Paper deep dive
Heterogeneous Agentic Coevolution
Wenqian Ye, Ziwei Guan, Zi Yang, Shivani Modi, Cici Cong Wu, Yunlong He, Yuxin Chen, Zhengyu Zhang, Liang Luo, Venkatesh Ranganathan, Jianhui Sun, Yantao Yao, Wenlin Chen, Buyun Zhang, Ellie Dingqiao Wen, Henry Kautz, Aidong Zhang
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Recent advances in self-improving LLM agents have demonstrated the potential of AI systems that drive their own intelligence. However, many domain-specific applications rely on specialized foundation models that possess strong domain expertise but remain dependent on human-designed improvements. Extending recursive self-improvement to these models presents a key challenge: as a specialist improves, the strategies used to advance it must also adapt. To address this challenge, we propose Heterogeneous Agentic CoEvolution (HACE), a framework that jointly evolves specialized foundation models and LLM agents through reciprocal learning. HACE improves the specialist through supervised learning while training the agent with on-policy reinforced self-training, guided by verified performance gains on the current specialist. The agent's exploration, in turn, informs subsequent specialist training, enabling both models to adapt to each other's progress. To sustain this reciprocal learning, HACE further employs evolutionary search to optimize their interaction harness, adapting how the models exchange information and select training experiences as their capabilities evolve. We evaluate HACE across three challenging domains: recommendation systems, protein function prediction in biology, and crystal stability prediction in material discovery. We show that HACE outperforms the evaluated self-improving methods on the primary task metrics across all three domains. These results shed light on heterogeneous coevolution as a universal approach to improving specialized foundation models through reciprocal interaction with a learning agent.
Tags
Links
- Source: https://arxiv.org/abs/2610.16078v1
- Canonical: https://arxiv.org/abs/2610.16078v1
Trouble viewing inline? Open PDF directly →
Full Text
106,294 characters extracted from source content.
Expand or collapse full text
Heterogeneous Agentic Coevolution Wenqian Ye Affiliation: Meta AI Affiliation: University of Virginia Work done at Meta Ziwei Guan Affiliation: Meta AI Zi Yang Affiliation: Meta AI Shivani Modi Affiliation: Meta AI Cici Cong Wu Affiliation: Meta AI Yunlong He Affiliation: Meta AI Yuxin Chen Affiliation: Meta AI Zhengyu Zhang Affiliation: Meta AI Liang Luo Affiliation: Meta AI Venkatesh Ranganathan Affiliation: Meta AI Jianhui Sun Affiliation: Meta AI Yantao Yao Affiliation: Meta AI Wenlin Chen Affiliation: Meta AI Buyun Zhang Affiliation: Meta AI Ellie Dingqiao Wen Affiliation: Meta AI Henry Kautz Affiliation: University of Virginia Aidong Zhang Affiliation: University of Virginia Abstract Recent advances in self-improving LLM agents have demonstrated the potential of AI systems that drive their own intelligence. However, many domain-specific applications rely on specialized foundation models that possess strong domain expertise but remain dependent on human-designed improvements. Extending recursive self-improvement to these models presents a key challenge: as a specialist improves, the strategies used to advance it must also adapt. To address this challenge, we propose Heterogeneous Agentic CoEvolution (HACE), a framework that jointly evolves specialized foundation models and LLM agents through reciprocal learning. HACE improves the specialist through supervised learning while training the agent with on-policy reinforced self-training, guided by verified performance gains on the current specialist. The agent’s exploration, in turn, informs subsequent specialist training, enabling both models to adapt to each other’s progress. To sustain this reciprocal learning, HACE further employs evolutionary search to optimize their interaction harness, adapting how the models exchange information and select training experiences as their capabilities evolve. We evaluate HACE across three challenging domains: recommendation systems, protein function prediction in biology, and crystal stability prediction in material discovery. We show that HACE outperforms the evaluated self-improving methods on the primary task metrics across all three domains. These results shed light on heterogeneous coevolution as a universal approach to improving specialized foundation models through reciprocal interaction with a learning agent. †correspondence: Wenqian Ye (wenqian@virginia.edu), Buyun Zhang (buyunz@meta.com) * 1 * 1.1 * 1.1.1 22footnotetext: These authors jointly supervised this work Figure 1: From human-directed optimization to heterogeneous agentic coevolution. HACE brings together specialized foundation models and LLM agents into a shared learning process, where each helps determine how the other improves. Across rounds, both learners and their interaction harness evolve, creating new opportunities for reciprocal improvement. 2 Introduction Lately, large language model (LLM) agents are beginning to automate the experimental work behind model development. Agents can revise their implementations, improve their interaction harnesses, and modify the procedures that generate further improvements (Zhang et al., 2026a; Lee et al., 2026; Zhang et al., 2026b). They can also continue training on experience collected during adaptation and exploration (Zweiger et al., 2025; Yuksekgonul et al., 2026). These advances suggest that model development itself can become an autonomous learning process. Extending this progress to scientific AI requires bringing the domain expertise of specialized foundation models into this process. In scientific AI, specialized foundation models generally learn reusable domain knowledge through large-scale pretraining. Protein language models learn biological regularities from sequences (Lin et al., 2023), while pretrained interatomic potential models learn the energies and forces governing material behavior (Deng et al., 2023). In recommendation, architectures such as HSTU (Zhai et al., 2024) and Wukong (Zhang et al., 2024) extend this scaling approach to large collections of user interactions. These models acquire domain expertise that often exceeds that of general LLM agents due to specific designs. Yet they still cannot self-evolve like LLM agents. Once supervised training converges and further training yields no improvement, researchers have to manually identify useful designs to improve the specialized foundation model, such as better feature engineering or algorithmic changes, as shown in Figure 1. Recent developments in agentic systems pave a feasible path to automate this process. Their ability to interpret feedback, explore alternatives, and revise executable procedures allows them to influence how another model learns. For example, Eureka (Ma et al., 2024) demonstrates this possibility by using LLMs to design reward programs for training robot policies autonomously. However, most self-evolving systems still rely on proprietary LLMs whose parameters remain fixed while agent code and interaction harnesses evolve (Zhang et al., 2026a; Lee et al., 2026; Zhang et al., 2026b). The experience collected during improvement thus does not update the underlying language model. For specialized foundation models, the challenge is for the LLM agent model to keep improving the specialized model as the specialized model advances. An intervention that improves the initial model can become redundant after further training. An agent that learns to address the initial model’s weaknesses, therefore, faces a changing task as those weaknesses are resolved. Continued progress requires the agent to learn continually with the model when it improves. We refer to this mechanism as reciprocal learning. To close the gap, we propose a novel framework, Heterogeneous Agentic Coevolution (HACE), which coevolves a specialized foundation model (i.e., the specialist) with a heterogeneous LLM agent which learns to guide the coevolution. The key idea is to improve the specialist through supervision while also training the agent to keep advancing it under an adaptive curriculum. Agent learning is guided by verified gains over the current specialist. The agent’s exploration, in turn, guides the selection of experiences for subsequent specialist training. As the specialist improves, it changes which agent contributions are useful. As the agent improves, it changes the learning opportunities available to the specialist. This reciprocal feedback loop seamlessly connects supervised specialist learning with on-policy reinforced self-training. We need to ensure that these models interact and develop their capabilities in a continuously improving direction. What the agent sees and which experiences reach specialist training determine the opportunities that either model can use. Thus, we also propose an evolutionary interaction harness in HACE that governs information exchange and training experience selection. Evolutionary search evaluates harness changes through their effects on task performance and specialist learning. A better harness can expose opportunities that the current models have not utilized, while model updates change which harness designs remain effective. HACE thus improves the specialist while learning how to organize its continued improvement. Domain expertise, agentic capabilities, and the process connecting their learning coevolve together. To show the effectiveness, we comprehensively evaluate HACE in three challenging domains, including recommendation systems, protein function prediction in biology, and crystal stability prediction in material discovery. In each domain, we begin with specialists already trained to convergence, where continued training under the original supervised procedure yields no further improvement. This setting stress-tests whether interaction with an evolving agent can create opportunities for learning beyond that plateau. Across diverse specialist architectures and large-scale datasets, HACE achieves state-of-the-art performance and outperforms all evaluated evolution methods. These results demonstrate that HACE enables specialists to continue improving beyond supervised convergence. HACE brings their existing domain expertise into autonomous self-improvement while developing the agentic capabilities needed to advance it. To summarize, our main contributions are as follows: • We introduce HACE, an evolutionary framework for autonomous self-improvement that brings specialized foundation models and LLM agents into a shared coevolution process. • We propose a novel reciprocal learning mechanism in which verified agent contributions guide agent learning, agent exploration also informs specialist training, and an evolving harness adaptively coordinates their reciprocal interaction. • We empirically demonstrate the effectiveness of HACE across three challenging domains: recommendation, protein function prediction, and material discovery. The results show that HACE can substantially push the domain-specific performance beyond the capability boundary. 