LLM-based judges have emerged as a scalable alternative to human evaluation and are increasingly used to assess, compare, and improve models. However, the reliability of LLM-based judges themselves is rarely scrutinized. As LLMs become more advanced, their responses grow more sophisticated, requiring stronger judges to evaluate them. Existing benchmarks primarily focus on a judge's alignment with human preferences, but often fail to account for more challenging tasks where crowdsourced human preference is a poor indicator of factual and logical correctness. To address this, we propose a novel evaluation framework to objectively evaluate LLM-based judges. Based on this framework, we propose JudgeBench, a benchmark for evaluating LLM-based judges on challenging response pairs spanning knowledge, reasoning, math, and coding. JudgeBench leverages a novel pipeline for converting existing difficult datasets into challenging response pairs with preference labels reflecting objective correctness. Our comprehensive evaluation on a collection of prompted judges, fine-tuned judges, multi-agent judges, and reward models shows that JudgeBench poses a significantly greater challenge than previous benchmarks, with many strong models (e.g., GPT-4o) performing just slightly better than random guessing. Overall, JudgeBench offers a reliable platform for assessing increasingly advanced LLM-based judges. Data and code are available at https://github.com/ScalerLab/JudgeBench .
Language is not monolithic. While benchmarks, including those designed for multiple languages, are often used as proxies to evaluate the performance of Large Language Models (LLMs), they tend to overlook the nuances of within-language variation, and thus fail to model the experience of speakers of non-standard dialects. Focusing on African American Vernacular English (AAVE), we present the first study aimed at objectively assessing the fairness and robustness of LLMs in handling dialects in canonical reasoning tasks, including algorithm, math, logic, and integrated reasoning. We introduce \textbf{ReDial} (\textbf{Re}asoning with \textbf{Dial}ect Queries), a benchmark containing 1.2K+ parallel query pairs in Standardized English and AAVE. We hire AAVE speakers, including experts with computer science backgrounds, to rewrite seven popular benchmarks, such as HumanEval and GSM8K. With ReDial, we evaluate widely used LLMs, including GPT, Claude, Llama, Mistral, and the Phi model families. Our findings reveal that \textbf{almost all of these widely used models show significant brittleness and unfairness to queries in AAVE}. Our work establishes a systematic and objective framework for analyzing LLM bias in dialectal queries. Moreover, it highlights how mainstream LLMs provide unfair service to dialect speakers in reasoning tasks, laying a critical foundation for relevant future research. Code and data can be accessed at https://github.com/fangru-lin/redial_dialect_robustness_fairness.
Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.
Mathematical reasoning remains a significant challenge for large language models (LLMs), despite progress in prompting techniques such as Chain-of-Thought (CoT). We present Chain of Mathematically Annotated Thought (CoMAT), which enhances reasoning through two stages: Symbolic Conversion (converting natural language queries into symbolic form) and Reasoning Execution (deriving answers from symbolic representations). CoMAT operates entirely with a single LLM and without external solvers. Across four LLMs, CoMAT outperforms traditional CoT on six out of seven benchmarks, achieving gains of 4.48% on MMLU-Redux (MATH) and 4.58% on GaoKao MCQ. In addition to improved performance, CoMAT ensures faithfulness and verifiability, offering a transparent reasoning process for complex mathematical tasks
Large language models (LLMs) like GPT-4, PaLM, and LLaMA have shown significant improvements in various reasoning tasks. However, smaller models such as Llama-3-8B and DeepSeekMath-Base still struggle with complex mathematical reasoning because they fail to effectively identify and correct reasoning errors. Recent reflection-based methods aim to address these issues by enabling self-reflection and self-correction, but they still face challenges in independently detecting errors in their reasoning steps. To overcome these limitations, we propose SuperCorrect, a novel two-stage framework that uses a large teacher model to supervise and correct both the reasoning and reflection processes of a smaller student model. In the first stage, we extract hierarchical high-level and detailed thought templates from the teacher model to guide the student model in eliciting more fine-grained reasoning thoughts. In the second stage, we introduce cross-model collaborative direct preference optimization (DPO) to enhance the self-correction abilities of the student model by following the teacher's correction traces during training. This cross-model DPO approach teaches the student model to effectively locate and resolve erroneous thoughts with error-driven insights from the teacher model, breaking the bottleneck of its thoughts and acquiring new skills and knowledge to tackle challenging problems. Extensive experiments consistently demonstrate our superiority over previous methods. Notably, our SuperCorrect-7B model significantly surpasses powerful DeepSeekMath-7B by 7.8%/5.3% and Qwen2.5-Math-7B by 15.1%/6.3% on MATH/GSM8K benchmarks, achieving new SOTA performance among all 7B models. Code: https://github.com/YangLing0818/SuperCorrect-llm
