AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block

The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 66. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-23 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued.

PATH-Bench: Path-Dependent Evaluation of Lifelong Agents

Lifelong LLM agents increasingly adapt through external learning states that store past interactions as retrievable memories or reusable skills, yet existing benchmarks rarely account for how the path of accumulated experience shapes what agents transfer and retain. In this work, we establish PATH-Bench, a benchmark for path-dependent evaluation of lifelong agents. PATH-Bench estimates directed task relationships via multi-model in-context learning, constructs probe-centered sequences with controlled helpful and interfering histories, and repeatedly evaluates probe tasks to measure average performance, forward transfer, backward transfer, and forgetting. We evaluate eight representative agents on single-turn code generation and multi-turn tool-use tasks under positive- and negative-dominant histories. Benchmark results show that experience utility depends jointly on how experience is represented and on the task's interaction structure, that strong transfer does not ensure retention, and that later experience can reshape gains acquired earlier in the learning path. Based on these findings, we propose Selective Experience Use (SEU), an agent harness that regulates how path-accumulated experience influences each new task, admitting helpful items while filtering out potential interference. SEU consistently reduces forgetting while improving forward transfer in the majority of settings. The PATH-Bench provides both a controlled evaluation framework and actionable guidance for designing more selective and robust lifelong agents.

MiniOpt: Reasoning to Model and Solve General Optimization Problems with Limited Resources

Achieving strong optimization generalization across diverse optimization problems while requiring limited training resources remains a challenging problem for optimization-oriented large language models (LLMs). Existing approaches typically rely on large-scale supervised datasets, costly reasoning annotations, and expensive intermediate step verification, resulting in substantial training overhead. To address these challenges, we propose MiniOpt, a reinforcement learning framework that learns to solve optimization problems through an "reasoning-to-model-and-solve" paradigm. MiniOpt decomposes optimization reasoning into structured optimization modeling and executable solver generation. Building upon this paradigm, we introduce OptReward, a reward function with hierarchical score structure that jointly evaluates formulation and solution, enabling effective policy learning without expert demonstrations. We further develop an optimization-oriented policy optimization strategy that improves exploration efficiency and stabilizes reinforcement learning for compact models. Extensive experiments show that MiniOpt-3B exhibits strong optimization generalization across various optimization types, problem scenarios, and task domains. For models with fewer than 10B parameters, MiniOpt series achieves the highest average solving accuracy (SA). For models with more than 10B parameters, MiniOpt still shows competitive performance. These results suggest that optimization-oriented reward design and reinforcement learning provide an effective pathway for developing compact optimization-specialized language models with strong optimization generalization capabilities. The code is available at https://github.com/Hsiang-1/MiniOpt.

IntElicit: Eliciting and Assessing Contextualized Creativity via Dialogue Policy Optimization

Contextualized assessment offers high ecological validity for evaluating creativity but introduces a critical challenge: observed performance may be confounded with cognitive proficiency (domain knowledge) and agency (willingness to engage). Meanwhile, in the age of generative AI, creative problem solving increasingly occurs in tool-mediated and human--AI interactive environments, making fully static assessment less aligned with contemporary creative practice. To address these issues, this paper proposes IntElicit, a framework for eliciting and assessing contextualized creativity via dialogue policy optimization. IntElicit functions as a constrained adaptive AI Interviewer: it provides non-directive knowledge and agency scaffolds in multi-turn interaction to reduce non-creative confounders, while preserving participants' responsibility for generating the creative content being evaluated. Specifically, to tackle sparse rewards and potential reward hacking (e.g., answer dictation) in open-ended educational dialogue, IntElicit introduces a decomposed process reward mechanism. This mechanism aligns the policy with pedagogical elicitation, rewarding prompts that draw out participant reasoning rather than producing optimal answers on their behalf. Extensive experiments, including participant simulation and a human subject study (N=64), show that IntElicit improves elicited creative outcomes over expert-designed baselines. Together, the results suggest that interactive elicitation can reveal creative potential that static FPSP-style assessment may miss, providing a formative and diagnostic lens for contextualized creativity assessment in AI-mediated learning contexts.