3 Related Work Agent and harness optimization. Recent work uses execution feedback to improve agent programs, interaction harnesses, and the procedures that propose further improvements (Hu et al., 2025; Zhang et al., 2026a; Lee et al., 2026; Luo et al., 2026; Zhang et al., 2026b; Cemri et al., 2026; Liu et al., 2026b). Other methods improve the underlying model through parameter updates. Reinforced self-training learns from responses selected by task rewards (Singh et al., 2024), while self-adaptation and test-time learning use generated experience or task feedback for further training (Zweiger et al., 2025; Yuksekgonul et al., 2026). HACE connects harness search with parameter learning in both a specialized foundation model and an LLM agent. The harness controls their interaction and the experience used to train each model. Coevolution and reciprocal learning. Coevolution can be traced to reciprocal evolutionary responses between interacting populations in biology (Janzen, 1980). In machine learning, many methods have connected the development of reward programs, environments, or optimization procedures with the learners they guide (Ma et al., 2024; Liu et al., 2026c; Wang et al., 2019; Dennis et al., 2020; Chen et al., 2026; Yang et al., 2026; Li et al., 2026). HACE studies this dependence between models with different capabilities and learning signals. Agent exploration can change which instances train the specialized model, whose predictions in turn determine which agent responses pass verification. Our analysis characterizes this dependence through the distribution of verified experience and identifies when an update to either learner changes its allocation across instances. Agent distillation. Agent distillation transfers reasoning and tool use from LLM agents to smaller language models (Liu et al., 2026a; Kang et al., 2025). MAD-OPD derives supervision from teachers that debate over the student’s current trajectories (Wang et al., 2026). These methods update the student while keeping teacher parameters fixed. HACE is distinct from this line of work and takes a step forward, where an LLM agent with lower overall domain performance can still improve on the specialized foundation model. Agent exploration guides instance selection for specialist training, while verified contributions guide response selection back for agent learning. Both models can therefore coevolve reciprocally without assigning either a fixed teacher role. 4 Heterogeneous Agentic Coevolution In this section, we formally present Heterogeneous Agentic Coevolution (HACE), a framework for reciprocal self-improvement between specialized foundation models and LLM agents. Figure 2 illustrates how an evolving harness connects specialist adaptation, joint selection, and agent learning within one coevolution loop. In Section 4.1, we describe how specialist predictions can guide agent exploration and how the resulting experience selects instances for specialist training with original task labels. In Section 4.2, we incorporate the proposed harness together with the adapted specialist, then use the retained pair to identify fresh responses for on-policy agent learning. These updates allow each learner to shape the training experience of the other precisely. In Section 4.3, we establish a source risk bound that relates specialist learning to the coverage of accepted experience within the coevolution. Figure 2: Overview of HACE. Agent exploration selects specialist training experience. Joint validation retains the specialist and harness, after which fresh verified responses train the agent and shape later exploration. The on-policy agent optimization step uses reinforced self-training. 4.1 Reciprocal Learning Let S denote the specialist, A the agent, and H the harness. Round t=0,1,…t=0,1,… starts from (St,Ht,At)(S_t,H_t,A_t). We propose a harness H′H by modifying a parent from an archive of accepted harnesses. Its fields control the task information and specialist context shown to the agent, how responses are interpreted, and which examples train each learner. Let ℐI index the finite nonempty training pool of labeled instances (xi,yi)(x_i,y_i). Under H′H , the agent receives a prompt constructed from a training input xix_i and the predictions of StS_t, without access to its label yiy_i. We evaluate each response τ against yiy_i using the same task constraints as the specialist. Let qiA(τ)q_i^A(τ) and qiSq_i^S denote their respective task qualities, with larger values preferred. The response contribution is ri(τ)=qiA(τ)−qiS.r_i(τ)=q_i^A(τ)-q_i^S. (1) A response passes verification when its prediction is valid, supervision is evaluable, and ri(τ)>0r_i(τ)>0. An agent with lower average task quality can still provide verified responses on individual instances. The specialist and harness used in each collection determine both quality terms. Appendix B.3 specifies the task quality and verification rules. The harness also determines which prediction candidates the agent can access. Additional sampling cannot recover a relevant candidate excluded from that view. Changing the view can therefore create opportunities that further sampling under the same harness would miss, while the curriculum determines which exploration outcomes receive the specialist’s training budget. The harness uses these outcomes to select a nonempty set of training indices ℐtS⊆ℐI_t^S . Its curriculum can prioritize verified improvements, unsuccessful exploration, or uncertain joint predictions, with uniform sampling as an option. Write SϕS_φ for the specialist with trainable parameters ϕφ, which may belong to the foundation model or its prediction head. A fixed optimization budget starting from StS_t produces S′S using ℒS,t(ϕ)=1|ℐtS|∑i∈ℐtSℓtask(Sϕ(xi),yi),L_S,t(φ)= 1|I_t^S| _i _t^S _task(S_φ(x_i),y_i), (2) where ℓtask _task is the original supervised task loss. Agent exploration determines where to train, while task labels determine what the specialist learns. The harness thus affects both current predictions and the specialist produced by adaptation. We evaluate H′H and S′S together after specialist adaptation to account for both effects. 4.2 Joint Selection Joint selection tests whether specialist adaptation under the harness improves joint predictions. For a candidate c in the view for instance i, let si(c)s_i(c) be its specialist score after any stored calibration, with larger values preferred, and ai(c)a_i(c) its agent score. Joint prediction ranks candidates by si(c)+λai(c),λ≥0.s_i(c)+λ a_i(c), λ≥ 0. (3) Here ai(c)=1/log2(u+1)a_i(c)=1/ _2(u+1) when c appears at position u in the agent ranking, and zero otherwise. The weight λ controls the agent’s influence, with zero recovering the specialist ordering within the same view. Training, validation, and final evaluation use disjoint splits. With AtA_t fixed, we compare the proposed and current pairs on the same validation batch ℬtB_t. Let Jℬt(S,H,A)J_B_t(S,H,A) be the mean joint task quality for state (S,H,A)(S,H,A). Both states use the same two folds, fitting their own weights from a fixed grid on one fold and evaluating on the other, then exchanging roles. The proposal is accepted when ΔJt=Jℬt(S′,H′,At)−Jℬt(St,Ht,At)>0. J_t=J_B_t(S ,H ,A_t)-J_B_t(S_t,H_t,A_t)>0. (4) Acceptance retains (S′,H′)(S ,H ) and records the harness and score. Otherwise, we restore (St,Ht)(S_t,H_t) and the specialist training state. Write (S¯t,H¯t)( S_t, H_t) for the pair retained after this decision. The retained pair now defines the context and reference for agent learning. A specialist update can change which responses improve on its predictions, and a harness update can change the information available to the agent. We therefore collect fresh responses with AtA_t under (S¯t,H¯t)( S_t, H_t), keep the original response with the largest positive contribution for each eligible instance, and apply the agent curriculum. This makes agent supervision depend on the current specialist’s remaining opportunities for improvement. Let tZ_t be the indexed dataset of selected prompt and response concatenations, with repeated records counted separately. For a record z, let |z||z| count its nonpadding tokens, zℓz_ be its token at position ℓ , and z<ℓz_< its preceding prefix. Write pθ(zℓ∣z<ℓ)p_θ(z_ z_< ) for the agent’s next-token probability before decoding transformations, with trainable parameters θ. For nonempty tZ_t, on-policy reinforced self-training (Singh et al., 2024) minimizes ℒA,t(θ)=−1|t|∑z∈t1|z|−1∑ℓ=2|z|logpθ(zℓ∣z<ℓ).L_A,t(θ)=- 1|Z_t| _z _t 1|z|-1 _ =2^|z| p_θ(z_ z_< ). (5) Each record has at least two tokens. The loss averages predicted tokens within each record, including prompt and response tokens, then averages records. Contribution scores select training responses without weighting this loss. The dataset stays fixed during fitting, and an empty dataset leaves the agent unchanged. Agent learning follows either selection outcome, giving the next state (St+1,Ht+1,At+1)=(S¯t,H¯t,At+1)(S_t+1,H_t+1,A_t+1)=( S_t, H_t,A_t+1). Its updated responses guide exploration in the next round. 4.3 Learning Guarantee The training distribution can change with model updates and harness selection. To study learning under that changing distribution, we theoretically analyze accepted exposure and the final specialist’s loss on the resulting experience. Let μi≥0 _i≥ 0, with ∑i∈ℐμi=1 _i _i=1, be fixed source probabilities. Fix a finite nonempty specialist class ℱF before exploration and a fixed loss ℓi(f)∈[0,1] _i(f)∈[0,1] for each i∈ℐi and f∈ℱf . This analysis loss is independent of the harness and may differ from ℓtask _task. The specialist’s source risk is Rμ(f)=∑i∈ℐμiℓi(f)R_μ(f)= _i _i _i(f). Let ℋtH_t be the observed history before round t, and let Wt,iW_t,i be the accepted training weight of instance i. A nonempty accepted update assigns total weight one uniformly to newly selected instances. All other weights are zero. These weights are known by the end of the round. After T rounds, define cumulative expected accepted exposure and loss by CT(i)=∑t<T[Wt,i∣ℋt],L^T(f)=∑t<T∑i∈ℐWt,iℓi(f),C_T(i)= _t<TE[W_t,i _t], L_T(f)= _t<T _i W_t,i _i(f), (6) where t<Tt<T denotes rounds 0,…,T−10,…,T-1. In L^T(ST) L_T(S_T), all accepted instances are evaluated at the final specialist ST∈ℱS_T , which may depend on the observed history. For v>0v>0, let UT(v)=∑i:CT(i)<vμiμiU_T(v)= _i:C_T(i)<v _i _i be the source mass with insufficient exposure. Theorem 1 (Learning from Evolving Experience). For any fixed δ∈(0,1)δ∈(0,1), with probability at least 1−δ1-δ over the learning process, every final specialist ST∈ℱS_T satisfies, simultaneously for all integers T≥1T≥ 1 and all v>0v>0, Rμ(ST)≤UT(v)+L^T(ST)+log(|ℱ|/δ)(1−e−1)v.R_μ(S_T)≤ U_T(v)+ L_T(S_T)+ (|F|/δ)(1-e^-1)v. (7) On the same event, Rμ(ST)→0R_μ(S_T)→ 0 whenever vT>0v_T>0 satisfies vT→∞v_T→∞, UT(vT)→0U_T(v_T)→ 0, and L^T(ST)/vT→0 L_T(S_T)/v_T→ 0. The bound separates reaching useful training instances from learning and retaining their information. Harness search, agent exploration, and joint selection determine accepted exposure. Specialist optimization determines the final loss on that experience, including whether later updates preserve earlier learning. Sufficient exposure across the source distribution and small loss relative to that exposure yield a small risk bound, even when selection depends on the full preceding history. Appendix B.1 gives the complete proof. Proposition 1 explains how agent fitting and reference changes affect verification before curriculum selection and acceptance determine exposure. 