Large Language Models (LLMs) have demonstrated impressive performance across various tasks. However, current training approaches combine standard cross-entropy loss with extensive data, human feedback, or ad hoc methods to enhance performance. These solutions are often not scalable or feasible due to their associated costs, complexity, or resource requirements. This study investigates the use of established semantic segmentation loss functions in natural language generation to create a versatile, practical, and scalable solution for fine-tuning different architectures. We evaluate their effectiveness in solving Math Word Problems and question answering across different models of varying sizes. For the analyzed tasks, we found that the traditional Cross-Entropy loss represents a sub-optimal choice, while models trained to minimize alternative (task-dependent) losses, such as Focal or Lov\'asz, achieve a mean improvement of +42% on exact match without requiring additional data or human feedback. These findings suggest a promising pathway for more efficient and accessible training processes.
Inference-time techniques are emerging as highly effective tools to enhance large language model (LLM) capabilities. However, best practices for developing systems that combine these techniques remain underdeveloped due to our limited understanding of the utility of individual inference-time techniques and the interactions between them. Additionally, efficiently and automatically searching the space of model choices, inference-time techniques, and their compositions is challenging due to the large design space. To address these challenges, we introduce Archon, a modular framework for selecting, combining, and stacking layers of inference-time techniques to construct optimized LLM systems for target benchmarks. Rather than relying on a single LLM called once, we leverage a diverse set of LLMs and inference-time techniques, creating LLM systems greater than the sum of their parts. Archon defines an extensible design space, encompassing techniques such as generation ensembling, repeated sampling, ranking, fusion, critiquing, verification, and unit testing. It transforms the problem of building LLM systems into a hyperparameter optimization objective. Given the available LLMs, inference-time techniques, and compute budget, Archon utilizes hyperparameter search techniques to discover optimized architectures for target benchmark(s). We evaluate Archon architectures across a range of instruction-following, reasoning, and coding benchmarks, including MT-Bench, Arena-Hard-Auto, AlpacaEval 2.0, MixEval, MixEval Hard, MATH, and CodeContests. Archon architectures outperform frontier models, such as GPT-4o and Claude 3.5 Sonnet, on these benchmarks, achieving an average accuracy increase of 15.1 percentage points by using all available LLMs. We make our code and datasets available publicly on Github: https://github.com/ScalingIntelligence/Archon.
Large Language Models (LLMs) have emerged as powerful tools for Text-to-SQL tasks, exhibiting remarkable reasoning capabilities. Different from tasks such as math word problems and commonsense reasoning, SQL solutions have a relatively fixed pattern. This facilitates the investigation of whether LLMs can benefit from categorical thinking, mirroring how humans acquire knowledge through inductive reasoning based on comparable examples. In this study, we propose that employing query group partitioning allows LLMs to focus on learning the thought processes specific to a single problem type, consequently enhancing their reasoning abilities across diverse difficulty levels and problem categories. Our experiments reveal that multiple advanced LLMs, when equipped with PTD-SQL, can either surpass or match previous state-of-the-art (SOTA) methods on the Spider and BIRD datasets. Intriguingly, models with varying initial performances have exhibited significant improvements, mainly at the boundary of their capabilities after targeted drilling, suggesting a parallel with human progress. Code is available at https://github.com/lrlbbzl/PTD-SQL.
In this report, we introduce the Qwen2.5-Coder series, a significant upgrade from its predecessor, CodeQwen1.5. This series includes six models: Qwen2.5-Coder-(0.5B/1.5B/3B/7B/14B/32B). As a code-specific model, Qwen2.5-Coder is built upon the Qwen2.5 architecture and continues pretrained on a vast corpus of over 5.5 trillion tokens. Through meticulous data cleaning, scalable synthetic data generation, and balanced data mixing, Qwen2.5-Coder demonstrates impressive code generation capabilities while retaining general and math skills. These models have been evaluated on a wide range of code-related tasks, achieving state-of-the-art (SOTA) performance across more than 10 benchmarks, including code generation, completion, reasoning, and repair, consistently outperforming larger models of the same model size. We believe that the release of the Qwen2.5-Coder series will advance research in code intelligence and, with its permissive licensing, support wider adoption by developers in real-world applications.