RealMath-Eval: Why SOTA Judges Struggle with Real Human Reasoning

While Large Language Models (LLMs) have achieved near-perfect performance in \emph{solving} high-school mathematics, their ability to \emph{evaluate} the diverse reasoning processes of real human students remains under-examined. To bridge this gap, we introduce \textbf{RealMath-Eval}, a rigorously annotated benchmark of 224 real-world exam responses from high schools. Our initial evaluation reveals that even state-of-the-art LLM judges struggle significantly on this task, exhibiting a high Mean Squared Error ($\sim$2.96) against expert human grading. To probe a plausible explanation, we contrast this performance with a control setting where the same judges evaluate synthetic LLM-generated solutions. We identify a stark ``Evaluation Gap'': judges are considerably more accurate and consistent on synthetic text (MSE $\sim$1.17) but struggle to generalize to authentic student reasoning. Through semantic embedding analysis, we find that synthetic errors suffer from a ``structural collapse'' into predictable, low-dimensional linear subspaces, whereas human errors form a more diverse error space. Furthermore, generative probability probes suggest that human reasoning involves significantly higher information-theoretic surprisal, indicating that student reasoning transitions are more out-of-distribution for current models. Finally, we find that surface-level style transfer fails to close this gap. Our findings suggest that current LLM evaluation pipelines relying heavily on synthetic data may not adequately capture the diversity of authentic student mathematical reasoning.

DEFINED: A Data-Efficient Computational Framework for Fine-Grained Creativity Assessment in Debate Scenarios

Human creativity has emerged as a critical competency in the era of large language models. Assessing creativity in complex, open-ended environments is a grand challenge in data mining, currently hindered by a reliance on standardized simple tasks and the scarcity of fine-grained expert data. As an ecologically valid assessment context, debate reflects multiple dimensions of creativity, encompassing both divergent thinking and convergent thinking. Moreover, debate is a data-rich domain, with a large volume of publicly accessible materials. Current mainstream automated scoring methods are poorly suited to complex settings such as debate, and therefore still rely on costly human evaluation. To this end, this paper proposes DEFINED, a data-efficient computational framework for fine-grained creativity assessment in debate scenarios. DEFINED operationalizes debate creativity through a hierarchical eight-dimensional metric system, implemented via a pre-trained autoregressive language model with a hierarchical scoring head that supports both fine-grained and coarse-grained evaluation. Statements and their associated expert scores were obtained from authentic debate competitions, and a constrained data augmentation strategy was employed to address the elite bias inherent in the original data. DEFINED adopts a mixed-granularity training strategy enabling robust learning from limited fine-grained supervision annotated by trained graduate experts. To rigorously validate ecological validity beyond synthetic benchmarks, we incorporate an empirical study with debate-naive participants, utilizing these authentic data to serve as a qualitative case study for mid-to-low proficiency populations. Across our evaluation protocol, our scoring model achieves accurate and stable scoring, outperforming prompt-based large language model evaluators and existing debate scoring methods.

CollabBench: Benchmarking and Unleashing Collaborative Ability of LLMs with Diverse Players via Proactive Engagement

While LLM-based agents excel at individual tasks, effective collaboration with realistic human partners remains challenging. Most of the existing conversation-level collaborative studies lack grounded interaction and behavioral execution, motivating the need for cooperative game environments that enable contextualized and immersive collaboration. To this end, this paper proposes CollabBench, a benchmark for evaluating and training collaborative agents in cooperative games. CollabBench features a Diverse Player Profile Simulation pipeline to model varied players behaviors, and a Collaborative Agentic Training paradigm that unifies reasoning, communication, and action via agentic rollouts, optimized with a hybrid reward balancing task efficiency and affective adaptation. We further extend classic environments to CWAH-MultiPlayer and Cook-MultiPlayer for systematic evaluation under diverse personalities. Experiments with efficiency and affective metrics show that our trained models outperform base models, achieving 19.5% higher efficiency and 24.4% improved affective performance. Further analysis reveals key collaborative limitations of existing models and offers insights for future collaborative training.