5 Experiments In this section, we empirically evaluate whether HACE can push the capability boundary with the converged specialized foundation models across three important domains (recommendation systems, protein function prediction, and material discovery), compared with other self-improving methods. Experimental setups. By default, we use Gemma4-E4B (Gemma Team, 2026) as the LLM agent. We also conduct more experiments with various LLM agents (Gemma4-E2B, Gemma4-26B-A4B) in Section 5.3. On the specialized foundation models, for recommendation tasks, we evaluate on MovieLens 1M (ML-1M) (Harper and Konstan, 2015) and four Amazon categories (McAuley et al., 2015) using three mainstream foundation models SASRec (Kang and McAuley, 2018), HSTU (Zhai et al., 2024), and Wukong (Zhang et al., 2024). For protein function prediction, we use frozen ESM-2 (Lin et al., 2023) and ProtT5-XL (Elnaggar et al., 2022) backbones on CAFA (Jiang et al., 2016). For crystal stability prediction in material discovery task, we use CHGNet (Deng et al., 2023) on WBM (Wang et al., 2021; Riebesell et al., 2025). Base denotes the initial specialists, which are already trained to convergence before coevolution. Unless stated otherwise, reported scores apply the fusion in Equation 3 with the weight selected by cross fitting, for Base and for all methods alike. At round 0 the agent is untrained and cross fitting selects a zero weight. Table 7 shows that HACE is agnostic to the specialist architectures and their adaptation scope, including causal attention models, a factorization network, protein language models, and a graph neural potential model. We compare against popular self-improving methods, including ReST-EM (Singh et al., 2024), Meta-Harness (Lee et al., 2026), OpenEvolve (Sharma, 2025), DGM (Zhang et al., 2026a), and T-Discover (Yuksekgonul et al., 2026). The recommendation task performs ranking on candidate items, the protein prediction task predicts the function terms, and the materials discovery task ranks candidate crystal structures by stability within a defined chemical system. These experiments measure improvements in prediction with their pretrained representations. Table 1: Main results on recommendation across five datasets and three architectures. Entries report HR@10 ↑ with standard deviations in subscript. Bold marks the best mean in each row. Dataset Base ReST-EM Meta-Harness OpenEvolve DGM T-Discover HACE SASRec ML-1M 0.2557±.00120.2557_ [rgb]0.4,0.4,0.4±.0012 0.2492±.00220.2492_ [rgb]0.4,0.4,0.4±.0022 0.2491±.00170.2491_ [rgb]0.4,0.4,0.4±.0017 0.2421±.00040.2421_ [rgb]0.4,0.4,0.4±.0004 0.2488±.00180.2488_ [rgb]0.4,0.4,0.4±.0018 0.2473±.00130.2473_ [rgb]0.4,0.4,0.4±.0013 0.2632±.00130.2632_ [rgb]0.4,0.4,0.4±.0013 Beauty 0.0599±.00040.0599_ [rgb]0.4,0.4,0.4±.0004 0.0597±.00100.0597_ [rgb]0.4,0.4,0.4±.0010 0.0631±.00070.0631_ [rgb]0.4,0.4,0.4±.0007 0.0638±.00030.0638_ [rgb]0.4,0.4,0.4±.0003 0.0634±.00050.0634_ [rgb]0.4,0.4,0.4±.0005 0.0633±.00040.0633_ [rgb]0.4,0.4,0.4±.0004 0.0672±.00040.0672_ [rgb]0.4,0.4,0.4±.0004 CDs 0.0576±.00050.0576_ [rgb]0.4,0.4,0.4±.0005 0.0587±.00100.0587_ [rgb]0.4,0.4,0.4±.0010 0.0604±.00040.0604_ [rgb]0.4,0.4,0.4±.0004 0.0611±.00020.0611_ [rgb]0.4,0.4,0.4±.0002 0.0592±.00080.0592_ [rgb]0.4,0.4,0.4±.0008 0.0587±.00070.0587_ [rgb]0.4,0.4,0.4±.0007 0.0647±.00060.0647_ [rgb]0.4,0.4,0.4±.0006 Sports 0.0315±.00030.0315_ [rgb]0.4,0.4,0.4±.0003 0.0325±.00010.0325_ [rgb]0.4,0.4,0.4±.0001 0.0323±.00020.0323_ [rgb]0.4,0.4,0.4±.0002 0.0331±.00040.0331_ [rgb]0.4,0.4,0.4±.0004 0.0333±.00030.0333_ [rgb]0.4,0.4,0.4±.0003 0.0323±.00020.0323_ [rgb]0.4,0.4,0.4±.0002 0.0356±.00030.0356_ [rgb]0.4,0.4,0.4±.0003 Toys 0.0596±.00010.0596_ [rgb]0.4,0.4,0.4±.0001 0.0648±.00070.0648_ [rgb]0.4,0.4,0.4±.0007 0.0657±.00080.0657_ [rgb]0.4,0.4,0.4±.0008 0.0662±.00080.0662_ [rgb]0.4,0.4,0.4±.0008 0.0661±.00090.0661_ [rgb]0.4,0.4,0.4±.0009 0.0646±.00080.0646_ [rgb]0.4,0.4,0.4±.0008 0.0706±.00030.0706_ [rgb]0.4,0.4,0.4±.0003 HSTU ML-1M 0.2604±.00030.2604_ [rgb]0.4,0.4,0.4±.0003 0.2563±.00050.2563_ [rgb]0.4,0.4,0.4±.0005 0.2569±.00070.2569_ [rgb]0.4,0.4,0.4±.0007 0.2573±.00050.2573_ [rgb]0.4,0.4,0.4±.0005 0.2572±.00060.2572_ [rgb]0.4,0.4,0.4±.0006 0.2588±.00130.2588_ [rgb]0.4,0.4,0.4±.0013 0.2685±.00030.2685_ [rgb]0.4,0.4,0.4±.0003 Beauty 0.0626±.00030.0626_ [rgb]0.4,0.4,0.4±.0003 0.0669±.00130.0669_ [rgb]0.4,0.4,0.4±.0013 0.0673±.00130.0673_ [rgb]0.4,0.4,0.4±.0013 0.0668±.00130.0668_ [rgb]0.4,0.4,0.4±.0013 0.0687±.00040.0687_ [rgb]0.4,0.4,0.4±.0004 0.0695±.00030.0695_ [rgb]0.4,0.4,0.4±.0003 0.0731±.00030.0731_ [rgb]0.4,0.4,0.4±.0003 CDs 0.0657±.00020.0657_ [rgb]0.4,0.4,0.4±.0002 0.0654±.00060.0654_ [rgb]0.4,0.4,0.4±.0006 0.0656±.00080.0656_ [rgb]0.4,0.4,0.4±.0008 0.0656±.00100.0656_ [rgb]0.4,0.4,0.4±.0010 0.0658±.00060.0658_ [rgb]0.4,0.4,0.4±.0006 0.0673±.00030.0673_ [rgb]0.4,0.4,0.4±.0003 0.0706±.00020.0706_ [rgb]0.4,0.4,0.4±.0002 Sports 0.0348±.00030.0348_ [rgb]0.4,0.4,0.4±.0003 0.0346±.00050.0346_ [rgb]0.4,0.4,0.4±.0005 0.0349±.00040.0349_ [rgb]0.4,0.4,0.4±.0004 0.0351±.00030.0351_ [rgb]0.4,0.4,0.4±.0003 0.0372±.00010.0372_ [rgb]0.4,0.4,0.4±.0001 0.0372±.00020.0372_ [rgb]0.4,0.4,0.4±.0002 0.0394±.00010.0394_ [rgb]0.4,0.4,0.4±.0001 Toys 0.0673±.00010.0673_ [rgb]0.4,0.4,0.4±.0001 0.0727±.00060.0727_ [rgb]0.4,0.4,0.4±.0006 0.0732±.00030.0732_ [rgb]0.4,0.4,0.4±.0003 0.0726±.00020.0726_ [rgb]0.4,0.4,0.4±.0002 0.0746±.00070.0746_ [rgb]0.4,0.4,0.4±.0007 0.0740±.00040.0740_ [rgb]0.4,0.4,0.4±.0004 0.0788±.00040.0788_ [rgb]0.4,0.4,0.4±.0004 Wukong ML-1M 0.2559±.00200.2559_ [rgb]0.4,0.4,0.4±.0020 0.2612±.00080.2612_ [rgb]0.4,0.4,0.4±.0008 0.2573±.00130.2573_ [rgb]0.4,0.4,0.4±.0013 0.2582±.00030.2582_ [rgb]0.4,0.4,0.4±.0003 0.2610±.00130.2610_ [rgb]0.4,0.4,0.4±.0013 0.2607±.00130.2607_ [rgb]0.4,0.4,0.4±.0013 0.2703±.00200.2703_ [rgb]0.4,0.4,0.4±.0020 Beauty 0.0734±.00020.0734_ [rgb]0.4,0.4,0.4±.0002 0.0810±.00040.0810_ [rgb]0.4,0.4,0.4±.0004 0.0805±.00040.0805_ [rgb]0.4,0.4,0.4±.0004 0.0795±.00040.0795_ [rgb]0.4,0.4,0.4±.0004 0.0788±.00040.0788_ [rgb]0.4,0.4,0.4±.0004 0.0772±.00040.0772_ [rgb]0.4,0.4,0.4±.0004 0.0856±.00020.0856_ [rgb]0.4,0.4,0.4±.0002 CDs 0.0726±.00050.0726_ [rgb]0.4,0.4,0.4±.0005 0.0771±.00020.0771_ [rgb]0.4,0.4,0.4±.0002 0.0798±.00040.0798_ [rgb]0.4,0.4,0.4±.0004 0.0780±.00040.0780_ [rgb]0.4,0.4,0.4±.0004 0.0785±.00040.0785_ [rgb]0.4,0.4,0.4±.0004 0.0770±.00040.0770_ [rgb]0.4,0.4,0.4±.0004 0.0842±.00050.0842_ [rgb]0.4,0.4,0.4±.0005 Sports 0.0388±.00050.0388_ [rgb]0.4,0.4,0.4±.0005 0.0408±.00020.0408_ [rgb]0.4,0.4,0.4±.0002 0.0398±.00020.0398_ [rgb]0.4,0.4,0.4±.0002 0.0390±.00020.0390_ [rgb]0.4,0.4,0.4±.0002 0.0410±.00020.0410_ [rgb]0.4,0.4,0.4±.0002 0.0395±.00020.0395_ [rgb]0.4,0.4,0.4±.0002 0.0438±.00070.0438_ [rgb]0.4,0.4,0.4±.0007 Toys 0.0759±.00020.0759_ [rgb]0.4,0.4,0.4±.0002 0.0785±.00040.0785_ [rgb]0.4,0.4,0.4±.0004 0.0752±.00040.0752_ [rgb]0.4,0.4,0.4±.0004 0.0793±.00040.0793_ [rgb]0.4,0.4,0.4±.0004 0.0820±.00040.0820_ [rgb]0.4,0.4,0.4±.0004 0.0770±.00040.0770_ [rgb]0.4,0.4,0.4±.0004 0.0866±.00060.0866_ [rgb]0.4,0.4,0.4±.0006 Evaluation metrics. Hit Rate (HR@10) measures the fraction of recommendation queries with the target item among the first ten predictions. NDCG uses binary relevance, discounts each relevant candidate at rank r by 1/log2(r+1)1/ _2(r+1), and normalizes by the ideal ranking within the same reference pool. We average NDCG@5 and NDCG@10 on CAFA, and NDCG@3 and NDCG@5 on WBM according to the baseline implementations. For protein function prediction, FmaxF_ is the maximum harmonic mean of precision and recall over prediction thresholds (Jiang et al., 2016). For material discovery, we use F1F_1 to measure their harmonic mean for identifying stable structures (Riebesell et al., 2025). Higher values indicate better performance for these metrics. Appendix C.1 defines all metrics in detail, including the additional measures reported there. Tables report means with standard deviations on three random seeds. Joint selection and fusion calibration only use the split validation data to avoid data contamination, as defined in Section 4.2. 5.1 Main Comparison Table 2: Main results on protein function prediction (CAFA) and material discovery (WBM). Higher is better for all metrics. Full scientific metrics appear in Tables 5 and 6. CAFA / ESM-2 CAFA / ProtT5-XL WBM / CHGNet Method NDCG ↑ Fmax↑F_ NDCG ↑ Fmax↑F_ NDCG ↑ F1↑F_1 Base 0.6285±.00190.6285_ [rgb]0.4,0.4,0.4±.0019 0.2218±.00140.2218_ [rgb]0.4,0.4,0.4±.0014 0.6241±.00110.6241_ [rgb]0.4,0.4,0.4±.0011 0.2221±.00050.2221_ [rgb]0.4,0.4,0.4±.0005 0.6388±.00110.6388_ [rgb]0.4,0.4,0.4±.0011 0.5159±.00030.5159_ [rgb]0.4,0.4,0.4±.0003 ReST-EM 0.6286±.00180.6286_ [rgb]0.4,0.4,0.4±.0018 0.2219±.00140.2219_ [rgb]0.4,0.4,0.4±.0014 0.6242±.00110.6242_ [rgb]0.4,0.4,0.4±.0011 0.2221±.00050.2221_ [rgb]0.4,0.4,0.4±.0005 0.6496±.00060.6496_ [rgb]0.4,0.4,0.4±.0006 0.5508±.00040.5508_ [rgb]0.4,0.4,0.4±.0004 Meta-Harness 0.6309±.00150.6309_ [rgb]0.4,0.4,0.4±.0015 0.2224±.00130.2224_ [rgb]0.4,0.4,0.4±.0013 0.6261±.00170.6261_ [rgb]0.4,0.4,0.4±.0017 0.2229±.00060.2229_ [rgb]0.4,0.4,0.4±.0006 0.6513±.00080.6513_ [rgb]0.4,0.4,0.4±.0008 0.5510±.00040.5510_ [rgb]0.4,0.4,0.4±.0004 OpenEvolve 0.6320±.00180.6320_ [rgb]0.4,0.4,0.4±.0018 0.2230±.00120.2230_ [rgb]0.4,0.4,0.4±.0012 0.6254±.00140.6254_ [rgb]0.4,0.4,0.4±.0014 0.2230±.00030.2230_ [rgb]0.4,0.4,0.4±.0003 0.6486±.00080.6486_ [rgb]0.4,0.4,0.4±.0008 0.5463±.00030.5463_ [rgb]0.4,0.4,0.4±.0003 DGM 0.6325±.00190.6325_ [rgb]0.4,0.4,0.4±.0019 0.2219±.00140.2219_ [rgb]0.4,0.4,0.4±.0014 0.6273±.00160.6273_ [rgb]0.4,0.4,0.4±.0016 0.2235±.00060.2235_ [rgb]0.4,0.4,0.4±.0006 0.6496±.00070.6496_ [rgb]0.4,0.4,0.4±.0007 0.5508±.00030.5508_ [rgb]0.4,0.4,0.4±.0003 T-Discover 0.6287±.00180.6287_ [rgb]0.4,0.4,0.4±.0018 0.2219±.00140.2219_ [rgb]0.4,0.4,0.4±.0014 0.6241±.00110.6241_ [rgb]0.4,0.4,0.4±.0011 0.2221±.00050.2221_ [rgb]0.4,0.4,0.4±.0005 0.6474±.00070.6474_ [rgb]0.4,0.4,0.4±.0007 0.5504±.00030.5504_ [rgb]0.4,0.4,0.4±.0003 HACE 0.6745±.00160.6745_ [rgb]0.4,0.4,0.4±.0016 0.2438±.00190.2438_ [rgb]0.4,0.4,0.4±.0019 0.6739±.00130.6739_ [rgb]0.4,0.4,0.4±.0013 0.2317±.00050.2317_ [rgb]0.4,0.4,0.4±.0005 0.6541±.00100.6541_ [rgb]0.4,0.4,0.4±.0010 0.5513±.00040.5513_ [rgb]0.4,0.4,0.4±.0004 Recommendation. HACE achieves the highest mean HR@10 in all 15 settings in Table 1. Relative gains over Base range from 2.9%2.9\% to 18.5%18.5\%, averaging 12.1%12.1\% across settings. On ML-1M, several evolution methods fall below the initial SASRec and HSTU models, while HACE improves all three architectures. On the four Amazon categories, Wukong has the strongest initial scores among the three architectures, yet HACE further improves its HR@10 by 12.9%12.9\% to 16.6%16.6\%. Contribution rewards train the agent against the current specialist throughout this process. Protein function prediction. Table 2 shows improvements with both protein backbones. HACE raises NDCG from 0.62850.6285 to 0.67450.6745 with ESM-2 and from 0.62410.6241 to 0.67390.6739 with ProtT5-XL. Existing evolution methods improve over Base by at most 0.00400.0040 and 0.00320.0032, respectively, compared with 0.04600.0460 and 0.04980.0498 for HACE. HACE also leads in FmaxF_ , AUPR, and Recall@10. These gains cover both the ordering of function terms and their classification. Table 5 also reports semantic distance, for which DGM has the lowest mean on ProtT5-XL. The ablations below examine how these gains depend on harness search and specialist optimization. Material Discovery. On WBM, all competing methods improve NDCG over Base, from 0.63880.6388 to at least 0.64740.6474. HACE reaches 0.65410.6541, exceeding the strongest competing mean by 0.00280.0028. HACE also achieves the best mean on all four additional metrics in Table 6. Removing specialist optimization reduces NDCG to 0.63950.6395, close to Base at 0.63880.6388. The contribution of supervised specialist updates is therefore particularly clear on this task. Supervised adaptation improves numerical predictions, while the ranking objective favors recovering stable candidates near the top. 5.2 Ablation Studies Table 3: Ablation studies on recommendation (Toys/SASRec), protein function prediction (CAFA/ESM-2 650M), and material discovery (WBM/CHGNet). Metrics are HR@10 for Toys and mean NDCG for CAFA and WBM. Leave-one-out component Toys CAFA WBM HACE 0.0706±.00030.0706_ [rgb]0.4,0.4,0.4±.0003 0.6745±.00160.6745_ [rgb]0.4,0.4,0.4±.0016 0.6541±.00100.6541_ [rgb]0.4,0.4,0.4±.0010 w/o harness optimization 0.0658±.00050.0658_ [rgb]0.4,0.4,0.4±.0005 0.6307±.00190.6307_ [rgb]0.4,0.4,0.4±.0019 0.6500±.00120.6500_ [rgb]0.4,0.4,0.4±.0012 w/o specialist optimization 0.0641±.00050.0641_ [rgb]0.4,0.4,0.4±.0005 0.6310±.00220.6310_ [rgb]0.4,0.4,0.4±.0022 0.6395±.00120.6395_ [rgb]0.4,0.4,0.4±.0012 w/o agent optimization 0.0622±.00060.0622_ [rgb]0.4,0.4,0.4±.0006 0.6520±.00160.6520_ [rgb]0.4,0.4,0.4±.0016 0.6502±.00110.6502_ [rgb]0.4,0.4,0.4±.0011 w/o joint selection 0.0650±.00060.0650_ [rgb]0.4,0.4,0.4±.0006 0.6531±.00200.6531_ [rgb]0.4,0.4,0.4±.0020 0.6497±.00130.6497_ [rgb]0.4,0.4,0.4±.0013 w/o agent-guided experience 0.0662±.00050.0662_ [rgb]0.4,0.4,0.4±.0005 0.6575±.00260.6575_ [rgb]0.4,0.4,0.4±.0026 0.6507±.00120.6507_ [rgb]0.4,0.4,0.4±.0012 We conduct leave-one-out ablation studies in Table 3 in three domains. Removing agent optimization causes the largest drop on Toys, from 0.07060.0706 to 0.06220.0622. On CAFA, removing harness or specialist optimization reduces NDCG by 0.04380.0438 and 0.04350.0435, respectively. Harness search controls the information exposed to the agent and the experience selected for adaptation, while specialist optimization learns from that experience using task labels. Agent optimization matters most on Toys and specialist optimization on WBM, indicating that their contributions vary by task. Removing agent-guided experience lowers performance even when specialist optimization remains enabled, by 0.00440.0044 on Toys, 0.01700.0170 on CAFA, and 0.00340.0034 on WBM. This decrease supports the role of agent exploration in selecting training experience, in addition to its role in prediction. Removing joint selection also reduces all three scores. Together, we show that all designed components contribute positively to the framework. 