped	Yes	Yes	Yes	Yes	No	No
Stopping Trigger	Stagnation	Stagnation	Stagnation	Stagnation	Max Iter	Max Iter
DAG Statistics
Total Nodes	38	37	35	35	40	40

Appendix

Table 14: DAGO optimization statistics per benchmark. Initial scores use one validation execution; fusion scores average three. Reward-distribution statistics are computed over fusion nodes only, and

\Delta

is measured in percentage points.

Percentile	GSM8K	MATH	HumanEval	MBPP	HotpotQA	DROP
Min (P0)	29.5%	38.7%	66.7%	0.0%	0.0%	34.6%
P25	92.8%	46.2%	84.8%	77.9%	69.2%	74.4%
Median (P50)	93.8%	48.7%	86.9%	81.0%	74.3%	79.0%
P75	94.3%	51.0%	87.9%	83.3%	75.3%	80.9%
Max (P100)	95.8%	55.7%	88.9%	86.0%	79.3%	84.0%

Appendix

Table 15: Percentiles of stored validation scores across all nodes (initial + fusion). Initial scores use one validation execution; fusion scores average three.

Dataset	Best Node	Parent IDs	Parent Scores	Unique Ancestors	Max Depth
GSM8K	11	{4,1,0,7,6}	{95.8,83.3,94.7,94.3,94.3}%	5	1
MATH	15	{1,9,2,3,4}	{44.5,42.9,46.2,50.4,38.7}%	5	1
HumanEval	12	{9,0,7,8,11}	{87.9,87.9,87.9,87.9,86.9}%	7	2
MBPP	23	{8,6,16,13,11}	{77.9,76.7,81.0,81.0,80.2}%	13	3
HotpotQA	29	{19,16,5,14,27}	{75.1,75.3,74.5,73.3,74.9}%	21	7
DROP	25	{23,14,8,9,17}	{79.3,78.5,34.6,82.9,75.6}%	18	7

Appendix

Table 16: Ancestry analysis of best-performing workflows per benchmark.

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We study the generation of agentic workflows that jointly optimize multiple objectives, such as accuracy, cost, latency, robustness, and consistency. Existing methods for workflow generation typically optimize accuracy alone or a weighted sum of objectives, so each trained generator commits to one fixed trade-off and must be retrained from scratch when preferences change. To alleviate this, we propose MoFlow, which generates workflows optimized across varied preferences. Specifically, MoFlow formulates workflow generation as a multi-objective Markov decision process and solves it by leveraging Convex-Hull Monte Carlo Tree Search with optimistic set-valued backups, where every node stores a set of reachable trade-offs rather than one weighted score. A single search thus approximately covers the Pareto front, from which MoFlow can return a workflow for any preference by lookup without retraining. We evaluate MoFlow against six strong baselines on six benchmarks spanning mathematics, code, and question answering. Since the baselines are single-scalar optimizers by design, an apples-to-apples comparison is difficult. We instead adopt an evaluation setup that favors the baselines, in that they are rerun for each testing preference, which MoFlow never sees. Even under this stringent setup, MoFlow achieves the highest average hypervolume.

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