Recent advancements in large language models (LLMs) have led to significant breakthroughs in mathematical reasoning capabilities. However, existing benchmarks like GSM8K or MATH are now being solved with high accuracy (e.g., OpenAI o1 achieves 94.8\% on MATH dataset), indicating their inadequacy for truly challenging these models. To bridge this gap, we propose a comprehensive and challenging benchmark specifically designed to assess LLMs' mathematical reasoning at the Olympiad level. Unlike existing Olympiad-related benchmarks, our dataset focuses exclusively on mathematics and comprises a vast collection of 4428 competition-level problems with rigorous human annotation. These problems are meticulously categorized into over 33 sub-domains and span more than 10 distinct difficulty levels, enabling a holistic assessment of model performance in Olympiad-mathematical reasoning. Furthermore, we conducted an in-depth analysis based on this benchmark. Our experimental results show that even the most advanced models, OpenAI o1-mini and OpenAI o1-preview, struggle with highly challenging Olympiad-level problems, with 60.54\% and 52.55\% accuracy, highlighting significant challenges in Olympiad-level mathematical reasoning.
Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .
Large Language Models (LLMs) exhibit impressive performance across various domains but still struggle with arithmetic reasoning tasks. Recent work shows the effectiveness of prompt design methods in enhancing reasoning capabilities. However, these approaches overlook crucial requirements for prior knowledge of specific concepts, theorems, and tricks to tackle most arithmetic reasoning problems successfully. To address this issue, we propose a novel and effective Teaching-Inspired Integrated Framework, which emulates the instructional process of a teacher guiding students. This method equips LLMs with essential concepts, relevant theorems, and similar problems with analogous solution approaches, facilitating the enhancement of reasoning abilities. Additionally, we introduce two new Chinese datasets, MathMC and MathToF, both with detailed explanations and answers. Experiments are conducted on nine benchmarks which demonstrates that our approach improves the reasoning accuracy of LLMs. With GPT-4 and our framework, we achieve new state-of-the-art performance on four math benchmarks (AddSub, SVAMP, Math23K and AQuA) with accuracies of 98.2% (+3.3%), 93.9% (+0.2%), 94.3% (+7.2%) and 81.1% (+1.2%). Our data and code are available at https://github.com/SallyTan13/Teaching-Inspired-Prompting.
The process of creating training data to teach models is currently driven by humans, who manually analyze model weaknesses and plan how to create data that improves a student model. Approaches using LLMs as annotators reduce human effort, but still require humans to interpret feedback from evaluations and control the LLM to produce data the student needs. Automating this labor-intensive process by creating autonomous data generation agents - or teachers - is desirable, but requires environments that can simulate the feedback-driven, iterative, closed loop of data creation. To enable rapid, scalable testing for such agents and their modules, we introduce DataEnvGym, a testbed of teacher environments for data generation agents. DataEnvGym frames data generation as a sequential decision-making task, involving an agent consisting of a data generation policy (which generates a plan for creating training data) and a data generation engine (which transforms the plan into data), inside an environment that provides student feedback. The agent's goal is to improve student performance. Students are iteratively trained and evaluated on generated data, and their feedback (in the form of errors or weak skills) is reported to the agent after each iteration. DataEnvGym includes multiple teacher environment instantiations across 3 levels of structure in the state representation and action space. More structured environments are based on inferred skills and offer more interpretability and curriculum control. We support 4 domains (math, code, VQA, and tool-use) and test multiple students and teachers. Example agents in our teaching environments can iteratively improve students across tasks and settings. Moreover, we show that environments teach different skill levels and test variants of key modules, pointing to future work in improving data generation agents, engines, and feedback mechanisms.