5.3 Agent and Model Scaling Table 4: Scaling experiment on protein function prediction task. NDCG averages cutoffs 5 and 10. Base denotes the initial specialists, which are already trained to convergence before coevolution; Appendix C.3 states the criterion we use. Agent variants use a shared 30-round configuration. Subscripts report standard deviations of within-run gains on three random seeds. Model Base HACE Reported Δ Specialist size Fixed agent Gemma-4 E4B ESM-2 35M 0.62830.6283 0.66920.6692 +0.0409±0.0007+0.0409_ [rgb]0.5,0.5,0.5± 0.0007 ESM-2 150M 0.62110.6211 0.66330.6633 +0.0422±0.0004+0.0422_ [rgb]0.5,0.5,0.5± 0.0004 ESM-2 650M 0.62850.6285 0.67450.6745 +0.0460±0.0006+0.0460_ [rgb]0.5,0.5,0.5± 0.0006 ESM-2 3B 0.63390.6339 0.67630.6763 +0.0424±0.0007+0.0424_ [rgb]0.5,0.5,0.5± 0.0007 Agent variant Fixed specialist ESM-2 650M Gemma-4 E2B 0.62850.6285 0.62970.6297 +0.0012±0.0018+0.0012_ [rgb]0.5,0.5,0.5± 0.0018 Gemma-4 E4B 0.62850.6285 0.67450.6745 +0.0460±0.0006+0.0460_ [rgb]0.5,0.5,0.5± 0.0006 Gemma-4 26B-A4B 0.62850.6285 0.63390.6339 +0.0054±0.0014+0.0054_ [rgb]0.5,0.5,0.5± 0.0014 In the specialist sweep with the agent fixed, HACE improves prediction with all four ESM-2 sizes in Table 4, from 35M to 3B parameters. The absolute NDCG gains remain between 0.04090.0409 and 0.04600.0460. The largest final score comes from the 3B model, while the largest gain occurs at 650M. The 35M model after HACE also exceeds the initial 3B model, with NDCG of 0.66920.6692 versus 0.63390.6339. In the agent comparison, E4B achieves the largest mean gain of 0.04600.0460, compared with 0.00120.0012 for E2B and 0.00540.0054 for 26B-A4B. These are within-run gains from a separate matched 30-round sweep with ESM-2 650M, detailed in Appendix C.3. The gains do not increase monotonically with agent size, suggesting that a larger agent does not necessarily contribute more to the specialist in this setting. 5.4 Evolution Dynamics Figure 3: Evolution dynamics of three domains. Best recorded scores during evolution, measured by HR@10, Recall@10, and mean NDCG@3 and NDCG@5. Dashed lines mark the frozen specialist. Improvement over rounds. Figure 3 shows HACE reaching the highest score in all three recorded trajectories. The best recorded recommendation score rises mostly in early rounds, while protein and materials reach new maxima after longer intervals. In HACE, each retained specialist and harness defines the starting point for the next round. Fresh agent responses are verified against the current specialist. This update order makes agent supervision depend on the retained specialist. The trajectories show when higher scoring configurations are found during search. Figure 4: Illustrations on events during coevolution. Upper panels compare specialist and combined predictions. Lower panels show recorded selection scores and their running maxima from one run in each domain, with selected updates annotated. Case studies on three domains. Figure 4 compares specialist and combined predictions on three domains. HACE recovers a matching product variant, places two relevant protein terms in the top five, and brings a second stable phase into the top three. Recorded changes include removing genre metadata, targeting difficult training instances, and hiding candidate ranks from the agent. These examples illustrate how the harness search and coevolution change both the context used for prediction and the experience used for reciprocal learning. 6 Conclusion We introduced HACE, a novel framework for reciprocal self-improvement between specialized foundation models and LLM agents. By evolving both the learners and their interaction with an updating harness, HACE makes each model a source of learning for the other. Our analysis identifies sufficient conditions for specialist learning from adaptively selected experience. Experiments in three domains demonstrate improvements beyond initially converged specialists across different architectures. These results show how heterogeneous AI models can support each other’s development. Future work can study sustained coevolution, collaboration among multiple specialists, and autonomous model development for scientific discovery. References Cemri et al. (2026) M. Cemri, S. Agrawal, A. Gupta, S. Liu, A. Cheng, Q. Mang, A. Naren, L. E. Erdogan, K. Sen, M. Zaharia, et al. Adaevolve: adaptive llm driven zeroth-order optimization. arXiv preprint arXiv:2602.20133. Cited by: §3. Chen et al. (2026) Z. Chen, T. Xiao, H. Zhu, Y. Yuan, L. Zhang, and J. Wang Co-harness: co-evolving harnesses and model for agents. In COLM 2026 The 2nd Workshop on Lifelong Agents: Learning, Aligning, and Evolving, External Links: Link Cited by: §3. Deng et al. (2023) B. Deng, P. Zhong, K. Jun, J. Riebesell, K. Han, C. J. Bartel, and G. Ceder CHGNet as a pretrained universal neural network potential for charge-informed atomistic modelling. Nature Machine Intelligence 5 (9), p. 1031–1041. External Links: Document Cited by: §2, §5. Dennis et al. (2020) M. Dennis, N. Jaques, E. Vinitsky, A. Bayen, S. Russell, A. Critch, and S. Levine Emergent complexity and zero-shot transfer via unsupervised environment design. In Advances in Neural Information Processing Systems, Vol. 33, p. 13049–13061. Cited by: §3. Elnaggar et al. (2022) A. Elnaggar, M. Heinzinger, C. Dallago, G. Rehawi, Y. Wang, L. Jones, T. Gibbs, T. Feher, C. Angerer, M. Steinegger, D. Bhowmik, and B. Rost ProtTrans: toward understanding the language of life through self-supervised learning. IEEE Transactions on Pattern Analysis and Machine Intelligence 44 (10), p. 7112–7127. External Links: Document Cited by: §5. Gemma Team (2026) Gemma Team Gemma 4 technical report. Note: arXiv preprint arXiv:2607.02770 External Links: 2607.02770 Cited by: §5. Harper and Konstan (2015) F. M. Harper and J. A. Konstan The MovieLens datasets: history and context. ACM Transactions on Interactive Intelligent Systems 5 (4), p. 19:1–19:19. External Links: Document, Link Cited by: §5. Howard et al. (2021) S. R. Howard, A. Ramdas, J. McAuliffe, and J. Sekhon Time-uniform, nonparametric, nonasymptotic confidence sequences. The Annals of Statistics 49 (2), p. 1055–1080. External Links: Document Cited by: §B.1. Hu et al. (2025) S. Hu, C. Lu, and J. Clune Automated design of agentic systems. In International Conference on Learning Representations, Cited by: §3. Janzen (1980) D. H. Janzen When is it coevolution?. Evolution 34 (3), p. 611–612. External Links: Document Cited by: §3. Järvelin and Kekäläinen (2002) K. Järvelin and J. Kekäläinen Cumulated gain-based evaluation of IR techniques. ACM Transactions on Information Systems 20 (4), p. 422–446. External Links: Document Cited by: §B.3. Jiang et al. (2016) Y. Jiang, T. R. Oron, W. T. Clark, et al. An expanded evaluation of protein function prediction methods shows an improvement in accuracy. Genome Biology 17 (1), p. 184. External Links: Document Cited by: §C.1, §5, §5. Kang et al. (2025) M. Kang, J. Jeong, S. Lee, J. Cho, and S. J. Hwang Distilling LLM agent into small models with retrieval and code tools. Note: arXiv preprint arXiv:2505.17612 External Links: 2505.17612 Cited by: §3. Kang and McAuley (2018) W. Kang and J. McAuley Self-attentive sequential recommendation. In 2018 IEEE International Conference on Data Mining (ICDM), p. 197–206. External Links: Document Cited by: §5. Lee et al. (2026) Y. Lee, R. Nair, Q. Zhang, K. Lee, O. Khattab, and C. Finn Meta-harness: end-to-end optimization of model harnesses. arXiv preprint arXiv:2603.28052. Cited by: §2, §2, §3, §5. Li et al. (2026) Z. Li, J. Zou, F. Fang, X. Ning, M. Ai, T. Wei, S. Chen, X. Yang, and J. He Heterogeneous scientific foundation model collaboration. arXiv preprint arXiv:2604.27351. Cited by: §3. Lin et al. (2023) Z. Lin, H. Akin, R. Rao, B. Hie, Z. Zhu, W. Lu, N. Smetanin, R. Verkuil, O. Kabeli, Y. Shmueli, A. dos Santos Costa, M. Fazel-Zarandi, T. Sercu, S. Candido, and A. Rives Evolutionary-scale prediction of atomic-level protein structure with a language model. Science 379 (6637), p. 1123–1130. External Links: Document Cited by: §2, §5. Liu et al. (2026a) J. Liu, Z. Kong, P. Dong, C. Yang, T. Li, Y. Xie, Y. Gong, X. Shen, P. Zhao, H. Tang, G. Yuan, W. Niu, W. Zhang, X. Lin, Y. Wang, and D. Huang Structured agent distillation for large language model agents. Note: arXiv preprint arXiv:2505.13820v5 External Links: 2505.13820v5 Cited by: §3. Liu et al. (2026b) S. Liu, S. Agarwal, M. Maheswaran, M. Cemri, Q. Mang, Z. Li, A. Naren, E. Boneh, A. Cheng, A. Du, M. Pan, K. Keutzer, A. Cheung, K. Sen, A. Dimakis, M. Zaharia, and I. Stoica EvoX: meta-evolution for automated discovery. In Third Conference on Language Modeling, Cited by: §3. Liu et al. (2026c) Z. Liu, X. Guo, X. Wei, H. Hao, and L. Yang Escher-Loop: mutual evolution by closed-loop self-referential optimization. Note: arXiv preprint arXiv:2604.23472 External Links: 2604.23472 Cited by: §3. Luo et al. (2026) Y. Luo, H. Jiang, J. Zou, X. Huang, W. Yan, H. Li, Z. Yue, J. Li, X. Chen, X. Zhao, et al. AutoDesign: meta-harness optimization for long-horizon agentic design. arXiv preprint arXiv:2608.13560. Cited by: §3. Ma et al. (2024) Y. J. Ma, W. Liang, G. Wang, D. Huang, O. Bastani, D. Jayaraman, Y. Zhu, L. Fan, and A. Anandkumar Eureka: human-level reward design via coding large language models. In International Conference on Learning Representations, Cited by: §2, §3. McAuley et al. (2015) J. McAuley, C. Targett, Q. Shi, and A. van den Hengel Image-based recommendations on styles and substitutes. In Proceedings of the 38th International ACM SIGIR Conference on Research and Development in Information Retrieval, p. 43–52. External Links: Document, Link Cited by: §5. Riebesell et al. (2025) J. Riebesell, R. E. A. Goodall, P. Benner, Y. Chiang, B. Deng, G. Ceder, M. Asta, A. A. Lee, A. Jain, and K. A. Persson A framework to evaluate machine learning crystal stability predictions. Nature Machine Intelligence 7 (6), p. 836–847. External Links: Document Cited by: §C.1, §5, §5. Sharma (2025) A. Sharma OpenEvolve: an open-source evolutionary coding agent. GitHub. External Links: Link Cited by: §5. Singh et al. (2024) A. Singh, J. D. Co-Reyes, R. Agarwal, A. Anand, P. Patil, X. Garcia, P. J. Liu, J. Harrison, J. Lee, K. Xu, A. Parisi, A. Kumar, A. Alemi, A. Rizkowsky, A. Nova, B. Adlam, B. Bohnet, G. Elsayed, H. Sedghi, I. Mordatch, I. Simpson, I. Gur, J. Snoek, J. Pennington, J. Hron, K. Kenealy, K. Swersky, K. Mahajan, L. Culp, L. Xiao, M. L. Bileschi, N. Constant, R. Novak, R. Liu, T. Warkentin, Y. Qian, Y. Bansal, E. Dyer, B. Neyshabur, J. Sohl-Dickstein, and N. Fiedel Beyond human data: scaling self-training for problem-solving with language models. Transactions on Machine Learning Research. Cited by: §3, §4.2, §5. Wang et al. (2021) H. Wang, S. Botti, and M. A. L. Marques Predicting stable crystalline compounds using chemical similarity. npj Computational Materials 7 (1), p. 12. External Links: Document, Link Cited by: §5. Wang et al. (2026) J. Wang, Y. Liu, J. Chen, X. Hu, Q. Zhang, Y. Cao, J. Wang, H. Yang, Y. Xie, and Q. Chen MAD-OPD: breaking the ceiling in on-policy distillation via multi-agent debate. Note: arXiv preprint arXiv:2605.01347 External Links: 2605.01347 Cited by: §3. Wang et al. (2019) R. Wang, J. Lehman, J. Clune, and K. O. Stanley POET: open-ended coevolution of environments and their optimized solutions. In Proceedings of the Genetic and Evolutionary Computation Conference, p. 142–151. External Links: Document Cited by: §3. Yang et al. (2026) S. Yang, Z. Ma, T. Huang, Y. Hu, Y. Wang, and X. Chu CoEvolve: training LLM agents via agent-data mutual evolution. In Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), p. 