Large Language Models' (LLM) reasoning can be improved using test-time aggregation strategies, i.e., generating multiple samples and voting among generated samples. While these improve performance, they often reach a saturation point. Refinement offers an alternative by using LLM-generated feedback to improve solution quality. However, refinement introduces 3 key challenges: (1) Excessive refinement: Uniformly refining all instances can over-correct and reduce the overall performance. (2) Inability to localize and address errors: LLMs have a limited ability to self-correct and struggle to identify and correct their own mistakes. (3) Insufficient refinement: Deciding how many iterations of refinement are needed is non-trivial, and stopping too soon could leave errors unaddressed. To tackle these issues, we propose MAgICoRe, which avoids excessive refinement by categorizing problem difficulty as easy or hard, solving easy problems with coarse-grained aggregation and hard ones with fine-grained and iterative multi-agent refinement. To improve error localization, we incorporate external step-wise reward model (RM) scores. Moreover, to ensure effective refinement, we employ a multi-agent loop with three agents: Solver, Reviewer (which generates targeted feedback based on step-wise RM scores), and the Refiner (which incorporates feedback). To ensure sufficient refinement, we re-evaluate updated solutions, iteratively initiating further rounds of refinement. We evaluate MAgICoRe on Llama-3-8B and GPT-3.5 and show its effectiveness across 5 math datasets. Even one iteration of MAgICoRe beats Self-Consistency by 3.4%, Best-of-k by 3.2%, and Self-Refine by 4.0% while using less than half the samples. Unlike iterative refinement with baselines, MAgICoRe continues to improve with more iterations. Finally, our ablations highlight the importance of MAgICoRe's RMs and multi-agent communication.
Chain-of-thought (CoT) via prompting is the de facto method for eliciting reasoning capabilities from large language models (LLMs). But for what kinds of tasks is this extra ``thinking'' really helpful? To analyze this, we conducted a quantitative meta-analysis covering over 100 papers using CoT and ran our own evaluations of 20 datasets across 14 models. Our results show that CoT gives strong performance benefits primarily on tasks involving math or logic, with much smaller gains on other types of tasks. On MMLU, directly generating the answer without CoT leads to almost identical accuracy as CoT unless the question or model's response contains an equals sign, indicating symbolic operations and reasoning. Following this finding, we analyze the behavior of CoT on these problems by separating planning and execution and comparing against tool-augmented LLMs. Much of CoT's gain comes from improving symbolic execution, but it underperforms relative to using a symbolic solver. Our results indicate that CoT can be applied selectively, maintaining performance while saving inference costs. Furthermore, they suggest a need to move beyond prompt-based CoT to new paradigms that better leverage intermediate computation across the whole range of LLM applications.
Large language models (LLMs) have exhibited remarkable few-shot learning capabilities and unified the paradigm of NLP tasks through the in-context learning (ICL) technique. Despite the success of ICL, the quality of the exemplar demonstrations can significantly influence the LLM's performance. Existing exemplar selection methods mainly focus on the semantic similarity between queries and candidate exemplars. On the other hand, the logical connections between reasoning steps can be beneficial to depict the problem-solving process as well. In this paper, we proposes a novel method named Reasoning Graph-enhanced Exemplar Retrieval (RGER). RGER first quires LLM to generate an initial response, then expresses intermediate problem-solving steps to a graph structure. After that, it employs graph kernel to select exemplars with semantic and structural similarity. Extensive experiments demonstrate the structural relationship is helpful to the alignment of queries and candidate exemplars. The efficacy of RGER on math and logit reasoning tasks showcases its superiority over state-of-the-art retrieval-based approaches. Our code is released at https://github.com/Yukang-Lin/RGER.
