Step-by-Step Optimization-like Reasoning in LLMs over Expanding Search Spaces
Nicolás Astorga, Nabeel Seedat, Mihaela van der Schaar
Abstract
Verifiable reward training has improved mathematical and coding reasoning, but these domains capture only part of step-by-step decision making. Many real-world tasks require finding a high-value feasible plan among many valid alternatives. We introduce OPT*, a scalable family of optimization-style tasks for training and evaluating LLM step-by-step optimization-like reasoning along a complexity axis: each task provides a feasibility checker and evaluator, while a complexity parameter expands the search space without requiring new human labels. This motivates studying these tasks in two regimes: (i) solver-guided online policy optimization, which uses a solver as a value oracle for partial states and applies rank-based reward shaping to reinforce better next steps, and (ii) search-based offline RL when such solvers are unavailable. Theoretically, we relate success in large search spaces to the information a reasoner extracts per unit of search budget. Empirically, we ablate the ingredients that make search efficient on OPT* and show that training on OPT* improves step-by-step optimization-like reasoning.
AI Impact Assessments
(1 models)Scientific Impact Assessment: "Step-by-Step Optimization-like Reasoning in LLMs over Expanding Search Spaces"
1. Core Contribution
The paper introduces OPT★, a formalized family of auto-verifiable, scalable optimization tasks designed to train and evaluate LLMs on step-by-step constrained decision-making. The key insight is that optimization problems naturally provide the ingredients needed for verifiable reward training—cheap feasibility checking, objective evaluation, and scalable difficulty—while requiring reasoning capabilities beyond what math/code benchmarks test. The paper contributes: (a) a formal task definition with complexity parameter α that controls search space size without additional annotation; (b) two complementary training recipes—offline RL via search-guided trajectory selection with feasibility pruning and action deduplication, and online RL via solver-guided policy optimization (νPO); and (c) an information-theoretic analysis relating success probability to information extracted per unit of search budget.
2. Methodological Rigor
Theoretical foundations. Theorem A.3 provides an information-theoretic bound connecting success probability to mutual information between the optimum and the search history, scaled by the effective number of optima. This is a clean Fano-style result, though it relies on the assumption that the optimal solution is uniformly distributed over an effective set (Assumption A.1), which may not hold for structured optimization instances where optima cluster. The effective branching factor b_eff provides a practical diagnostic, though its connection to the theorem is more illustrative than tight.
Experimental design. The experiments are well-structured around five questions (a-e), covering search efficiency, solver vs. self-improvement, domain transfer, generalization, and curriculum effects. However, several concerns arise:
Reproducibility. The paper provides extensive implementation details (Appendices C-F), hyperparameters, and prompt examples. The task generation procedures are clearly specified and should be reproducible.
3. Potential Impact
Practical value. The framework addresses a genuine gap: many real-world tasks (scheduling, routing, resource allocation) require optimization-like reasoning that math/code benchmarks don't capture. The plug-and-play components [C1-C2]—feasibility pruning and action deduplication—are simple, general, and immediately applicable to any structured search method over LLM outputs.
Benchmark utility. OPT★'s key advantage is infinite scalability along the complexity axis without annotation cost. This is valuable for RL-based training where data hunger is a bottleneck. The six classical task families (role assignment, scheduling, MaxSAT, TSP, 2D packing, QAP) cover diverse reasoning modalities.
Training methodology. The solver-guided online RL (νPO) provides a clean recipe for using domain solvers as value oracles for intermediate states, with rank-based reward shaping. This is a practical contribution for domains where solvers exist but are too expensive for deployment.
Broader influence. The generalization results—training on optimization tasks improving math benchmark performance (AMC'23, AIME)—are intriguing but modest. The spatial reasoning transfer (polyomino → spatial QA) is more convincing, showing that optimization training can develop transferable sub-skills.
4. Timeliness & Relevance
The paper addresses a timely gap in the LLM reasoning literature. While verifiable reward training has driven remarkable progress on math and code, extending these methods to constrained multi-step planning is an active frontier. The optimization framing is natural and well-motivated. The work connects to the growing interest in process reward models, MCTS for LLMs, and curriculum learning for reasoning.