23015–23036. External Links: Link, Document Cited by: §3. Yuksekgonul et al. (2026) M. Yuksekgonul, D. Koceja, X. Li, F. Bianchi, J. McCaleb, X. Wang, J. Kautz, Y. Choi, J. Zou, C. Guestrin, and Y. Sun Learning to discover at test time. In International Conference on Machine Learning, Cited by: §2, §3, §5. Zhai et al. (2024) J. Zhai, L. Liao, X. Liu, Y. Wang, R. Li, X. Cao, L. Gao, Z. Gong, F. Gu, J. He, Y. Lu, and Y. Shi Actions speak louder than words: trillion-parameter sequential transducers for generative recommendations. In Proceedings of the 41st International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 235, p. 58484–58509. Cited by: §2, §5. Zhang et al. (2024) B. Zhang, L. Luo, Y. Chen, J. Nie, X. Liu, S. Li, Y. Zhao, Y. Hao, Y. Yao, E. D. Wen, J. Park, M. Naumov, and W. Chen Wukong: towards a scaling law for large-scale recommendation. In Proceedings of the 41st International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 235, p. 59421–59434. Cited by: §2, §5. Zhang et al. (2026a) J. Zhang, S. Hu, C. Lu, R. Lange, and J. Clune Darwin Gödel Machine: open-ended evolution of self-improving agents. In International Conference on Learning Representations, Cited by: §2, §2, §3, §5. Zhang et al. (2026b) J. Zhang, B. Zhao, W. Yang, J. Foerster, J. Clune, M. Jiang, S. Devlin, and T. Shavrina Hyperagents. Note: arXiv preprint arXiv:2603.19461 External Links: 2603.19461 Cited by: §2, §2, §3. Zweiger et al. (2025) A. Zweiger, J. Pari, H. Guo, Y. Kim, and P. Agrawal Self-adapting language models. In Advances in Neural Information Processing Systems, Vol. 38, p. 74084–74115. External Links: Document Cited by: §2, §3. Appendix A Limitations HACE coevolves a single specialized foundation model with a single LLM agent. We do not study coevolution among multiple specialists or agents, which may require different mechanisms for interaction and learning. The compared methods differ in which components they can update. Some support parameter updates, others evolve the harness or agent program with the specialist held fixed, and several assume a closed-weight LLM. We follow each method’s original formulation rather than introducing specialist adaptation beyond its design. These comparisons therefore evaluate the methods as complete systems, while Table 3 separately examines the contribution of each HACE component. Our findings are based on the evaluated domains, architectures, and computational budgets. Further work can study how long coevolution sustains useful learning and how to allocate computation among harness search, specialist adaptation, and agent training. Appendix B Proofs and Implementation Details Throughout this appendix, log is the natural logarithm and an explicitly written log2 _2 uses base two. For a finite set or indexed dataset, |B||B| counts its elements or records, while |z||z| counts tokens. The indicator B1\B\ is one when condition B holds and zero otherwise. Empty sums are zero and empty products are one. Unless specified otherwise, ∑i _i means ∑i∈ℐ _i and ∑t<T _t<T means ∑t=0T−1 _t=0^T-1. B.1 Specialist Learning Let train=(xi,yi):i∈ℐD_train=\(x_i,y_i):i \ be the labeled training pool defined in Section 4.1. Equation 2 gives the supervised objective on the selected training indices ℐtSI_t^S. The specialist’s state includes its trainable parameters ϕφ and stored score calibration. Adaptation starts from StS_t and uses a fixed optimization budget. When supervised replay is used, the following term is added to Equation 2, with both terms using original task labels, ℛreplay,t(ϕ)=∑i∈ℐγt,iℓtask(Sϕ(xi),yi).R_replay,t(φ)= _i _t,i _task(S_φ(x_i),y_i). (8) The nonnegative weights γt,i _t,i are fixed during the specialist update and are zero unless instance i is retained from earlier specialist training. Their sum sets the total replay strength, and equal positive weights give uniform replay over the retained instances. Setting all weights to zero disables replay. This defines the term without assuming a particular retention policy. For the proof of Theorem 1, condition on the fixed source pool and initial configuration. The source probabilities μi _i, specialist class ℱF, and losses ℓi(f) _i(f) are fixed before exploration. The analysis loss may differ from the optimization loss ℓtask _task. Interpret ℋtH_t as the sigma field of information available before round t. These histories increase with t, and ℋt+1H_t+1 includes that round’s proposal, responses, specialist training, validation decision, and agent update. The accepted weights are measurable with respect to ℋt+1H_t+1, are nonnegative, and satisfy ∑iWt,i≤1 _iW_t,i≤ 1. Under the uniform allocation used here, Wt,i=1/|ℐtS|W_t,i=1/|I_t^S| on selected instances in a nonempty accepted update and zero otherwise. Conditional expectation given ℋtH_t averages over the remaining round randomness, allowing dependence between selection and acceptance. The final specialist may depend on the full observed history. Replay may affect its final loss but is not counted as additional exposure. We use CT(i)C_T(i), L^T(f) L_T(f), and UT(v)U_T(v) as defined in Section 4.3. Proof of Theorem 1. Fix a predictor f∈ℱf before the random exploration history. Define its accepted loss in round t and the conditional mean of that loss by Xt(f)=∑iWt,iℓi(f),mt(f)=[Xt(f)∣ℋt].X_t(f)= _iW_t,i _i(f), m_t(f)=E[X_t(f) _t]. (9) The weight and loss bounds give 0≤Xt(f)≤10≤ X_t(f)≤ 1 and 0≤mt(f)≤10≤ m_t(f)≤ 1. Since the task losses are fixed, linearity of conditional expectation gives ∑t<Tmt(f)=∑iCT(i)ℓi(f),∑t<TXt(f)=L^T(f). _t<Tm_t(f)= _iC_T(i) _i(f), _t<TX_t(f)= L_T(f). (10) We first compare the conditional expected losses with their realized sum, uniformly over rounds. For 0≤x≤10≤ x≤ 1, convexity gives e−x≤1−(1−e−1)xe^-x≤ 1-(1-e^-1)x. Taking conditional expectations and using 1−u≤e−u1-u≤ e^-u yields [e−Xt(f)∣ℋt]≤1−(1−e−1)mt(f)≤e−(1−e−1)mt(f).E[e^-X_t(f) _t]≤ 1-(1-e^-1)m_t(f)≤ e^-(1-e^-1)m_t(f). (11) Define M0(f)=1M_0(f)=1 and MT(f)=exp((1−e−1)∑t<Tmt(f)−∑t<TXt(f)).M_T(f)= ((1-e^-1) _t<Tm_t(f)- _t<TX_t(f) ). (12) This process is nonnegative and measurable with respect to ℋTH_T. Equation 11 implies [MT+1(f)∣ℋT] [M_T+1(f) _T] =MT(f)e(1−e−1)mT(f)[e−XT(f)∣ℋT] =M_T(f)e^(1-e^-1)m_T(f)E[e^-X_T(f) _T] (13) ≤MT(f). ≤ M_T(f). Thus it is a supermartingale, meaning that its conditional expected next value is at most its current value. This exponential construction gives a probability bound valid over time (Howard et al., 2021). The following stopping argument establishes that bound directly. For a threshold a>1a>1, let σa _a be the first integer T≥0T≥ 0 with MT(f)≥aM_T(f)≥ a, and set σa=∞ _a=∞ if no such round exists. Whether a crossing has occurred by round t is known from ℋtH_t, so σa _a is a stopping time. For a positive integer n, write σa∧n=minσa,n _a n= \ _a,n\. The stopped process satisfies Mσa∧n(f)=1+∑t=0n−1σa>t(Mt+1(f)−Mt(f)).M_ _a n(f)=1+ _t=0^n-11\ _a>t\ (M_t+1(f)-M_t(f) ). (14) Each indicator is known under ℋtH_t, so each expected summand is nonpositive by Equation 13. Hence [Mσa∧n(f)]≤1E[M_ _a n(f)]≤ 1. The process is at least a whenever a crossing occurs by n and is nonnegative otherwise. Therefore aPr(σa≤n)≤1a ( _a≤ n)≤ 1. Letting n→∞n→∞ gives Pr(∃T≥0 with MT(f)≥a)≤1a. (∃\,T≥ 0 with M_T(f)≥ a )≤ 1a. (15) Choose a=|ℱ|/δa=|F|/δ and take a union bound over the fixed class. With probability at least 1−δ1-δ, no predictor’s process crosses this threshold at any round. On this event, taking logarithms and applying Equation 10 gives, for all f∈ℱf and T≥1T≥ 1, (1−e−1)∑iCT(i)ℓi(f)≤L^T(f)+log(|ℱ|/δ).(1-e^-1) _iC_T(i) _i(f)≤ L_T(f)+ (|F|/δ). (16) No independence across rounds is used. This bound controls task risk wherever accepted exposure is sufficiently large. Fix any v>0v>0 on this event. Instances with CT(i)<vμiC_T(i)<v _i contribute at most UT(v)U_T(v) to the source risk, because their losses are at most one. For each remaining instance, μi≤CT(i)/v _i≤ C_T(i)/v. Summing and using Equation 16 gives Rμ(f)≤UT(v)+1v∑iCT(i)ℓi(f)≤UT(v)+L^T(f)+log(|ℱ|/δ)(1−e−1)v.R_μ(f)≤ U_T(v)+ 1v _iC_T(i) _i(f)≤ U_T(v)+ L_T(f)+ (|F|/δ)(1-e^-1)v. (17) This deterministic implication holds for every v>0v>0 on the same event. Since that event already covers every predictor and round, we may substitute the adaptive final specialist ST∈ℱS_T . The result also holds at any finite stopping time chosen from the observed history. Treating only the retrospectively selected STS_T as fixed would not justify the probability bound. Finally, for any positive levels vT→∞v_T→∞ such that UT(vT)→0U_T(v_T)→ 0 and L^T(ST)/vT→0 L_T(S_T)/v_T→ 0, all terms in the upper bound tend to zero. Nonnegativity then gives Rμ(ST)→0R_μ(S_T)→ 0. The levels may depend on the observed history because the same event covers every v. This conclusion applies to trajectories satisfying these conditions. For a fixed finite class, a zero risk limit requires minf∈ℱRμ(f)=0 _f R_μ(f)=0. ∎ The exposure term connects the bound to the training allocation. When v is proportional to T, the term from class size decreases as 1/T1/T, while the other terms depend on exposure and retained loss. To relate exposure to selection, suppose the nonempty training set has at most B≥1B≥ 1 instances and uses uniform weights. On the joint event that instance i is selected and the proposal is accepted, its weight is at least 1/B1/B. Thus [Wt,i∣ℋt]≥1BPr(i∈ℐtS,ΔJt>0∣ℋt).E[W_t,i _t]≥ 1B (i _t^S,\ J_t>0 _t ). (18) Summing gives the corresponding lower bound for CT(i)C_T(i). The joint probability preserves dependence between inclusion and acceptance. A high overall acceptance rate need not expose every instance, and verification need not lead to inclusion after curriculum truncation. Small losses immediately after each update do not ensure small final loss when later updates forget those instances. Replay may affect this final loss. If acceptance has conditional probability zero in every round, then CT(i)=0C_T(i)=0 and UT(v)=1U_T(v)=1, so the bound gives no learning conclusion. Expected exposure is generally not observable from one trace, and |ℱ||F| counts a class fixed before exploration rather than adaptively visited checkpoints. A population bound requires assumptions on how the source pool is drawn. Let P be a distribution of labeled task instances. Suppose the source pool consists of n=|ℐ|≥1n=|I|≥ 1 independent draws Zi=(xi,yi)Z_i=(x_i,y_i) from P, indexed by i=1,…,ni=1,…,n, and set μi=1/n _i=1/n. Assume ℱF is fixed independently of these draws. For a generic draw Z∼PZ P, let ℓ(f,Z)∈[0,1] (f,Z)∈[0,1] be a fixed loss and set ℓi(f)=ℓ(f,Zi) _i(f)= (f,Z_i). Define RP(f)=Z∼P[ℓ(f,Z)],Rμ(f)=1n∑i=1nℓ(f,Zi).R_P(f)=E_Z P[ (f,Z)], R_μ(f)= 1n _i=1^n (f,Z_i). (19) Under the remaining assumptions of Theorem 1, with probability at least 1−δ1-δ over the pool and learning process, every final specialist satisfies, simultaneously for all T≥1T≥ 1 and v>0v>0, RP(ST)≤UT(v)+L^T(ST)+log(2|ℱ|/δ)(1−e−1)v+log(2|ℱ|/δ)2n.R_P(S_T)≤ U_T(v)+ L_T(S_T)+ (2|F|/δ)(1-e^-1)v+ (2|F|/δ)2n. (20) Proof. For this calculation, let ξ be a centered random variable supported in an interval of width one and let η≥0η≥ 0 be an exponential moment parameter. Define ψ(η)=log[eηξ]ψ(η)= [e^ηξ]. Differentiation shows that ψ′(η)ψ (η) is the variance of ξ under probabilities reweighted by eηξe^ηξ. This distribution retains the same support, and its variance is at most the expected squared distance to the interval midpoint, hence at most 1/41/4. Since ψ(0)=ψ′(0)=0ψ(0)=ψ (0)=0, integrating twice gives ψ(η)=∫0η(η−s)ψ′(s)s≤η2/8,ψ(η)= _0^η(η-s)ψ (s)\,ds≤η^2/8, (21) where s is the integration variable. Apply this inequality to RP(f)−ℓ(f,Zi)R_P(f)- (f,Z_i) for a fixed f. Independence and exponential Markov inequality give, for every ϵ>0ε>0 and η>0η>0, Pr(RP(f)−Rμ(f)≥ϵ)≤exp(−ηnϵ+nη2/8). (R_P(f)-R_μ(f)≥ε)≤ (-η nε+nη^2/8). (22) Choosing η=4ϵη=4ε gives an upper bound of e−2nϵ2e^-2nε^2. Taking ϵ=log(2|ℱ|/δ)/(2n)ε= (2|F|/δ)/(2n) and a union bound over ℱF therefore yields, with probability at least 1−δ/21-δ/2, RP(f)≤Rμ(f)+log(2|ℱ|/δ)2nfor every f∈ℱ.R_P(f)≤ R_μ(f)+ (2|F|/δ)2n every f . (23) Conditional on the realized pool and initial configuration, apply Theorem 1 with failure probability δ/2δ/2. Averaging over the pool and