Finetuning large language models on instruction data is crucial for enhancing pre-trained knowledge and improving instruction-following capabilities. As instruction datasets proliferate, selecting optimal data for effective training becomes increasingly important. This work addresses the question: How can we determine the optimal subset of data for effective training? While existing research often emphasizes local criteria like instance quality for subset selection, we argue that a global approach focused on data diversity is more critical. Our method employs k-means clustering to ensure the selected subset effectively represents the full dataset. We propose an iterative refinement method inspired by active learning techniques to resample instances from clusters, reassessing each cluster's importance and sampling weight in every training iteration. This approach reduces the effect of outliers and automatically filters out clusters containing low-quality data. Through extensive evaluation across natural language reasoning, general world knowledge, code and math reasoning tasks, and by fine-tuning models from various families, we observe consistent improvements, achieving a 7% increase over random selection and a 3.8% improvement over state-of-the-art sampling methods. Our work highlights the significance of diversity-first sampling when finetuning LLMs to enhance performance across a broad array of evaluation tasks. Our code is available at https://github.com/for-ai/iterative-data-selection.
To make sense of massive data, we often fit simplified models and then interpret the parameters; for example, we cluster the text embeddings and then interpret the mean parameters of each cluster. However, these parameters are often high-dimensional and hard to interpret. To make model parameters directly interpretable, we introduce a family of statistical models -- including clustering, time series, and classification models -- parameterized by natural language predicates. For example, a cluster of text about COVID could be parameterized by the predicate "discusses COVID". To learn these statistical models effectively, we develop a model-agnostic algorithm that optimizes continuous relaxations of predicate parameters with gradient descent and discretizes them by prompting language models (LMs). Finally, we apply our framework to a wide range of problems: taxonomizing user chat dialogues, characterizing how they evolve across time, finding categories where one language model is better than the other, clustering math problems based on subareas, and explaining visual features in memorable images. Our framework is highly versatile, applicable to both textual and visual domains, can be easily steered to focus on specific properties (e.g. subareas), and explains sophisticated concepts that classical methods (e.g. n-gram analysis) struggle to produce.
Variational Autoencoders are widespread in Machine Learning, but are typically explained with dense math notation or static code examples. This paper presents VAE Explainer, an interactive Variational Autoencoder running in the browser to supplement existing static documentation (e.g., Keras Code Examples). VAE Explainer adds interactions to the VAE summary with interactive model inputs, latent space, and output. VAE Explainer connects the high-level understanding with the implementation: annotated code and a live computational graph. The VAE Explainer interactive visualization is live at https://xnought.github.io/vae-explainer and the code is open source at https://github.com/xnought/vae-explainer.
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However, existing efforts are largely limited to creating formalized versions of proofs from extensive online mathematical corpora, struggling to keep pace with the rapidly evolving nature of mathematics. To bridge the gap between traditional and computerized proof techniques, this paper explores the use of Large Language Models (LLMs) to generate formal proof steps and complete formalized proofs. By converting natural language (NL) mathematical proofs into formalized versions, this work introduces the basic structure and tactics of the Lean 4 language. The goal is to determine how AI can be leveraged to assist the mathematical formalization process and improve its performance. Several examples are provided that demonstrate solving problems using both traditional and Lean 4-based approaches. Ultimately, this paper presents an explanation of the foundations of Lean 4 and comparative analyses of the mathematical formalization process using traditional and AI-augmented techniques. The findings indicate that AI- powered tools have significant potential to accelerate and enhance the formalization of mathematical proofs, paving the way for more efficient and reliable theorem-proving for AI for Math in the future.
With the blossom of large language models (LLMs), inference efficiency becomes increasingly important. Various approximation methods are proposed to reduce the cost at inference time. Contextual Sparsity (CS) is appealing for its training-free nature and its ability to reach a higher compression ratio seemingly without quality degradation. However, after a comprehensive evaluation of contextual sparsity methods on various complex generation tasks, we find that although CS succeeds in prompt-understanding tasks, CS significantly degrades the model performance for reasoning, deduction, and knowledge-based tasks. Despite the gap in end-to-end accuracy, we observed that sparse models often share general problem-solving logic and require only a few token corrections to recover the original model performance. This paper introduces Sirius, an efficient correction mechanism, which significantly recovers CS models quality on reasoning tasks while maintaining its efficiency gain. Sirius is evaluated on 6 models with 8 difficult generation tasks in reasoning, math, and coding and shows consistent effectiveness and efficiency. Also, we carefully develop a system implementation for Sirius and show that Sirius achieves roughly 20% reduction in latency for 8B model on-chip and 35% reduction for 70B model offloading. We open-source our implementation of Sirius at https://github.com/Infini-AI-Lab/Sirius.git.