5. Strengths & Limitations
Strengths:
Limitations:
Overall assessment: The paper makes a solid conceptual contribution by identifying optimization tasks as a scalable source for training step-by-step reasoning. The practical components ([C1-C2], νPO) are useful but incremental. The theoretical analysis is clean but unsurprising. The experimental evidence supports the main claims but at modest scale. This is a well-organized framework paper that opens a promising research direction rather than delivering a breakthrough result.
Generated Jun 5, 2026
Comparison History (17)
Paper 2 is likely higher impact: it introduces a scalable task family (OPT*) that generalizes beyond math/code to planning-like optimization, provides an evaluation/training framework with controllable search-space complexity, and connects performance to information efficiency with theoretical framing. This can influence benchmarking, RLHF/verifiable training, and agentic planning across domains. Paper 1 is strong and practical for federated personalization, but its contributions are more incremental within federated LoRA, with narrower cross-field reach than a new reasoning/task paradigm for LLM optimization and search.
SHARP addresses the critical credit assignment problem in multi-agent LLM systems using Shapley values, providing a principled and practical framework with strong empirical results (23.66% and 14.05% improvements). It tackles a fundamental bottleneck in the rapidly growing multi-agent LLM field. Paper 2 introduces interesting optimization-style reasoning tasks but is more incremental, extending verifiable reward training to a new domain. SHARP's broader applicability to multi-agent systems, methodological rigor via game-theoretic foundations, and immediate practical impact give it an edge.
Paper 1 introduces a novel, structured approach to analyzing the internal reasoning topology of Large Language Models, addressing a critical gap in current LLM evaluation which relies heavily on opaque token counts and final accuracy. By converting reasoning traces into verifiable graphs, it offers a foundational interpretability and diagnostic tool that has broad implications across AI safety, cognitive modeling, and model development, giving it a broader potential scientific impact than the specialized optimization tasks in Paper 2.
Paper 1 likely has higher scientific impact due to broader novelty and generality: it proposes a scalable task family (OPT*) to study optimization-like step-by-step reasoning across expanding search spaces without new labels, introduces training regimes (solver-guided online PO and solver-free offline RL), and offers theoretical framing relating performance to information gained per search budget. This can transfer to many real-world planning/optimization domains beyond math/proving. Paper 2 is highly timely and strong empirically, but is more domain-specific (Lean theorem proving) and primarily an engineering/system advance, limiting breadth despite impressive results.
Paper 1 addresses a critical bottleneck in current AI research: improving step-by-step reasoning and optimization capabilities in LLMs. By providing a scalable framework (OPT*) and novel training paradigms, it offers immediate, broad utility for developing advanced AI agents. While Paper 2 provides valuable insights into human-AI interaction, Paper 1's methodological contributions to enhancing core LLM capabilities are likely to drive broader downstream applications and higher immediate scientific impact across the rapidly moving field of agentic AI.
Paper 2 addresses a critical bottleneck in the current frontier of AI: the computational and memory inefficiency of Large Reasoning Models. By proposing a dynamic KV cache eviction method based on decision-critical tokens, it offers highly practical, immediate real-world applications that can significantly reduce inference costs. While Paper 1 introduces a novel reasoning framework, Paper 2's potential to optimize the deployment of state-of-the-art models gives it broader and more immediate impact.
Paper 1 addresses a foundational challenge in AI: extending LLM reasoning beyond math and coding to general optimization and planning. Its scalable framework and theoretical insights into search and RL align with the frontier of reasoning research, offering broad, cross-domain applicability. Paper 2, while highly practical and addressing critical efficiency issues in autonomous driving, is more domain-specific and relies on established knowledge distillation paradigms, resulting in a narrower overall scientific impact.
Paper 2 proposes a visionary paradigm—biomedical world models—that could transform drug discovery, clinical trials, surgical planning, and fundamental biology by enabling prospective simulation rather than static pattern recognition. Its breadth of impact across biomedicine, AI, and clinical practice is enormous, and the concept addresses a critical gap in current foundation model capabilities. While Paper 1 makes solid contributions to LLM reasoning via optimization tasks and RL training, its scope is narrower (optimization-style reasoning benchmarks). Paper 2's timeliness, given the convergence of foundation models and biomedicine, gives it substantially higher potential impact.