While state-of-the-art LLMs have shown poor logical and basic mathematical reasoning, recent works try to improve their problem-solving abilities using prompting techniques. We propose giving "hints" to improve the language model's performance on advanced mathematical problems, taking inspiration from how humans approach math pedagogically. We also test robustness to adversarial hints and demonstrate their sensitivity to them. We demonstrate the effectiveness of our approach by evaluating various diverse LLMs, presenting them with a broad set of problems of different difficulties and topics from the MATH dataset and comparing against techniques such as one-shot, few-shot, and chain of thought prompting.
One-to-one tutoring is one of the most efficient methods of teaching. With the growing popularity of Large Language Models (LLMs), there have been efforts to create LLM based conversational tutors which can expand the benefits of one to one tutoring to everyone. However, current LLMs are trained primarily to be helpful assistants and lack crucial pedagogical skills. For example, they often quickly reveal the solution to the student and fail to plan for a richer multi turn pedagogical interaction. To use LLMs in pedagogical settings, they need to be steered to use effective teaching strategies: a problem we introduce as Pedagogical Steering. We develop StratL, an algorithm to optimize LLM prompts and steer it to follow a predefined multi-turn tutoring plan represented as a transition graph. As a case study, we create a prototype tutor for high school math following Productive Failure (PF), an advanced and effective learning design. To validate our approach in a real-world setting, we run a field study with 17 high school students in Singapore and show that StratL succeeds in steering the LLM to follow the PF tutoring strategy. Finally, we highlight challenges in Pedagogical Steering of LLMs and offer opportunities for further improvements by publishing a dataset of PF problems and our code.
As a manner to augment pre-trained large language models (LLM), knowledge injection is critical to develop vertical domain large models and has been widely studied. Although most current approaches, including parameter-efficient fine-tuning (PEFT) and block expansion methods, uniformly apply knowledge across all LLM layers, it raises the question: are all layers equally crucial for knowledge injection? We begin by evaluating the importance of each layer in finding the optimal layer range for knowledge injection. Intuitively, the more important layers should play a more critical role in knowledge injection and deserve a denser injection. We observe performance dips in question-answering benchmarks after the removal or expansion of the shallow layers, and the degradation shrinks as the layer gets deeper, indicating that the shallow layers hold the key to knowledge injection. This insight leads us to propose the S strategy, a post-pretraining strategy of selectively enhancing shallow layers while pruning the less effective deep ones. Based on this strategy, we introduce Llama Slayer-8B and Llama Slayer-8B-Instruct. We experimented on the corpus of code $\&$ math and demonstrated the effectiveness of our strategy. Further experiments across different LLM, Mistral-7B, and a legal corpus confirmed the general applicability of the approach, underscoring its wide-ranging efficacy. Our code is available at: \https://github.com/txchen-USTC/Llama-Slayer
Benchmarks are critical for measuring progress of math reasoning abilities of Large Language Models (LLMs). However, existing widely-used benchmarks such as GSM8K have been rendered less useful as multiple cutting-edge LLMs achieve over 94% accuracy. While harder benchmarks have been proposed, their creation is often manual and expensive. We present Scheherazade, an automated approach for producing challenging mathematical reasoning benchmarks by logically chaining mathematical reasoning problems. We propose two different chaining methods, forward chaining and backward chaining, which require reasoning forward and backward through the chain respectively. We apply Scheherazade on GSM8K to create GSM8K-Scheherazade and evaluate 3 frontier LLMs and OpenAI's o1-preview on it. We show that while frontier models' performance declines precipitously at only a few questions chained, a preliminary evaluation suggests o1-preview performance persists up to 5 questions chained backwards. In addition, while all other models perform worse when problems are chained backwards, o1-preview performs better on backward-chained benchmarks. We will release the dataset and code publicly.
