Generalization in LLM Problem Solving: The Case of the Shortest Path
The study shows language models exhibit strong spatial transfer in shortest path problems but fail in length scaling due to recursive instability.
Key Findings
Methodology
The study constructs a synthetic environment based on shortest-path planning to analyze systematic generalization in language models. It explores two orthogonal axes of generalization: spatial transfer and length scaling. By controlling training data, paradigms, and inference strategies, the study independently assesses their impact on model generalization.
Key Results
- Result 1: Models show over 90% success in spatial transfer tasks but significantly drop in performance in length scaling tasks, indicating recursive instability as the main cause.
- Result 2: Increasing question diversity rather than solution diversity under limited training budgets enhances spatial transfer capability.
- Result 3: Reinforcement learning improves training stability but does not surpass the best supervised fine-tuning performance, especially in length scaling tasks.
Significance
This study reveals the generalization capabilities and limitations of language models in solving composable optimization problems, particularly highlighting recursive instability when handling longer paths. These findings are significant for applying language models in complex reasoning tasks, suggesting researchers consider these factors when designing training data and inference strategies.
Technical Contribution
The technical contributions include constructing a controlled synthetic environment that allows independent assessment of training data, paradigms, and inference strategies on model generalization. Additionally, the study identifies recursive instability as a major limitation in handling length scaling tasks.
Novelty
This study is the first to systematically evaluate language models' generalization capabilities in composable optimization problems within a controlled synthetic environment, focusing on spatial transfer and length scaling. This approach differs from previous natural language benchmarks, providing clearer generalization assessments.
Limitations
- Limitation 1: Models perform poorly in length scaling tasks primarily due to recursive instability, limiting their application in more complex tasks.
- Limitation 2: Although reinforcement learning improves training stability, it does not extend the model's capability boundaries.
Future Work
Future research could explore enhancing model performance in length scaling tasks through improved training data and inference strategies. Additionally, further analysis of the root causes of recursive instability and the development of new algorithms to overcome this limitation are needed.
AI Executive Summary
The systematic generalization capabilities of language models in solving composable optimization problems have been a subject of intense research interest. Existing studies indicate that the reasoning abilities of language models are influenced by multiple factors, including training data, paradigms, and inference strategies. However, the complex interactions of these factors make it difficult to interpret the causes of model failures. To address this issue, researchers have constructed a synthetic environment based on shortest-path planning. This environment allows researchers to evaluate model performance along two orthogonal axes of generalization: spatial transfer and length scaling.
In spatial transfer tasks, models need to solve shortest path problems on entirely unseen maps. Experimental results show that models exhibit strong generalization capabilities in this task, achieving success rates above 90%. However, in length scaling tasks, where models must handle paths longer than those encountered during training, performance significantly declines. This failure is primarily attributed to recursive instability, where models become unstable when recursively applying learned rules.
The study further analyzes how different stages of the learning pipeline influence systematic problem-solving abilities. Results indicate that data coverage sets the capability limits; while reinforcement learning improves training stability, it does not extend the model's capability boundaries; inference-time scaling enhances performance but cannot rescue failures in length scaling tasks.
The significance of this study lies in its revelation of the generalization capabilities and limitations of language models in solving composable optimization problems, particularly highlighting recursive instability when handling longer paths. These findings are significant for applying language models in complex reasoning tasks, suggesting researchers consider these factors when designing training data and inference strategies.
Future research could explore enhancing model performance in length scaling tasks through improved training data and inference strategies. Additionally, further analysis of the root causes of recursive instability and the development of new algorithms to overcome this limitation are needed.
Deep Analysis
Background
Language models have achieved significant progress in natural language processing, particularly in generation and understanding tasks. However, their systematic generalization capabilities in complex reasoning tasks remain controversial. Existing studies indicate that the reasoning abilities of models are influenced by multiple factors, including the properties of training data, training paradigms (such as supervised fine-tuning and reinforcement learning), and inference strategies. The complex interactions of these factors make it difficult to interpret the causes of model failures. