OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization

Recent large-scale language models (LLMs) with long Chain-of-Thought reasoning-such as DeepSeek-R1-have achieved impressive results on Olympiad-level mathematics benchmarks. However, they often rely on a narrow set of strategies and struggle with problems that require a novel way of thinking. To systematically investigate these limitations, we introduce OMEGA-Out-of-distribution Math Problems Evaluation with 3 Generalization Axes-a controlled yet diverse benchmark designed to evaluate three axes of out-of-distribution generalization, inspired by Boden's typology of creativity: (1) Exploratory-applying known problem solving skills to more complex instances within the same problem domain; (2) Compositional-combining distinct reasoning skills, previously learned in isolation, to solve novel problems that require integrating these skills in new and coherent ways; and (3) Transformative-adopting novel, often unconventional strategies by moving beyond familiar approaches to solve problems more effectively. OMEGA consists of programmatically generated training-test pairs derived from templated problem generators across geometry, number theory, algebra, combinatorics, logic, and puzzles, with solutions verified using symbolic, numerical, or graphical methods. We evaluate frontier (or top-tier) LLMs and observe sharp performance degradation as problem complexity increases. Moreover, we fine-tune the Qwen-series models across all generalization settings and observe notable improvements in exploratory generalization, while compositional generalization remains limited and transformative reasoning shows little to no improvement. By isolating and quantifying these fine-grained failures, OMEGA lays the groundwork for advancing LLMs toward genuine mathematical creativity beyond mechanical proficiency.

Infini-gram mini: Exact n-gram Search at the Internet Scale with FM-Index

Language models are trained mainly on massive text data from the Internet, and it becomes increasingly important to understand this data source. Exact-match search engines enable searching in large text corpora -- counting string appearances and retrieving the enclosing documents -- yet the high storage overhead hinders their application on Internet-scale data. We present Infini-gram mini, an efficient and scalable system that can make petabyte-level text corpora searchable. Based on the FM-index data structure (Ferragina and Manzini, 2000), which simultaneously indexes and compresses text, our system creates indexes with size only 44% of the corpus. Infini-gram mini greatly improves upon the best existing implementation of FM-index in terms of indexing speed (18$\times$) and memory use during both indexing (3.2$\times$ reduction) and querying (down to a negligible amount). We index 46TB of Internet text in 50 days with a single 128-core CPU node (or 19 hours if using 75 such nodes). We show one important use case of Infini-gram mini in a large-scale analysis of benchmark contamination. We find several core LM evaluation benchmarks to be heavily contaminated in Internet crawls (up to 40% in SQuAD), which could lead to overestimating the capabilities of language models if trained on such data. We host a benchmark contamination bulletin to share the contamination rate of many core and community-contributed benchmarks. We also release a web interface and an API endpoint to serve general search queries on Infini-gram mini indexes.

Spurious Rewards: Rethinking Training Signals in RLVR

We show that reinforcement learning with verifiable rewards (RLVR) can elicit strong mathematical reasoning in certain models even with spurious rewards that have little, no, or even negative correlation with the correct answer. For example, RLVR improves MATH-500 performance for Qwen2.5-Math-7B in absolute points by 21.4% (random reward), 13.8% (format reward), 24.1% (incorrect label), 26.0% (1-shot RL), and 27.1% (majority voting) -- nearly matching the 29.1% gained with ground truth rewards. However, the spurious rewards that work for Qwen often fail to yield gains with other model families like Llama3 or OLMo2. In particular, we find code reasoning -- thinking in code without actual code execution -- to be a distinctive Qwen2.5-Math behavior that becomes significantly more frequent after RLVR, from 65% to over 90%, even with spurious rewards. Overall, we hypothesize that, given the lack of useful reward signal, RLVR must somehow be surfacing useful reasoning representations learned during pretraining, although the exact mechanism remains a topic for future work. We suggest that future RLVR research should possibly be validated on diverse models rather than a single de facto choice, as we show that it is easy to get significant performance gains on Qwen models even with completely spurious reward signals.

ScienceMeter: Tracking Scientific Knowledge Updates in Language Models

Large Language Models (LLMs) are increasingly used to support scientific research, but their knowledge of scientific advancements can quickly become outdated. We introduce ScienceMeter, a new framework for evaluating scientific knowledge update methods over scientific knowledge spanning the past, present, and future. ScienceMeter defines three metrics: knowledge preservation, the extent to which models' understanding of previously learned papers are preserved; knowledge acquisition, how well scientific claims from newly introduced papers are acquired; and knowledge projection, the ability of the updated model to anticipate or generalize to related scientific claims that may emerge in the future. Using ScienceMeter, we examine the scientific knowledge of LLMs on claim judgment and generation tasks across a curated dataset of 15,444 scientific papers and 30,888 scientific claims from ten domains including medicine, biology, materials science, and computer science. We evaluate five representative knowledge update approaches including training- and inference-time methods. With extensive experiments, we find that the best-performing knowledge update methods can preserve only 85.9% of existing knowledge, acquire 71.7% of new knowledge, and project 37.7% of future knowledge. Inference-based methods work for larger models, whereas smaller models require training to achieve comparable performance. Cross-domain analysis reveals that performance on these objectives is correlated. Even when applying on specialized scientific LLMs, existing knowledge update methods fail to achieve these objectives collectively, underscoring that developing robust scientific knowledge update mechanisms is both crucial and challenging.

Critical Batch Size Revisited: A Simple Empirical Approach to Large-Batch Language Model Training

The right batch size is important when training language models at scale: a large batch size is necessary for fast training, but a batch size that is too large will harm token efficiency. To navigate this tradeoff, McCandlish et al. (2018) suggest that a critical batch size (CBS), below which training will not substantially degrade loss, can be estimated based on the gradient noise scale during training. While their method has been adopted in practice, e.g., when training GPT-3, strong assumptions are required to justify gradient noise as a proxy for the CBS, which makes it unclear whether their approach should be trusted in practice, limiting its applicability. In this paper, we introduce a simple, empirical approach to directly measure the CBS and show how the CBS evolves over training. Applying our approach to the OLMo models, we find that CBS is near 0 at initialization, increases rapidly at first, and then plateaus as training progresses. Furthermore, we find that this trend holds across different model sizes (1B and 7B), suggesting CBS from small training runs can inform larger-scale training runs. Our findings about how the CBS changes over training motivate batch size warmup as a natural way to reliably train language models at large batch size: start the batch size small and increase it as the CBS grows. To validate this claim, we use batch size warmup to train OLMo 1B to slightly better loss than the original training run with 43% fewer gradient steps. This shows how our framework can be applied to reliably train language models at larger batch sizes, increasing data parallelism without compromising performance.

ParaPO: Aligning Language Models to Reduce Verbatim Reproduction of Pre-training Data

Language models (LMs) can memorize and reproduce segments from their pretraining data verbatim even in non-adversarial settings, raising concerns about copyright, plagiarism, privacy, and creativity. We introduce Paraphrase Preference Optimization (ParaPO), a post-training method that fine-tunes LMs to reduce unintentional regurgitation while preserving their overall utility. ParaPO trains LMs to prefer paraphrased versions of memorized segments over the original verbatim content from the pretraining data. To maintain the ability to recall famous quotations when appropriate, we develop a variant of ParaPO that uses system prompts to control regurgitation behavior. In our evaluation on Llama3.1-8B, ParaPO consistently reduces regurgitation across all tested datasets (e.g., reducing the regurgitation metric from 17.3 to 12.9 in creative writing), whereas unlearning methods used in prior work to mitigate regurgitation are less effective outside their targeted unlearned domain (from 17.3 to 16.9). When applied to the instruction-tuned Tulu3-8B model, ParaPO with system prompting successfully preserves famous quotation recall while reducing unintentional regurgitation (from 8.7 to 6.3 in creative writing) when prompted not to regurgitate. In contrast, without ParaPO tuning, prompting the model not to regurgitate produces only a marginal reduction (8.7 to 8.4).

OLMoTrace: Tracing Language Model Outputs Back to Trillions of Training Tokens

We present OLMoTrace, the first system that traces the outputs of language models back to their full, multi-trillion-token training data in real time. OLMoTrace finds and shows verbatim matches between segments of language model output and documents in the training text corpora. Powered by an extended version of infini-gram (Liu et al., 2024), our system returns tracing results within a few seconds. OLMoTrace can help users understand the behavior of language models through the lens of their training data. We showcase how it can be used to explore fact checking, hallucination, and the creativity of language models. OLMoTrace is publicly available and fully open-source.

Steering off Course: Reliability Challenges in Steering Language Models

Steering methods for language models (LMs) have gained traction as lightweight alternatives to fine-tuning, enabling targeted modifications to model activations. However, prior studies primarily report results on a few models, leaving critical gaps in understanding the robustness of these methods. In this work, we systematically examine three prominent steering methods -- DoLa, function vectors, and task vectors. In contrast to the original studies, which evaluated a handful of models, we test up to 36 models belonging to 14 families with sizes ranging from 1.5B to 70B parameters. Our experiments reveal substantial variability in the effectiveness of the steering approaches, with a large number of models showing no improvement and at times degradation in steering performance. Our analysis demonstrate fundamental flaws in the assumptions underlying these methods, challenging their reliability as scalable steering solutions.

EvalTree: Profiling Language Model Weaknesses via Hierarchical Capability Trees

An ideal model evaluation should achieve two goals: identifying where the model fails and providing actionable improvement guidance. Toward these goals for Language Model (LM) evaluations, we formulate the problem of generating a weakness profile, a set of weaknesses expressed in natural language, given an LM's performance on every individual instance in a benchmark. We introduce a suite of quantitative assessments to compare different weakness profiling methods. We also propose a weakness profiling method EvalTree. It constructs a capability tree where each node represents a capability described in natural language and is linked to a subset of benchmark instances that specifically evaluate this capability; it then extracts nodes where the LM performs poorly to generate a weakness profile. On the MATH and WildChat benchmarks, we show that EvalTree outperforms baseline weakness profiling methods by identifying weaknesses more precisely and comprehensively. Weakness profiling further enables weakness-guided data collection, and training data collection guided by EvalTree-identified weaknesses improves LM performance more than other data collection strategies. We also show how EvalTree exposes flaws in Chatbot Arena's human-voter-based evaluation practice. To facilitate future work, we release our code and an interface that allows practitioners to interactively explore the capability trees built by EvalTree.

s1: Simple test-time scaling

Test-time scaling is a promising new approach to language modeling that uses extra test-time compute to improve performance. Recently, OpenAI's o1 model showed this capability but did not publicly share its methodology, leading to many replication efforts. We seek the simplest approach to achieve test-time scaling and strong reasoning performance. First, we curate a small dataset s1K of 1,000 questions paired with reasoning traces relying on three criteria we validate through ablations: difficulty, diversity, and quality. Second, we develop budget forcing to control test-time compute by forcefully terminating the model's thinking process or lengthening it by appending "Wait" multiple times to the model's generation when it tries to end. This can lead the model to double-check its answer, often fixing incorrect reasoning steps. After supervised finetuning the Qwen2.5-32B-Instruct language model on s1K and equipping it with budget forcing, our model s1 exceeds o1-preview on competition math questions by up to 27% (MATH and AIME24). Further, scaling s1 with budget forcing allows extrapolating beyond its performance without test-time intervention: from 50% to 57% on AIME24. Our model, data, and code are open-source at https://github.com/simplescaling/s1.