Paper 1 introduces a novel framework (OPT*) that addresses a fundamental gap in LLM reasoning—optimization-like step-by-step decision making beyond math/coding. It offers both theoretical foundations and practical training methodologies (solver-guided RL and search-based offline RL), with broad applicability across combinatorial optimization domains. Paper 2, while technically solid and practically useful for LLM serving infrastructure, is more incremental—optimizing KV cache management. Paper 1 has greater breadth of impact, opening a new research direction for LLM reasoning in optimization tasks, whereas Paper 2 addresses systems-level engineering improvements with narrower scope.
Paper 1 addresses a highly timely and impactful challenge: enhancing the reasoning and decision-making capabilities of LLMs in optimization tasks. Its framework for verifiable reward training has broad applicability across many domains requiring complex planning. In contrast, Paper 2 focuses on a classical search algorithm optimization for specific longest-path problems, which, while methodologically rigorous, has a narrower scope and less widespread real-world impact compared to advancements in foundational AI models.
While Paper 1 offers an elegant solution to AI text attribution, Paper 2 tackles a fundamental bottleneck in foundation models: step-by-step reasoning and planning in complex environments. By introducing a scalable framework (OPT*) for training LLMs on optimization tasks via search and RL, Paper 2 directly contributes to the frontier of agentic AI and System 2 thinking. Improving verifiable reasoning over expanding search spaces has massive implications across mathematics, coding, operations research, and autonomous decision-making, giving it a higher potential ceiling for broad scientific and technological impact.
Paper 1 resolves a major ongoing debate in LLM research regarding self-correction capabilities, demonstrating that failures are driven by chat-template artifacts rather than cognitive deficits. This is a highly novel, paradigm-shifting insight with immediate, broadly applicable real-world implications for prompt design and agentic workflows. While Paper 2 offers a valuable benchmark for optimization-like reasoning, Paper 1's fundamental mechanistic discovery about model behavior gives it broader cross-disciplinary impact and higher immediate relevance.
Paper 1 likely has higher impact due to broader relevance and timeliness: it proposes a scalable task family (OPT*) for training/evaluating step-by-step optimization-like reasoning in LLMs with minimal new labeling, aligning with current LLM/RL research priorities. Its framework (solver-guided online policy optimization vs offline search-based RL) and theoretical lens on information extracted per search budget could generalize across many planning and decision-making domains. Paper 2 is methodologically solid and impactful for constrained optimization (notably power flow), but is more domain- and architecture-specific with narrower cross-field reach.
Paper 2 has higher estimated impact due to clear novelty in bridging perception-to-planning via modality-gap-aware self-distillation, strong real-world relevance (robotics, embodied agents, UI automation), and convincing empirical gains with diagnostics across multiple model scales. Its approach is broadly applicable wherever symbolic supervision is available at training time but not at inference, making it timely for multimodal agent research. Paper 1 is methodologically interesting (scalable OPT* tasks, solver-guided RL) but is more specialized to synthetic optimization-like benchmarks and depends on oracle/solver availability, potentially limiting immediate cross-domain adoption.
Paper 2 introduces a novel framework (OPT*) that addresses a fundamental gap in LLM reasoning—optimization-style step-by-step decision making beyond math/coding. It offers broader impact through scalable task generation, theoretical grounding connecting search budgets to information extraction, and practical training methods (solver-guided RL and search-based offline RL). Paper 1, while addressing an important safety concern in memory-augmented agents, is primarily a measurement/evaluation study with narrower scope. Paper 2's contributions to training methodology and reasoning capabilities have wider applicability across fields.
Paper 2 addresses a fundamental and broadly applicable challenge—memory management for LLM agents across diverse tasks—with a novel biologically-inspired approach (reconstructive memory vs. retrieval). It demonstrates strong empirical gains (up to 23%) with practical efficiency benefits. Paper 1, while rigorous and interesting, targets a more specialized niche (optimization-style reasoning tasks for LLMs) with a narrower scope of applicability. Paper 2's graph-based associative memory framework has broader potential impact across many agent applications and connects to established cognitive science principles, giving it wider interdisciplinary appeal.
Paper 2 addresses a fundamental and highly timely challenge in AI: enhancing LLM step-by-step reasoning and planning in large search spaces using RL. Its theoretical framework and scalable task design offer broad foundational impact across any domain requiring complex decision-making. Paper 1, while practically valuable, focuses on the narrower domain of time series analysis, making its potential scientific impact more localized compared to the general algorithmic advancements proposed in Paper 2.