There is a growing need for pluralistic alignment methods that can steer language models towards individual attributes and preferences. One such method, Self-Supervised Alignment with Mutual Information (SAMI), uses conditional mutual information to encourage the connection between behavioral preferences and model responses. We conduct two experiments exploring SAMI in multi-task settings. First, we compare SAMI to Direct Preference Optimization (DPO) on a multi-task benchmark (MT-Bench), using a stronger model to generate training data for a weaker one across diverse categories (humanities, STEM, extraction, coding, math, reasoning, and roleplay). Our results indicate that one iteration of SAMI has a 57% win rate against DPO, with significant variation in performance between task categories. Second, we examine SAMI's impact on mathematical accuracy (GSM-8K) relative to supervised fine-tuning (SFT). While SAMI increases zero-shot performance by 1.1%, SFT is more effective with a 3.2% boost. However, SAMI shows interesting scaling trends. When given 10 attempts, SAMI improves accuracy by 3.9%, while SFT achieves a 10.1% increase. Combining SAMI with SFT yields an additional improvement of 1.3% in multi-attempt settings, though single-attempt accuracy remains unchanged.
Reward Models (RMs) play a crucial role in aligning LLMs with human preferences, enhancing their performance by ranking outputs during inference or iterative training. However, the degree to which an RM generalizes to new tasks is often not known a priori (e.g. some RMs may excel at scoring creative writing vs. math reasoning). Therefore, using only one fixed RM while training LLMs can be suboptimal. Moreover, optimizing LLMs with multiple RMs simultaneously can be prohibitively computationally-intensive and challenging due to conflicting signals from different RMs, potentially degrading performance. To address these challenges, we introduce LASeR (Learning to Adaptively Select Rewards), which iteratively trains LLMs using multiple RMs, selecting and utilizing the most well-suited RM for each instance to rank outputs and generate preference data, framed as a multi-armed bandit problem. Our results on commonsense and math reasoning tasks demonstrate that LASeR can boost iterative LLM optimization by optimizing for multiple RMs, improving the absolute average accuracy of Llama-3-8B over three datasets by 2.67% over training with ensemble RM scores while also showing superior training efficiency (e.g., a 2x speedup). Moreover, on WildChat, a benchmark of instruction-following prompts, we find that using Llama-3-8B LASeR leads to a 71.45% AlpacaEval win rate over sequentially optimizing multiple RMs. Extending to long-context generation tasks, we find that on Llama-3-8B, LASeR achieves an average improvement of 2.64 F1 and 2.42 F1 on single- and multi-document QA over random RM selection when used with best-of-n sampling. LASeR is robust to noisy rewards and generalizes to multiple settings. Finally, LASeR's RM selection changes depending on the underlying task or instance and we verify the presence of conflicting preferences from multiple RMs that can be mitigated using LASeR.
Knowledge tracing (KT) is a popular approach for modeling students' learning progress over time, which can enable more personalized and adaptive learning. However, existing KT approaches face two major limitations: (1) they rely heavily on expert-defined knowledge concepts (KCs) in questions, which is time-consuming and prone to errors; and (2) KT methods tend to overlook the semantics of both questions and the given KCs. In this work, we address these challenges and present KCQRL, a framework for automated knowledge concept annotation and question representation learning that can improve the effectiveness of any existing KT model. First, we propose an automated KC annotation process using large language models (LLMs), which generates question solutions and then annotates KCs in each solution step of the questions. Second, we introduce a contrastive learning approach to generate semantically rich embeddings for questions and solution steps, aligning them with their associated KCs via a tailored false negative elimination approach. These embeddings can be readily integrated into existing KT models, replacing their randomly initialized embeddings. We demonstrate the effectiveness of KCQRL across 15 KT algorithms on two large real-world Math learning datasets, where we achieve consistent performance improvements.
Large language models (LLMs) are increasingly applied to complex reasoning tasks that require executing several complex steps before receiving any reward. Properly assigning credit to these steps is essential for enhancing model performance. Proximal Policy Optimization (PPO), a state-of-the-art reinforcement learning (RL) algorithm used for LLM finetuning, employs value networks to tackle credit assignment. However, value networks face challenges in predicting the expected cumulative rewards accurately in complex reasoning tasks, often leading to high-variance updates and suboptimal performance. In this work, we systematically evaluate the efficacy of value networks and reveal their significant shortcomings in reasoning-heavy LLM tasks, showing that they barely outperform a random baseline when comparing alternative steps. To address this, we propose VinePPO, a straightforward approach that leverages the flexibility of language environments to compute unbiased Monte Carlo-based estimates, bypassing the need for large value networks. Our method consistently outperforms PPO and other RL-free baselines across MATH and GSM8K datasets with fewer gradient updates (up to 9x), less wall-clock time (up to 3.0x). These results emphasize the importance of accurate credit assignment in RL finetuning of LLM and demonstrate VinePPO's potential as a superior alternative.