Mean-Field Diffuser: Scaling Offline MARL to Thousands of Agents

Diffusion-based planning has achieved strong results in single-agent offline reinforcement learning, yet scaling to many-agent systems remains intractable due to the curse of dimensionality in the joint trajectory space. We introduce MF-Diffuser, a framework that lifts trajectory planning to the Wasserstein space of trajectory distributions, where the propagation of chaos ensures a small representative subset of agents captures the full population dynamics. Our approach features a value-weighted chaotic entropy objective that reconciles generative fidelity with return maximization, and a hierarchical coarse-to-fine strategy that progressively grows the agent population during denoising. We establish end-to-end suboptimality bounds with four interpretable terms, revealing that mean-field approximation error scales as $O(H^2/\sqrt{N})$ while offline distribution shift provably does not grow with population size $N$, and prove the generated policy is an approximate mean-field Nash equilibrium with explicit convergence guarantees. Experiments on three mean-field RL benchmarks -- spanning stage games, sequential dynamics, and adversarial team competition -- show MF-Diffuser achieves the best return in the majority of settings, with the largest gains on suboptimal offline data and at extreme scales ($N \geq 10^3$).

AgentSchool: An LLM-Powered Multi-Agent Simulation for Education

Despite the rapid deployment of LLMs into classrooms, validating educational AI remains uniquely intractable: interventions act on developing learners whose cognitive and social trajectories are irreversibly shaped, while real-world trials are slow, ethically constrained, and institutionally locked. LLM-based educational simulators have emerged as a potential remedy, but many still collapse learning into persona-conditioned role-play and, when optimized only to reproduce existing classrooms, can structurally penalize the institutional novelty that pedagogical reform requires. In this work, we introduce AgentSchool, an LLM-driven multi-agent simulator that models learning as state transition rather than prompted behavior. AgentSchool couples cognitively growable student agents -- equipped with weighted subject knowledge graphs, thinking-workflow pools, and explicit misconceptions -- with adaptive teacher agents that plan, scaffold, and reflect along the Zone of Proximal Development, embedded in a configurable scenery generator that situates instruction within both formal and informal learning fields, and a multi-scale simulator that decouples interaction scale, temporal granularity, and simulation duration. Experiments show that structured student agents produce more differentiated mastery and misconception traces than a baseline simulator, while teacher-agent comparisons show backbone-dependent patterns consistent with ZPD-informed adaptation. Further, AgentSchool generates plausible traces of peripheral participation, clique formation, aggressor-induced cohesion, and opinion-leader emergence consistent with classroom social theories. Beyond its role as an educational research instrument, AgentSchool frames education as a socially meaningful testbed for long-horizon memory, multi-agent coordination, and future institutional reasoning under organizational pressure.

OptSkills: Learning Generalizable Optimization Skills from Problem Archetypes via Cluster-Based Distillation

Leveraging Large Language Models (LLMs) to automatically formulate and solve optimization problems from natural language has emerged as an efficient paradigm for automated optimization. However, existing methods still exhibit limited generalization: they are sensitive to superficial narrative variations, reuse experience mainly at the case level, and struggle to adapt to shifted or emerging problem types. We propose OptSkills, an archetype-centric skill learning and reasoning agent system for optimization modeling and solving. To improve robust generalization, our system clusters problems by their underlying archetypes rather than surface narratives. To improve in-distribution generalization, it explores diverse modeling paradigms and solver configurations within each cluster, then distills successful trajectories into reusable workflow-level skills. To improve out-of-distribution generalization, it refines existing skills or expands the skill library using newly obtained trajectories. Our system achieves a state-of-the-art micro-averaged accuracy of 68.27% on datasets encompassing diverse problem types and scenarios. In addition, on MIPLIB-NL, a highly challenging large-scale and high-dimensional benchmark, it achieves 26.91% accuracy, outperforming DeepSeek-V3.2-Thinking by 4.53%. After skill learning on Nano-CO, it reaches 72.79% on the OOD NLCO benchmark. Code and skills are available at https://github.com/fujiwaranoM0kou/OptSkills.