On the Learning Dynamics of RLVR at the Edge of Competence

Reinforcement learning with verifiable rewards (RLVR) has been a main driver of recent breakthroughs in large reasoning models. Yet it remains a mystery how rewards based solely on final outcomes can help overcome the long-horizon barrier to extended reasoning. To understand this, we develop a theory of the training dynamics of RL for transformers on compositional reasoning tasks. Our theory characterizes how the effectiveness of RLVR is governed by the smoothness of the difficulty spectrum. When data contains abrupt discontinuities in difficulty, learning undergoes grokking-type phase transitions, producing prolonged plateaus before progress recurs. In contrast, a smooth difficulty spectrum leads to a relay effect: persistent gradient signals on easier problems elevate the model's capabilities to the point where harder ones become tractable, resulting in steady and continuous improvement. Our theory explains how RLVR can improve performance at the edge of competence, and suggests that appropriately designed data mixtures can yield scalable gains. As a technical contribution, our analysis develops and adapts tools from Fourier analysis on finite groups to our setting. We validate the predicted mechanisms empirically via synthetic experiments.

Constraint-Rectified Training for Efficient Chain-of-Thought

Chain-of-Thought (CoT) has significantly enhanced the reasoning capabilities of Large Language Models (LLMs), especially when combined with reinforcement learning (RL) based post-training methods. While longer reasoning traces can improve answer quality and unlock abilities such as self-correction, they also incur high inference costs and often introduce redundant steps, known as overthinking. Recent research seeks to develop efficient reasoning strategies that balance reasoning length and accuracy, either through length-aware reward design or prompt-based calibration. However, these heuristic-based approaches may suffer from severe accuracy drop and be very sensitive to hyperparameters. To address these problems, we introduce CRT (Constraint-Rectified Training), a principled post-training framework based on reference-guarded constrained optimization, yielding a more stable and interpretable formulation for efficient reasoning. CRT alternates between minimizing reasoning length and rectifying accuracy only when performance falls below the reference, enabling stable and effective pruning of redundant reasoning. We further extend CRT with a two-stage training scheme that first discovers the shortest reliable reasoning patterns and then refines accuracy under a learnt length budget, preventing the re-emergence of verbose CoT. Our comprehensive evaluation shows that this framework consistently reduces token usage while maintaining answer quality at a robust and reliable level. Further analysis reveals that CRT improves reasoning efficiency not only by shortening responses but also by reducing internal language redundancy, leading to a new evaluation metric. Moreover, CRT-based training naturally yields a sequence of intermediate checkpoints that span a spectrum of explanation lengths while preserving correctness, enabling fine-grained control over reasoning verbosity without retraining.

Reward Modeling for Reinforcement Learning-Based LLM Reasoning: Design, Challenges, and Evaluation

Large Language Models (LLMs) demonstrate transformative potential, yet their reasoning remains inconsistent and unreliable. Reinforcement learning (RL)-based fine-tuning is a key mechanism for improvement, but its effectiveness is fundamentally governed by reward design. Despite its importance, the relationship between reward modeling and core LLM challenges--such as evaluation bias, hallucination, distribution shift, and efficient learning--remains poorly understood. This work argues that reward modeling is not merely an implementation detail but a central architect of reasoning alignment, shaping what models learn, how they generalize, and whether their outputs can be trusted. We introduce Reasoning-Aligned Reinforcement Learning (RARL), a unifying framework that systematizes diverse reward paradigms for multi-step reasoning. Within this framework, we present a taxonomy of reward mechanisms, analyze reward hacking as a pervasive failure mode, and examine how reward signals unify challenges ranging from inference-time scaling to hallucination mitigation. We further critically evaluate existing benchmarks, highlighting vulnerabilities such as data contamination and reward misalignment, and outline directions for more robust evaluation. By integrating fragmented research threads and clarifying the interplay between reward design and fundamental reasoning capabilities, this work provides a foundational roadmap for building reasoning models that are robust, verifiable, and trustworthy.

Learnable Chernoff Baselines for Inference-Time Alignment

We study inference-time reward-guided alignment for generative models. Existing methods often rely on either architecture-specific adaptations or computationally costly inference procedures. We introduce Learnable Chernoff Baselines (LCBs) as a method for efficiently and approximately sampling from the exponentially tilted kernels that arise from KL-regularized reward alignment. Using only black-box sampling access to the pretrained model, LCBs implement a form of rejection sampling with adaptively selected acceptance probabilities, which allows fine-grained control over inference-compute scaling. We establish total-variation guarantees to the ideal aligned model, and demonstrate in both continuous and discrete diffusion settings that LCB sampling closely matches ideal rejection sampling while using substantially fewer queries to the pretrained model.

Bridging Online and Offline RL: Contextual Bandit Learning for Multi-Turn Code Generation

Recently, there have been significant research interests in training large language models (LLMs) with reinforcement learning (RL) on real-world tasks, such as multi-turn code generation. While online RL tends to perform better than offline RL, its higher training cost and instability hinders wide adoption. In this paper, we build on the observation that multi-turn code generation can be formulated as a one-step recoverable Markov decision process and propose contextual bandit learning with offline trajectories (Cobalt), a new method that combines the benefits of online and offline RL. Cobalt first collects code generation trajectories using a reference LLM and divides them into partial trajectories as contextual prompts. Then, during online bandit learning, the LLM is trained to complete each partial trajectory prompt through single-step code generation. Cobalt outperforms two multi-turn online RL baselines based on GRPO and VeRPO, and substantially improves R1-Distill 8B and Qwen3 8B by up to 9.0 and 6.2 absolute Pass@1 scores on LiveCodeBench. Also, we analyze LLMs' in-context reward hacking behaviors and augment Cobalt training with perturbed trajectories to mitigate this issue. Overall, our results demonstrate Cobalt as a promising solution for iterative decision-making tasks like multi-turn code generation. Our code and data are available at https://github.com/OSU-NLP-Group/cobalt.

ConvexBench: Can LLMs Recognize Convex Functions?

Convex analysis is a modern branch of mathematics with many applications. As Large Language Models (LLMs) start to automate research-level math and sciences, it is important for LLMs to demonstrate the ability to understand and reason with convexity. We introduce \cb, a scalable and mechanically verifiable benchmark for testing \textit{whether LLMs can identify the convexity of a symbolic objective under deep functional composition.} Experiments on frontier LLMs reveal a sharp compositional reasoning gap: performance degrades rapidly with increasing depth, dropping from an F1-score of $1.0$ at depth $2$ to approximately $0.2$ at depth $100$. Inspection of models' reasoning traces indicates two failure modes: \textit{parsing failure} and \textit{lazy reasoning}. To address these limitations, we propose an agentic divide-and-conquer framework that (i) offloads parsing to an external tool to construct an abstract syntax tree (AST) and (ii) enforces recursive reasoning over each intermediate sub-expression with focused context. This framework reliably mitigates deep-composition failures, achieving substantial performance improvement at large depths (e.g., F1-Score $= 1.0$ at depth $100$).

Mixture-of-Transformers Learn Faster: A Theoretical Study on Classification Problems

Mixture-of-Experts (MoE) models improve transformer efficiency but lack a unified theoretical explanation, especially when both feed-forward and attention layers are allowed to specialize. To this end, we study the Mixture-of-Transformers (MoT), a tractable theoretical framework in which each transformer block acts as an expert governed by a continuously trained gating network. This design allows us to isolate and study the core learning dynamics of expert specialization and attention alignment. In particular, we develop a three-stage training algorithm with continuous training of the gating network, and show that each transformer expert specializes in a distinct class of tasks and that the gating network accurately routes data samples to the correct expert. Our analysis shows how expert specialization reduces gradient conflicts and makes each subtask strongly convex. We prove that the training drives the expected prediction loss to near zero in $O(\log(\epsilon^{-1}))$ iteration steps, significantly improving over the $O(\epsilon^{-1})$ rate for a single transformer. We further validate our theoretical findings through extensive real-data experiments, demonstrating the practical effectiveness of MoT. Together, these results offer the first unified theoretical account of transformer-level specialization and learning dynamics, providing practical guidance for designing efficient large-scale models.

Monitoring State Transitions in Markovian Systems with Sampling Cost

We consider a node-monitor pair, where the node's state varies with time. The monitor needs to track the node's state at all times; however, there is a fixed cost for each state query. So the monitor may instead predict the state using time-series forecasting methods, including time-series foundation models (TSFMs), and query only when prediction uncertainty is high. Since query decisions influence prediction accuracy, determining when to query is nontrivial. A natural approach is a greedy policy that predicts when the expected prediction loss is below the query cost and queries otherwise. We analyze this policy in a Markovian setting, where the optimal (OPT) strategy is a state-dependent threshold policy minimizing the time-averaged sum of query cost and prediction losses. We show that, in general, the greedy policy is suboptimal and can have an unbounded competitive ratio, but under common conditions such as identically distributed transition probabilities, it performs close to OPT. For the case of unknown transition probabilities, we further propose a projected stochastic gradient descent (PSGD)-based learning variant of the greedy policy, which achieves a favorable predict-query tradeoff with improved computational efficiency compared to OPT.

Large Language Models Achieve Gold Medal Performance at International Astronomy & Astrophysics Olympiad

While task-specific demonstrations show early success in applying large language models (LLMs) to automate some astronomical research tasks, they only provide incomplete views of all necessary capabilities in solving astronomy problems, calling for more thorough understanding of LLMs' strengths and limitations. So far, existing benchmarks and evaluations focus on simple question-answering that primarily tests astronomical knowledge and fails to evaluate the complex reasoning required for real-world research in the discipline. Here, we address this gap by systematically benchmarking five state-of-the-art LLMs on the International Olympiad on Astronomy and Astrophysics (IOAA) exams, which are designed to examine deep conceptual understanding, multi-step derivations, and multimodal analysis. With average scores of 85.6% and 84.2%, Gemini 2.5 Pro and GPT-5 (the two top-performing models) not only achieve gold medal level performance but also rank in the top two among ~200-300 participants in all four IOAA theory exams evaluated (2022-2025). In comparison, results on the data analysis exams show more divergence. GPT-5 still excels in the exams with an 88.5% average score, ranking top 10 among the participants in the four most recent IOAAs, while other models' performances drop to 48-76%. Furthermore, our in-depth error analysis underscores conceptual reasoning, geometric reasoning, and spatial visualization (52-79% accuracy) as consistent weaknesses among all LLMs. Hence, although LLMs approach peak human performance in theory exams, critical gaps must be addressed before they can serve as autonomous research agents in astronomy.

Multi-head Transformers Provably Learn Symbolic Multi-step Reasoning via Gradient Descent

Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.