Research into the external behaviors and internal mechanisms of large language models (LLMs) has shown promise in addressing complex tasks in the physical world. Studies suggest that powerful LLMs, like GPT-4, are beginning to exhibit human-like cognitive abilities, including planning, reasoning, and reflection. In this paper, we introduce a research line and methodology called LLM Psychology, leveraging human psychology experiments to investigate the cognitive behaviors and mechanisms of LLMs. We migrate the Typoglycemia phenomenon from psychology to explore the "mind" of LLMs. Unlike human brains, which rely on context and word patterns to comprehend scrambled text, LLMs use distinct encoding and decoding processes. Through Typoglycemia experiments at the character, word, and sentence levels, we observe: (I) LLMs demonstrate human-like behaviors on a macro scale, such as lower task accuracy and higher token/time consumption; (II) LLMs exhibit varying robustness to scrambled input, making Typoglycemia a benchmark for model evaluation without new datasets; (III) Different task types have varying impacts, with complex logical tasks (e.g., math) being more challenging in scrambled form; (IV) Each LLM has a unique and consistent "cognitive pattern" across tasks, revealing general mechanisms in its psychology process. We provide an in-depth analysis of hidden layers to explain these phenomena, paving the way for future research in LLM Psychology and deeper interpretability.
Large Language Models (LLMs) have exhibited exceptional performance across a broad range of tasks and domains. However, they still encounter difficulties in solving mathematical problems due to the rigorous and logical nature of mathematics. Previous studies have employed techniques such as supervised fine-tuning (SFT), prompt engineering, and search-based methods to improve the mathematical problem-solving abilities of LLMs. Despite these efforts, their performance remains suboptimal and demands substantial computational resources. To address this issue, we propose a novel approach, BEATS, to enhance mathematical problem-solving abilities. Our method leverages newly designed prompts that guide the model to iteratively rewrite, advance by one step, and generate answers based on previous steps. Additionally, we introduce a new back-verification technique that uses LLMs to validate the correctness of the generated answers. Furthermore, we employ a pruning tree search to optimize search time while achieving strong performance. Notably, our method improves Qwen2-7b-Instruct's score from 36.94 to 61.52, outperforming GPT4's 42.5 on the MATH benchmark.
Parameter-Efficient Fine-Tuning (PEFT) methods, particularly Low-Rank Adaptation (LoRA), effectively reduce the number of trainable parameters in Large Language Models (LLMs). However, as model scales continue to grow, the demand for computational resources remains a significant challenge. Existing LoRA variants often struggle to strike an optimal balance between adaptability (model performance and convergence speed) and efficiency (computational overhead, memory usage, and initialization time). This paper introduces FOSSIL(\textbf{F}ramework for \textbf{O}ptimal \textbf{S}hard \textbf{S}haring \textbf{I}ntegration in \textbf{L}oRA), a novel PEFT approach that addresses this trade-off through a simple shard-sharing mechanism. FOSSIL leverages the insight that a low-rank adaptation can be achieved by decomposing the weight matrix into multiple fragment matrices and utilizing a shared, trainable common fragment. This method constructs the low-rank update matrix through the replication of these shared, partitioned shards. We also propose a hardware-efficient and broadly applicable implementation for FOSSIL. Extensive experiments conducted on a range of tasks, alongside a systematic analysis of computational performance, demonstrate FOSSIL's superiority. The results show that FOSSIL significantly outperforms standard LoRA and its prominent variants in both model performance metrics and computational efficiency, including initialization speed and training throughput. By effectively balancing expressive power and resource utilization, FOSSIL offers a compelling solution for efficiently adapting large-scale models.