initial configuration keeps its failure probability at most δ/2δ/2. A union bound combines this event with Equation 23, without requiring independence between the events. On their intersection, which has probability at least 1−δ1-δ, adding the bounds proves Equation 20 for every T and v. Repeated training on a fixed pool does not reduce the final sample term. Population consistency additionally requires that term to vanish, for example by increasing independent source data while controlling class size. The source risk result permits a class chosen after fixing the pool but before exploration. This population extension requires the stronger independence condition stated above. B.2 Agent Learning and Verification We next analyze the responses that provide learning experience, conditioning on the collection state and using the fixed source probabilities μi _i. Write AθA_θ for the agent language model with trainable parameters θ, including an adapter when used, and let θt _t be the parameters of AtA_t. For a fixed specialist and harness, pip_i is the nonempty prompt constructed from xix_i and specialist predictions, excluding yiy_i. The decoded distribution πθ(τ∣pi) _θ(τ p_i) gives the probability of the complete response sequence τ after decoding and stopping. We collect m≥1m≥ 1 responses per instance, where m is an integer. They are independent conditional on the prompt and any shared task context. Fix a collection round t, its specialist and harness, and the task context used to form each prompt. For an instance i with evaluable supervision, define Et(i)=τ∈∣τ is valid and ri(τ)>0,αt(i)=πθt(Et(i)∣pi).E_t(i)=\τ τ is valid and r_i(τ)>0\, _t(i)= _ _t(E_t(i) p_i). (24) Here T is the countable space of finite response token sequences. For any response distribution π, the notation π(E)π(E) means ∑τ∈Eπ(τ) _τ∈ Eπ(τ). For an instance without evaluable supervision, set Et(i)=∅E_t(i)= and αt(i)=0 _t(i)=0. Let I be the source instance index, with Pr(I=i)=μi (I=i)= _i, and let V be the event that at least one of its m collected responses passes verification. Conditional independence of those responses gives Pr(V∣I=i)=1−(1−αt(i))m (V I=i)=1-(1- _t(i))^m. Therefore, whenever Pr(V)>0 (V)>0, Bayes’ rule gives Pr(I=i∣V)=μi[1−(1−αt(i))m]∑jμj[1−(1−αt(j))m]. (I=i V)= _i[1-(1- _t(i))^m] _j _j[1-(1- _t(j))^m]. (25) This describes source sampling followed only by verification. A zero denominator means that no source instance can provide a verified response under this collection procedure. If the responses share a random context C, independence must instead hold conditional on that context. Define αt(i,C) _t(i,C) by Equation 24 using the prompt and verifier determined by C. The collection probability becomes Pr(V∣I=i)=[1−(1−αt(i,C))m∣I=i]. (V I=i)=E [1-(1- _t(i,C))^m I=i ]. (26) Equation 25 then uses this conditional expectation in place of each bracketed collection probability. For m>1m>1, the function a↦1−(1−a)ma 1-(1-a)^m is concave on [0,1][0,1], since its second derivative is −m(m−1)(1−a)m−2-m(m-1)(1-a)^m-2 in the interior. Thus Jensen’s inequality yields Pr(V∣I=i)≤1−(1−[αt(i,C)∣I=i])m. (V I=i)≤ 1- (1-E[ _t(i,C) I=i] )^m. (27) Averaging the response probability before computing the collection probability can therefore overestimate the chance of verification. Equality holds for m=1m=1. We next compare verification before agent fitting, after fitting under the same reference, and after the next specialist and harness selection. Fix one instance and its agent collection context under the retained pair (S¯t,H¯t)( S_t, H_t), and suppress the instance index. Write EtE_t for its verified response set, and let πt _t and π~t π_t be the decoded response distributions before and after agent fitting under that same prompt, specialist, and harness. Set αt=πt(Et)>0 _t= _t(E_t)>0 and α~t=π~t(Et)>0 α_t= π_t(E_t)>0. Let qtq_t be a distribution on selected responses in EtE_t, including their training weights. Every response with positive qtq_t mass must have positive probability under both πt _t and π~t π_t. For distributions q and p, define KL(q∥p)=∑τ:q(τ)>0q(τ)log[q(τ)/p(τ)]KL(q\|p)= _τ:q(τ)>0q(τ) [q(τ)/p(τ)], with value +∞+∞ if p is zero on a response with positive q mass. All divergences and log likelihood expectations below are assumed finite. Define gt=qtlogπ~t(τ)πt(τ),kt=KL(qt∥πt(⋅∣Et)),g_t=E_q_t π_t(τ) _t(τ), k_t=KL (q_t\| _t(· E_t) ), (28) where πt(τ∣Et)=πt(τ)/αt _t(τ E_t)= _t(τ)/ _t on EtE_t and is zero elsewhere. The conditional distribution after fitting is defined in the same way. Thus gtg_t measures the mean decoded likelihood gain on selected responses, while ktk_t measures how their distribution differs from the old distribution conditional on verification. The next agent collection uses the fitted agent after the next specialist and harness selection. Let πt+1 _t+1 and Et+1E_t+1 be its decoded distribution and verified response set for the same instance, and set αt+1=πt+1(Et+1) _t+1= _t+1(E_t+1). These probabilities condition on the realized parameters, specialist, harness, and collection context. Define ηt=TV(π~t,πt+1)+π~t(Et∖Et+1), _t=TV( π_t, _t+1)+ π_t(E_t E_t+1), (29) where TV(p,q)=supB⊆|p(B)−q(B)|TV(p,q)= _B |p(B)-q(B)|. The two terms measure the change in response distribution and the old probability of responses that no longer pass verification. Proposition 1 (Agent Verification). Under the conditions above, logα~tαt=gt−kt+KL(qt∥π~t(⋅∣Et))≥gt−kt, α_t _t=g_t-k_t+KL (q_t\| π_t(· E_t) )≥ g_t-k_t, (30) and αt+1≥[αtegt−kt−ηt]+, _t+1≥ [ _te^g_t-k_t- _t ]_+, (31) where [x]+=maxx,0[x]_+= \x,0\. In particular, gt>ktg_t>k_t is sufficient for improved verification under the fixed reference. The stronger condition ηt<αt(egt−kt−1) _t< _t(e^g_t-k_t-1) is sufficient for αt+1>αt _t+1> _t after the reference changes. Proof. Expanding the conditional distributions gives kt k_t =qtlogqt(τ)πt(τ)+logαt, =E_q_t q_t(τ) _t(τ)+ _t, KL(qt∥π~t(⋅∣Et)) (q_t\| π_t(· E_t) ) =qtlogqt(τ)π~t(τ)+logα~t. =E_q_t q_t(τ) π_t(τ)+ α_t. Subtracting the first equality from the second proves the identity in Equation 30. To prove its inequality, concavity of the logarithm gives, for any distributions q and p with the stated support and integrability conditions, qlogp(τ)q(τ)≤log∑τ:q(τ)>0p(τ)≤0.E_q p(τ)q(τ)≤ _τ:q(τ)>0p(τ)≤ 0. (32) Hence the KL divergence is nonnegative. Exponentiating yields α~t≥αtegt−kt α_t≥ _te^g_t-k_t. For the next collection, total variation and the decomposition of EtE_t imply αt+1 _t+1 ≥π~t(Et+1)−TV(π~t,πt+1) ≥ π_t(E_t+1)-TV( π_t, _t+1) ≥α~t−π~t(Et∖Et+1)−TV(π~t,πt+1) ≥ α_t- π_t(E_t E_t+1)-TV( π_t, _t+1) =α~t−ηt. = α_t- _t. Combining this inequality with the fixed reference bound and probability nonnegativity proves Equation 31. The two strict improvement conditions follow directly. ∎ If qt=πt(⋅∣Et)q_t= _t(· E_t), then kt=0k_t=0. Response selection and length weights generally change this distribution. For m conditionally independent responses in the next collection, monotonicity of a↦1−(1−a)ma 1-(1-a)^m also gives Pr(at least one verified response in round t+1) (at least one verified response in round t+1) (33) ≥1−(1−[αtegt−kt−ηt]+)m. ≥ 1- (1- [ _te^g_t-k_t- _t ]_+ )^m. The probability conditions on the fixed instance, collection state, and context. The lower bound inside the parentheses lies in [0,1][0,1] because ηt≥0 _t≥ 0 and αtegt−kt≤α~t≤1 _te^g_t-k_t≤ α_t≤ 1. With random shared context, apply this inequality conditionally and then average as in Equation 26. To connect this result to the training update, we decompose Equation 5 into its prompt and response terms. The dataset tZ_t, token indices, and raw next-token probabilities pθp_θ are defined in Section 4.2. The loss includes prompt tokens after the first token and all response tokens present in the record, including a stopping token when present. Contribution scores select records without weighting their losses, and the dataset remains fixed during fitting. Separating prompt and response tokens shows which likelihood gains this objective measures. Fix a nonempty training dataset Z and suppress the update round. Let θ and θ~ θ be the parameters before and after fitting. Its indexed records are zj=pj|τjz_j=p_j\| _j for j=1,…,||j=1,…,|Z|, where ∥\| denotes token concatenation. Each record has at least two tokens. Let wj=1/(|zj|−1)w_j=1/(|z_j|-1). The prompt pjp_j is nonempty, and pj,ℓp_j, and pj,<ℓp_j,< denote its token at position ℓ and preceding prefix. The same indexing applies to zjz_j. Define the raw response token product Pθ(τj∣pj)=∏ℓ=|pj|+1|zj|pθ(zj,ℓ∣zj,<ℓ).P_θ( _j p_j)= _ =|p_j|+1^|z_j|p_θ(z_j, z_j,< ). (34) Splitting the predicted tokens at the prompt boundary rewrites Equation 5 exactly as ℒA(θ)=−1||∑jwj[∑ℓ=2|pj|logpθ(pj,ℓ∣pj,<ℓ)+logPθ(τj∣pj)].L_A(θ)=- 1|Z| _jw_j [ _ =2^|p_j| p_θ(p_j, p_j,< )+ P_θ( _j p_j) ]. (35) An empty prompt token sum is zero. The first token has no preceding token in the training sequence and is excluded from the implemented loss. Assume that both parameter values assign positive probabilities to every predicted token in these records. The dataset and its length weights remain fixed during this comparison. Partition the record indices into groups uJ_u with the same source instance, prompt, and verifier, where u ranges over the nonempty groups. Write pup_u for their common prompt. Their weighted response distributions and shares of total weight are qu(τ)=∑j∈u,τj=τwj∑j∈uwj,ρu=∑j∈uwj∑jwj.q_u(τ)= _j _u,\, _j=τw_j _j _uw_j, _u= _j _uw_j _jw_j. (36) Thus ∑uρu=1 _u _u=1, and each quq_u has finite support consisting of responses that pass one fixed verifier. Subtracting the two losses and collecting response terms by group gives ℒA(θ)−ℒA(θ~)= _A(θ)-L_A( θ)= 1||∑jwj∑ℓ=2|pj|logpθ~(pj,ℓ∣pj,<ℓ)pθ(pj,ℓ∣pj,<ℓ) 1|Z| _jw_j _ =2^|p_j| p_ θ(p_j, p_j,< )p_θ(p_j, p_j,< ) (37) +∑jwj||∑uρuqulogPθ~(τ∣pu)Pθ(τ∣pu). + _jw_j|Z| _u _uE_q_u P_ θ(τ p_u)P_θ(τ p_u). The first term accounts for fitting prompt tokens. The second averages response gains over groups, with larger record weights for shorter concatenations. A lower total loss therefore does not establish an improved response likelihood for each instance. The proposition uses decoded probabilities πθ _θ, whereas the loss uses raw token probabilities pθp_θ. If both decoded distributions are positive on the finite support of quq_u, direct cancellation gives qulogπθ~(τ∣pu)πθ(τ∣pu)= _q_u _ θ(τ p_u) _θ(τ p_u)= qulogPθ~(τ∣pu)Pθ(τ∣pu) _q_u P_ θ(τ p_u)P_θ(τ p_u) (38) +qulogπθ~(τ∣pu)Pθ(τ∣pu)πθ(τ∣pu)Pθ~(τ∣pu). +E_q_u _ θ(τ p_u)P_θ(τ p_u) _θ(τ p_u)P_ θ(τ p_u). The left side is the decoded likelihood gain gtg_t for that group. The final term is zero when decoding samples directly from the model and the recorded token product contains every generation probability factor, including any stopping token. Temperature scaling, probability truncation, or an omitted stopping factor can change it. If decoding gives zero probability to a selected response, the finite likelihood and divergence conditions do not hold. Proposition 1 explains how agent fitting and later reference changes affect verification. The curriculum then selects specialist training instances, and pair acceptance determines their contribution to the exposure weights in Theorem 1. Increased verification alone does not establish that theorem’s exposure or retained loss conditions. B.3 Task and Implementation Details Our experiments use candidate predictions in recommendation, protein function prediction, and material discovery. For an instance i, let iC_i be the fixed reference candidate pool and Yi⊆iY_i _i the relevant candidates determined by the original label yiy_i. The harness exposes a view iS,H⊆iC_i^S,H _i. The definitions below apply to training and validation instances alike. To evaluate an individual response, the parser extracts an ordered list of distinct candidate indices from the view. A response is valid if this list is nonempty. It is then completed with omitted candidates in specialist order. A response with no valid indices receives quality zero without completion. For any ordering ρ of distinct candidates, let |ρ||ρ| be its length and ρ(r)ρ(r) the candidate at rank r. At a positive integer cutoff k, binary normalized discounted cumulative