The rapid development of large language models (LLMs) has spurred extensive research into their domain-specific capabilities, particularly mathematical reasoning. However, most open-source LLMs focus solely on mathematical reasoning, neglecting the integration with visual injection, despite the fact that many mathematical tasks rely on visual inputs such as geometric diagrams, charts, and function plots. To fill this gap, we introduce \textbf{MultiMath-7B}, a multimodal large language model that bridges the gap between math and vision. \textbf{MultiMath-7B} is trained through a four-stage process, focusing on vision-language alignment, visual and math instruction-tuning, and process-supervised reinforcement learning. We also construct a novel, diverse and comprehensive multimodal mathematical dataset, \textbf{MultiMath-300K}, which spans K-12 levels with image captions and step-wise solutions. MultiMath-7B achieves state-of-the-art (SOTA) performance among open-source models on existing multimodal mathematical benchmarks and also excels on text-only mathematical benchmarks. Our model and dataset are available at {\textcolor{blue}{\url{https://github.com/pengshuai-rin/MultiMath}}}.
Large language models (LLMs) have obtained promising results in mathematical reasoning, which is a foundational skill for human intelligence. Most previous studies focus on improving and measuring the performance of LLMs based on textual math reasoning datasets (e.g., MATH, GSM8K). Recently, a few researchers have released English multimodal math datasets (e.g., MATHVISTA and MATH-V) to evaluate the effectiveness of large multimodal models (LMMs). In this paper, we release a Chinese multimodal math (CMM-Math) dataset, including benchmark and training parts, to evaluate and enhance the mathematical reasoning of LMMs. CMM-Math contains over 28,000 high-quality samples, featuring a variety of problem types (e.g., multiple-choice, fill-in-the-blank, and so on) with detailed solutions across 12 grade levels from elementary to high school in China. Specifically, the visual context may be present in the questions or opinions, which makes this dataset more challenging. Through comprehensive analysis, we discover that state-of-the-art LMMs on the CMM-Math dataset face challenges, emphasizing the necessity for further improvements in LMM development. We also propose a Multimodal Mathematical LMM (Math-LMM) to handle the problems with mixed input of multiple images and text segments. We train our model using three stages, including foundational pre-training, foundational fine-tuning, and mathematical fine-tuning. The extensive experiments indicate that our model effectively improves math reasoning performance by comparing it with the SOTA LMMs over three multimodal mathematical datasets.
In this paper, we investigate the presence of additive bias in Large Language Models (LLMs), drawing a parallel to the cognitive bias observed in humans where individuals tend to favor additive over subtractive changes. Using a series of controlled experiments, we tested various LLMs, including GPT-3.5 Turbo, Claude 3.5 Sonnet, Mistral, Math$\Sigma$tral, and Llama 3.1, on tasks designed to measure their propensity for additive versus subtractive modifications. Our findings demonstrate a significant preference for additive changes across all tested models. For example, in a palindrome creation task, Llama 3.1 favored adding letters 97.85% of the time over removing them. Similarly, in a Lego tower balancing task, GPT-3.5 Turbo chose to add a brick 76.38% of the time rather than remove one. In a text summarization task, Mistral 7B produced longer summaries in 59.40% to 75.10% of cases when asked to improve its own or others' writing. These results indicate that, similar to humans, LLMs exhibit a marked additive bias, which might have implications when LLMs are used on a large scale. Addittive bias might increase resource use and environmental impact, leading to higher economic costs due to overconsumption and waste. This bias should be considered in the development and application of LLMs to ensure balanced and efficient problem-solving approaches.
In this paper, we propose Singular Values and Orthonormal Regularized Singular Vectors Adaptation, or SORSA, a novel PEFT method. Each SORSA adapter consists of two main parts: trainable principal singular weights $W_p = U_p \text{diag}(S_p) V^\top_p$, and frozen residual weights $W_r = U_r \text{diag}(S_r) V^\top_r$. These parts are initialized by performing singular value decomposition (SVD) on pre-trained weights. Moreover, we implement and analyze an orthonormal regularizer, which we prove could decrease the condition number of $W_p$ and make the optimization more efficient. SORSA adapters could be merged during inference, thus eliminating any inference latency. We also introduce a method to analyze the variation of the parameters by performing SVD and discuss and analyze SORSA's superiority in minimizing the alteration in the SVD aspect. After all, SORSA shows a faster convergence than LoRA and PiSSA in our experiments. On the GSM-8K benchmark, Llama 2 7B adapted using SORSA achieved 56.03% accuracy, surpassing LoRA (42.30%), AdaLoRA (47.30%), Full FT (49.05%), and PiSSA (53.07%). On the MATH benchmark, SORSA achieved 10.36% accuracy, outperforming LoRA (5.50%), AdaLoRA (6.48%), Full FT (7.22%), and PiSSA (7.44%). We conclude that SORSA offers a new perspective on parameter-efficient fine-tuning, demonstrating remarkable performance.
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.
The use of Large Language Models (LLMs) in mathematical reasoning has become a cornerstone of related research, demonstrating the intelligence of these models and enabling potential practical applications through their advanced performance, such as in educational settings. Despite the variety of datasets and in-context learning algorithms designed to improve the ability of LLMs to automate mathematical problem solving, the lack of comprehensive benchmarking across different datasets makes it complicated to select an appropriate model for specific tasks. In this project, we present a benchmark that fairly compares seven state-of-the-art in-context learning algorithms for mathematical problem solving across five widely used mathematical datasets on four powerful foundation models. Furthermore, we explore the trade-off between efficiency and performance, highlighting the practical applications of LLMs for mathematical reasoning. Our results indicate that larger foundation models like GPT-4o and LLaMA 3-70B can solve mathematical reasoning independently from the concrete prompting strategy, while for smaller models the in-context learning approach significantly influences the performance. Moreover, the optimal prompt depends on the chosen foundation model. We open-source our benchmark code to support the integration of additional models in future research.
Multimodal Large Language Models (MLLMs) excel in solving text-based mathematical problems, but they struggle with mathematical diagrams since they are primarily trained on natural scene images. For humans, visual aids generally enhance problem-solving, but MLLMs perform worse as information shifts from textual to visual modality. This decline is mainly due to their shortcomings in aligning images and text. To tackle aforementioned challenges, we propose Math-PUMA, a methodology focused on Progressive Upward Multimodal Alignment. This approach is designed to improve the mathematical reasoning skills of MLLMs through a three-stage training process, with the second stage being the critical alignment stage. We first enhance the language model's mathematical reasoning capabilities with extensive set of textual mathematical problems. We then construct a multimodal dataset with varying degrees of textual and visual information, creating data pairs by presenting each problem in at least two forms. By leveraging the Kullback-Leibler (KL) divergence of next-token prediction distributions to align visual and textual modalities, consistent problem-solving abilities are ensured. Finally, we utilize multimodal instruction tuning for MLLMs with high-quality multimodal data. Experimental results on multiple mathematical reasoning benchmarks demonstrate that the MLLMs trained with Math-PUMA surpass most open-source MLLMs. Our approach effectively narrows the performance gap for problems presented in different modalities. The code and data are available at: \url{https://github.com/wwzhuang01/Math-PUMA}.
Most large language models are fine-tuned using either expensive human-annotated data or GPT-4 generated data which cannot guarantee performance in certain domains. We argue that although the web-crawled data often has formatting errors causing semantic inaccuracies, it can still serve as a valuable source for high-quality supervised fine-tuning in specific domains without relying on advanced models like GPT-4. To this end, we create a paired training dataset automatically by aligning web-crawled data with a smaller set of high-quality data. By training a language model on this dataset, we can convert web data with irregular formats into high-quality ones. Our experiments show that training with the model-transformed data yields better results, surpassing training with only high-quality data by an average score of 9.4% in Chinese math problems. Additionally, our 7B model outperforms several open-source models larger than 32B and surpasses well-known closed-source models such as GPT-3.5, highlighting the efficacy of our approach.