gain is (Järvelin and Kekäläinen, 2002) NDCGk(ρ,Yi)=∑r=1min(k,|ρ|)ρ(r)∈Yi/log2(r+1)∑r=1min(k,|Yi|)1/log2(r+1),Yi≠∅,0,Yi=∅.NDCG_k(ρ,Y_i)= cases _r=1 (k,|ρ|)1\ρ(r)∈ Y_i\/ _2(r+1) _r=1 (k,|Y_i|)1/ _2(r+1),&Y_i≠ ,\\[8.0pt] 0,&Y_i= . cases (39) For a valid response τ, its quality qiA(τ)q_i^A(τ) is this score on its completed ordering at the harness’s verification cutoff kHk_H. The specialist quality qiSq_i^S is the same score on the specialist ordering of the same view. Verification requires Yi≠∅Y_i≠ and qiA(τ)>qiSq_i^A(τ)>q_i^S. An empty view has no valid responses. Empty target sets are excluded from verification and calibration fitting, but receive zero in system evaluation. The normalization uses all relevant candidates in the reference pool, so removing relevant candidates from a view does not lower the denominator. When Yi≠∅Y_i≠ , qiA(τ)≤∑r=1min(kH,|Yi∩iS,H|)1/log2(r+1)∑r=1min(kH,|Yi|)1/log2(r+1).q_i^A(τ)≤ _r=1 (k_H,|Y_i _i^S,H|)1/ _2(r+1) _r=1 (k_H,|Y_i|)1/ _2(r+1). (40) At most |Yi∩iS,H||Y_i _i^S,H| relevant candidates are available, and placing them first maximizes the discounted gain. This proves the bound and shows it is attainable by an ordering of the view. If the specialist attains this ceiling, further response sampling in that view cannot produce a positive contribution. Changing the harness can change the available view. Joint selection orders each view by the combined score in Equation 3. The agent ranking is the ordered candidate list returned by the evaluation response procedure, and specialist scores include any stored calibration. Each compared state fits λ separately over the grid 0,0.25,0.5,1,2,4,8\0,0.25,0.5,1,2,4,8\. Let K be the task’s finite nonempty set of positive integer evaluation cutoffs. Let valD_val and testD_test be the validation and test pools, both disjoint from trainD_train and from each other. The comparison indices ℬtB_t refer to a batch from valD_val. Split this batch into the same two nonempty folds for both compared states. On each fitting fold, choose the grid value maximizing mean NDCG averaged over K, using only instances with nonempty target sets. Use zero if the fitting fold has no such instances. Evaluate on the other fold, then exchange their roles. All parsing, ordering, and tie rules remain fixed during the comparison. At zero weight, the joint ordering agrees with the specialist ordering within the same view. Let ρi _i be the joint ordering for evaluation instance i, formed with the weight fitted on the opposite fold. Then the score in Equation 4 is Jℬt(S,H,A)=1|ℬt|∑i∈ℬt1||∑k∈NDCGk(ρi,Yi).J_B_t(S,H,A)= 1|B_t| _i _t 1|K| _k NDCG_k( _i,Y_i). (41) The orderings depend on the evaluated (S,H,A)(S,H,A) and that state’s fitted weights. This instance average weights the fold means by fold size and includes empty target sets with their zero scores. Pair selection compares joint predictions, while response verification compares an individual interpreted response with the specialist. These evaluation rules determine the round procedure. Sample two archived harnesses and select the one with the higher score recorded at acceptance. Use a single entry directly, or the current harness if the archive is empty. Mutate one or two categorical fields to obtain H′H . Under (St,H′,At)(S_t,H ,A_t), the verified curriculum prioritizes instances with at least one verified response, and the unsuccessful curriculum prioritizes those with none. Both select in pool order, truncate at the training budget, and fill shortfalls from remaining instances. The uncertainty curriculum prefers smaller differences between the two largest scores si(c)+ai(c)s_i(c)+a_i(c). Uniform sampling bypasses exploration. The selected instances train S′S with Equation 2. With AtA_t fixed, evaluate the proposed and current pairs using Equation 41. Any specialist calibration uses a validation draw disjoint from this comparison batch. Accept a strictly positive improvement and record the harness and score, or restore the specialist parameters, optimizer, calibration, and harness. The retained pair then supplies the context for agent learning. Under (S¯t,H¯t)( S_t, H_t), collect fresh responses with AtA_t. Keep each eligible instance’s original response with the largest positive contribution. The agent curriculum retains all such pairs or the half with the largest contributions, keeping at least one when available. The retained original tokens form tZ_t for Equation 5. An empty dataset leaves the agent unchanged. Set St+1=S¯tS_t+1= S_t and Ht+1=H¯tH_t+1= H_t. The next round starts from (St+1,Ht+1,At+1)(S_t+1,H_t+1,A_t+1). Labels enter specialist supervision, verification, and evaluation, and are excluded from prompts. Validation data are reused across rounds, and test data are reserved for final evaluation. Figure 2 uses SD_S for the labeled examples indexed by ℐtSI_t^S. An archive entry (Hi,Ji)(H_i,J_i) stores a harness and its score at acceptance, where i indexes archive entries in that panel. The figure abbreviates Jℬt(S,H,At)J_B_t(S,H,A_t) as J(S,H,At)J(S,H;A_t) and ΔJt J_t as ΔJ J. Sampling τ∼Atτ A_t means sampling from πθt(⋅∣pi) _ _t(· p_i) under the retained pair. Superscripts in S(k)S^(k), H(k)H^(k), and A(k)A^(k) count component versions. The specialist and harness advance on acceptance, and the agent advances when it is updated. Appendix C Additional Experiments C.1 Evaluation Details We fix disjoint training, validation, and test sets before coevolution. Specialist adaptation, agent learning, and training instance selection use only trainD_train. At each round, we compare the proposed and current specialist and harness pairs on the same batch sampled from valD_val, holding the agent fixed. Both pairs use the same two folds, but fit their own λ on one fold and are evaluated on the other. We then exchange the folds and average the resulting instance scores for joint selection. Internal specialist calibration uses a separate validation batch disjoint from the comparison batch. Validation data guide selection throughout coevolution, while testD_test is reserved for final evaluation and does not inform training, calibration, or selection. For recommendation, we partition users across the three sets and train the initial specialist on histories excluding each user’s last two interactions. During coevolution, we use the second to last interactions of training users for learning and those of validation users for calibration and selection. Final evaluation uses the last interaction of each test user. For protein function prediction, we follow the CAFA temporal split, restrict inputs to information available at t0t_0, and keep instances sharing a protein name in the same set. For material discovery, we partition by chemical system, keeping all candidate structures from the same system in one set. Below we describe the evaluation metrics. Ranking metrics. Hit rate at cutoff k (HR@k) is the fraction of recommendation queries whose held out target item appears among the first k predictions. We report HR@10 in the main comparison. Normalized discounted cumulative gain (NDCG) assigns a binary relevance label to each candidate. For a ranking π, its discounted cumulative gain is DCG@k(π)=∑r=1min(k,|π|)rel(πr)log2(r+1).DCG@k(π)= _r=1 (k,|π|) rel( _r) _2(r+1). (42) NDCG@k divides this sum by the largest possible DCG@k within the same fixed reference pool. It rewards both recovering relevant candidates and placing them earlier. The reported CAFA score averages NDCG@5 and NDCG@10, while the WBM score averages NDCG@3 and NDCG@5. Protein function metrics. Precision is the fraction of predicted function terms that are correct, and recall is the fraction of relevant terms recovered. Recall@10 measures the fraction of relevant terms recovered among the first ten predictions. At prediction threshold t, their harmonic mean is F1(t)=2P(t)R(t)/(P(t)+R(t))F_1(t)=2P(t)R(t)/(P(t)+R(t)). The maximum score over thresholds is Fmax=maxtF1(t)F_ = _tF_1(t). The area under the precision–recall curve (AUPR) summarizes the tradeoff between precision and recall across thresholds. Minimum semantic distance (SminS_ ) measures errors using the information added by each ontology term given its parents (Jiang et al., 2016). At each threshold, remaining uncertainty ru(t)ru(t) is the average information in missed terms, and misinformation mi(t)mi(t) is the average information in incorrectly predicted terms. The metric is Smin=mintru(t)2+mi(t)2.S_ = _t ru(t)^2+mi(t)^2. (43) Lower values indicate less semantic error. Materials metrics. For crystal stability prediction, F1F_1 is the harmonic mean of precision and recall for the stable class. The discovery acceleration factor (DAF) divides precision by the fraction of stable structures in the evaluation pool (Riebesell et al., 2025). A value of one matches the expected precision of random selection. Mean absolute error (MAE) averages the absolute differences between predicted and reference target values. The coefficient of determination (R2R^2) equals one minus the ratio of squared prediction error to the squared error of predicting the mean reference value. It equals one for perfect predictions, zero for this constant baseline, and is negative when predictions have greater squared error. Higher values are better for all metrics except SminS_ and MAE. C.2 Complete Scientific Results Table 5: Protein function prediction on CAFA with two frozen backbones. Method Fmax↑F_ AUPR↑ Smin↓S_ NDCG↑ Recall@10↑ ESM-2 Base 0.2218±0.00140.2218_ [rgb]0.5,0.5,0.5± 0.0014 0.1078±0.00120.1078_ [rgb]0.5,0.5,0.5± 0.0012 12.557±0.06012.557_ [rgb]0.5,0.5,0.5± 0.060 0.6285±0.00190.6285_ [rgb]0.5,0.5,0.5± 0.0019 0.7179±0.00140.7179_ [rgb]0.5,0.5,0.5± 0.0014 ReST-EM 0.2219±0.00140.2219_ [rgb]0.5,0.5,0.5± 0.0014 0.1079±0.00120.1079_ [rgb]0.5,0.5,0.5± 0.0012 12.557±0.06012.557_ [rgb]0.5,0.5,0.5± 0.060 0.6286±0.00180.6286_ [rgb]0.5,0.5,0.5± 0.0018 0.7180±0.00120.7180_ [rgb]0.5,0.5,0.5± 0.0012 Meta-Harness 0.2224±0.00130.2224_ [rgb]0.5,0.5,0.5± 0.0013 0.1076±0.00120.1076_ [rgb]0.5,0.5,0.5± 0.0012 12.553±0.05912.553_ [rgb]0.5,0.5,0.5± 0.059 0.6309±0.00150.6309_ [rgb]0.5,0.5,0.5± 0.0015 0.7245±0.00070.7245_ [rgb]0.5,0.5,0.5± 0.0007 OpenEvolve 0.2230±0.00120.2230_ [rgb]0.5,0.5,0.5± 0.0012 0.1087±0.00110.1087_ [rgb]0.5,0.5,0.5± 0.0011 12.549±0.06012.549_ [rgb]0.5,0.5,0.5± 0.060 0.6320±0.00180.6320_ [rgb]0.5,0.5,0.5± 0.0018 0.7222±0.00160.7222_ [rgb]0.5,0.5,0.5± 0.0016 DGM 0.2219±0.00140.2219_ [rgb]0.5,0.5,0.5± 0.0014 0.1075±0.00120.1075_ [rgb]0.5,0.5,0.5± 0.0012 12.557±0.06012.557_ [rgb]0.5,0.5,0.5± 0.060 0.6325±0.00190.6325_ [rgb]0.5,0.5,0.5± 0.0019 0.7261±0.00160.7261_ [rgb]0.5,0.5,0.5± 0.0016 T-Discover 0.2219±0.00140.2219_ [rgb]0.5,0.5,0.5± 0.0014 0.1078±0.00130.1078_ [rgb]0.5,0.5,0.5± 0.0013 12.557±0.06012.557_ [rgb]0.5,0.5,0.5± 0.060 0.6287±0.00180.6287_ [rgb]0.5,0.5,0.5± 0.0018 0.7185±0.00130.7185_ [rgb]0.5,0.5,0.5± 0.0013 HACE 0.2438±0.00190.2438_ [rgb]0.5,0.5,0.5± 0.0019 0.1258±0.00150.1258_ [rgb]0.5,0.5,0.5± 0.0015 12.494±0.04812.494_ [rgb]0.5,0.5,0.5± 0.048 0.6745±0.00160.6745_ [rgb]0.5,0.5,0.5± 0.0016 0.7615±0.00150.7615_ [rgb]0.5,0.5,0.5± 0.0015 ProtT5-XL Base 0.2221±0.00050.2221_ [rgb]0.5,0.5,0.5± 0.0005 0.1078±0.00020.1078_ [rgb]0.5,0.5,0.5± 0.0002 13.127±0.07413.127_ [rgb]0.5,0.5,0.5± 0.074 0.6241±0.00110.6241_ [rgb]0.5,0.5,0.5± 0.0011 0.7122±0.00150.7122_ [rgb]0.5,0.5,0.5± 0.0015 ReST-EM 0.2221±0.00050.2221_ [rgb]0.5,0.5,0.5± 0.0005 0.1079±0.00030.1079_ [rgb]0.5,0.5,0.5± 0.0003 13.127±0.07413.127_ [rgb]0.5,0.5,0.5± 0.074 0.6242±0.00110.6242_ [rgb]0.5,0.5,0.5± 0.0011 0.7131±0.00160.7131_ [rgb]0.5,0.5,0.5± 0.0016 Meta-Harness 0.2229±0.00060.2229_ [rgb]0.5,0.5,0.5± 0.0006 0.1085±0.00020.1085_ [rgb]0.5,0.5,0.5± 0.0002 13.122±0.07513.122_ [rgb]0.5,0.5,0.5± 0.075 0.6261±0.00170.6261_ [rgb]0.5,0.5,0.5± 0.0017 0.7156±0.00200.7156_ [rgb]0.5,0.5,0.5± 0.0020 OpenEvolve 0.2230±0.00030.2230_ [rgb]0.5,0.5,0.5± 0.0003 0.1082±0.00030.1082_ [rgb]0.5,0.5,0.5± 0.0003 13.121±0.07213.121_ [rgb]0.5,0.5,0.5± 0.072 0.6254±0.00140.6254_ [rgb]0.5,0.5,0.5± 0.0014 0.7139±0.00150.7139_ [rgb]0.5,0.5,0.5± 0.0015 DGM 0.2235±0.00060.2235_ [rgb]0.5,0.5,0.5± 0.0006 0.1088±0.00010.1088_ [rgb]0.5,0.5,0.5± 0.0001 13.118±0.07513.118_ [rgb]0.5,0.5,0.5± 0.075 0.6273±0.00160.6273_ [rgb]0.5,0.5,0.5± 0.0016 0.7175±0.00160.7175_ [rgb]0.5,0.5,0.5± 0.0016 T-Discover 0.2221±0.00050.2221_ [rgb]0.5,0.5,0.5± 0.0005 0.1079±0.00030.1079_ [rgb]0.5,0.5,0.5± 0.0003 13.127±0.07413.127_ [rgb]0.5,0.5,0.5± 0.074 0.6241±0.00110.6241_ [rgb]0.5,0.5,0.5± 0.0011 0.7135±0.00160.7135_ [rgb]0.5,0.5,0.5± 0.0016 HACE 0.2317±0.00050.2317_ [rgb]0.5,0.5,0.5± 0.0005 0.1177±0.00100.1177_ [rgb]0.5,0.5,0.5± 0.0010 13.127±0.07213.127_ [rgb]0.5,0.5,0.5± 0.072 0.6739±0.00130.6739_ [rgb]0.5,0.5,0.5± 0.0013 0.7553±0.00120.7553_ [rgb]0.5,0.5,0.5± 0.0012 Table 6: Material discovery task on WBM with CHGNet. Method NDCG↑ F1↑F_1 DAF↑ MAE↓ R2↑R^2 Base 0.6388±0.00110.6388_ [rgb]0.5,0.5,0.5± 0.0011 0.5159±0.00030.5159_ [rgb]0.5,0.5,0.5± 0.0003 1.644±0.0031.644_ [rgb]0.5,0.5,0.5± 0.003 0.1049±0.00020.1049_ [rgb]0.5,0.5,0.5± 0.0002 −1.289±0.001-1.289_ [rgb]0.5,0.5,0.5± 0.001 ReST-EM 0.6496±0.00060.6496_ [rgb]0.5,0.5,0.5± 0.0006 0.5508±0.00040.5508_ [rgb]0.5,0.5,0.5± 0.0004 2.172±0.0052.172_ [rgb]0.5,0.5,0.5± 0.005 0.0756±0.00020.0756_ [rgb]0.5,0.5,0.5± 0.0002 −0.323±0.001-0.323_ [rgb]0.5,0.5,0.5± 0.001 Meta-Harness 0.6513±0.00080.6513_ [rgb]0.5,0.5,0.5± 0.0008 0.5510±0.00040.5510_ [rgb]0.5,0.5,0.5± 0.0004 2.171±0.0032.171_ [rgb]0.5,0.5,0.5± 0.003 0.0754±0.00030.0754_ [rgb]0.5,0.5,0.5± 0.0003 −0.281±0.001-0.281_ [rgb]0.5,0.5,0.5± 0.001 OpenEvolve 0.6486±0.00080.6486_ [rgb]0.5,0.5,0.5± 0.0008 0.5463±0.00030.5463_ [rgb]0.5,0.5,0.5± 0.0003 2.193±0.0042.193_ [rgb]0.5,0.5,0.5± 0.004 0.0755±0.00020.0755_ [rgb]0.5,0.5,0.5± 0.0002 −0.264±0.001-0.264_ [rgb]0.5,0.5,0.5± 0.001 DGM 0.6496±0.00070.6496_ [rgb]0.5,0.5,0.5± 0.0007 0.5508±0.00030.5508_ [rgb]0.5,0.5,0.5± 0.0003 2.183±0.0032.183_ [rgb]0.5,0.5,0.5± 0.003 0.0748±0.00030.0748_ [rgb]0.5,0.5,0.5± 0.0003 −0.251±0.001-0.251_ [rgb]0.5,0.5,0.5± 0.001 T-Discover 0.6474±0.00070.6474_ [rgb]0.5,0.5,0.5± 0.0007 0.5504±0.00030.5504_ [rgb]0.5,0.5,0.5± 0.0003 2.203±0.0032.203_ [rgb]0.5,0.5,0.5± 0.003 0.0748±0.00020.0748_ [rgb]0.5,0.5,0.5± 0.0002 −0.284±0.001-0.284_ [rgb]0.5,0.5,0.5± 0.001 HACE 0.6541±0.00100.6541_ [rgb]0.5,0.5,0.5± 0.0010 0.5513±0.00040.5513_ [rgb]0.5,0.5,0.5± 0.0004 2.242±0.0042.242_ [rgb]0.5,0.5,0.5± 0.004 0.0729±0.00030.0729_ [rgb]0.5,0.5,0.5± 0.0003 −0.203±0.001-0.203_ [rgb]0.5,0.5,0.5± 0.001 C.3 More Experimental Details Agent and model scaling. Table 4 varies specialist size and agent size in separate sweeps. ESM-2 backbone representations remain frozen while the prediction head is trained. The agent variants share a 30-round configuration with ESM-2 650M. In this sweep, Base is the initial specialist score, shared across the agent variants, and Δ is the reported mean within-run gain. The specialist sweep uses a separate configuration. Subscripts report the standard deviation of the gains. The HACE means for ESM-2 35M, 150M, and 3B are obtained by adding the reported Base means and gains. Both the specialist sweep and all agent variants use three random seeds. Convergence on base. We begin with specialists already trained to convergence under their original schedules. Continued supervised training alone does not by itself indicate which further updates help, and our ablations show that the acceptance gate in Equation 4 is what makes continued training productive. This setting tests whether interaction with an evolving agent can create additional opportunities beyond that plateau. We define convergence as follows: continuing the original supervised schedule for an additional 33 epochs changes the reported metric by less than 0.0010.001 in every domain. Evolution records. The lower panels of Figure 4 show selection scores from one recorded run in each domain. Grey points are the available per-round scores, and the lines track their running maxima. The recommendation trajectory covers 30 rounds, while the protein and materials trajectories cover 50 rounds. The CAFA record contains scores for 18 rounds. Missing entries are omitted. These trajectories describe the scores observed during selection, while the main tables report aggregate evaluation results. The recommendation case was collected in a separate logging run with the same configuration, seed, and data as its trajectory. The upper panels compare specialist and combined predictions on the same instance. The protein example is recorded at round 40 and the materials example at round 0. Hyperparameters. We sample 192 training instances per round. For each instance, the agent generates m responses with temperature 1.0 and top-p 0.95, where m=8m=8 for recommendation and crystal stability and m=4m=4 for protein function. The generation batch size is 96. Prompts exceeding 2560 tokens are truncated by retaining the first 853 and the last 1707 tokens. For joint selection, we sample |ℬt|=min(256,|val|)|B_t|= (256,|D_val|) validation instances. We train the agent with LoRA adapters (r=16r=16, α=32α=32) using AdamW with a learning rate of 10−510^-5, gradient clipping at 1.0, and a micro-batch size of 2 sequences. Specialist adaptation in Equation 2 uses 200 steps for recommendation and crystal stability and 750 steps for protein function, all at a learning rate of 10−410^-4. Hardware. All experiments are conducted on NVIDIA B200 GPU clusters, where each node contains 8 GPUs. Table 7: HACE is agnostic to specialist architectures and adaptation on different learning components. Domain Model Architecture Updated parameters Recommendation SASRec Causal self-attention Transformer Model parameters HSTU Causal sequential transducer with pointwise aggregated attention Model parameters Wukong Stacked factorization-machine network Model parameters Protein ESM-2 Encoder-only protein Transformer Prediction head ProtT5-XL Protein T5 with encoder-decoder pretraining Prediction head Materials CHGNet Charge-informed graph neural potential Model parameters Table 8: How exploration guides the two learners. Relative improvement determines eligibility for different learning objectives. Exploration outcome Agent learning Specialist learning At least one sampled ranking improves the specialist Best positive response is eligible for the agent curriculum Instance is eligible for the verified curriculum No sampled ranking improves the specialist No response from this instance enters the agent update Instance is eligible for the unsuccessful curriculum Specialist selection uses exploration before joint selection. Agent learning uses fresh responses afterward. The specialist always trains on task labels. Uniform and uncertainty curricula, as well as budget completion, may select instances from either row. Appendix D Harness Engineering D.1 Prompts for Recommendation We use a common recommendation interface for MovieLens 1M and the Amazon Beauty, CDs, Sports, and Toys datasets. For each user, it combines an interaction history with a candidate ranking from SASRec, HSTU, or Wukong. Harness search controls the history length, available item metadata, and the partition between leading predictions and candidate alternatives. The prompt uses the same item terminology across all five datasets. Prompt: Recommendation Systems Use the user’s interaction history and the available item metadata to select the k candidate alternatives that best match the user’s preferences. The specialist’s leading predictions provide additional context. Interaction history recent interactions in sequence, with item titles and available metadata Specialist’s leading predictions leading items in specialist order, with available metadata Candidate alternatives 1. item title and available metadata 2. item title and available metadata … Output Return exactly k distinct IDs from Candidate alternatives, ordered from best to worst and separated by spaces. Return no other text. When k=1k=1 and the leading block is nonempty, the following instruction uses the same inputs and allows the agent to retain the specialist’s selection. Otherwise, the ranking prompt above applies. Prompt: Recommendation with a Single Selection Use the user’s interaction history and the available item metadata to identify the best candidate alternative. Compare it with the lowest ranked item among the specialist’s leading predictions. Output If this alternative is a better match for the user, return its integer ID. Otherwise, return 0. Return one integer and no other text. Table 9: Recommendation harness fields. “Fixed” denotes a value outside the search space. Field Initial Candidate values Role n_candidates 50 Fixed Specialist shortlist size n_history 15 5, 10, 15, 25, 40 Number of history entries top_k 5 1, 3, 5, 10 Requested ranking length promote_from 10 0, 5, 10, 20 Leading candidates shown separately n_promote 1 Fixed Promotion count when k=1k=1 show_year True True, False Release year metadata show_genre True True, False Category metadata D.2 Prompts for Protein Function Prediction For protein function prediction, the candidates are Gene Ontology terms. The prompt combines protein metadata and available annotations with the specialist’s ranking. Harness search controls the number of annotations, the length of term definitions, and the metadata shown to the agent. The agent returns candidate IDs using the same ranking format as in recommendation. Prompt: Protein Function Prediction Use the protein information, available annotations, and Gene Ontology term descriptions to select the k candidate alternatives most likely to apply to this protein. The specialist’s leading predictions provide additional context. Protein information Protein: protein name Organism: organism, when enabled Target ontology aspect: aspect, when enabled Available annotations: known terms Specialist’s leading predictions GO terms in specialist order, with available definitions Candidate alternatives 1. GO term name and available definition 2. GO term name and available definition … Output Return exactly k distinct IDs from Candidate alternatives, ordered from most to least likely and separated by spaces. Return no other text. Table 10: Presentation fields of the protein harness. Experience selection is described in Appendix B.3. Field Initial Candidate values Role n_candidates 50 Fixed Specialist shortlist size n_known 10 0, 5, 10, 20 Number of known annotations top_k 5 1, 3, 5, 10 Requested ranking length promote_from 10 0, 5, 10, 20 Leading candidates shown separately def_chars 120 0, 120, 300 Definition length, zero for name only show_aspect True True, False Target ontology aspect show_organism True True, False Source organism D.3 Prompts for Material Discovery For material discovery, the agent compares candidate crystals within a chemical system. The prompt provides compositions and available structure prototypes, together with the specialist’s leading predictions. Harness search controls the prototype length and whether element information and specialist ranks are shown. Candidate IDs identify the agent’s selections independently of the specialist ranks. Prompt: Crystal Stability Prediction Use the chemical system, candidate compositions, and available structural descriptors to select the k candidate alternatives most likely to be thermodynamically stable. The specialist’s leading predictions provide additional context. Chemical system System: chemical system Elements: element set, when enabled Specialist’s leading predictions formulas in specialist order, with available structure prototypes Candidate alternatives 1. formula, structure prototype, and specialist rank when enabled 2. formula, structure prototype, and specialist rank when enabled … Output Return exactly k distinct IDs from Candidate alternatives, ordered from most to least likely and separated by spaces. Use the IDs at the start of the candidate rows. Return no other text. Table 11: Presentation fields of the crystal stability harness. Prototype lengths are measured in characters. Field Initial Candidate values Role n_candidates 20 Fixed Specialist shortlist size top_k 3 1, 2, 3, 5 Requested ranking length promote_from 2 0, 1, 2, 3 Leading candidates shown separately proto_chars 24 0, 24, 48 Structure prototype length show_elements True True, False Element set show_rank True True, False Specialist rank per candidate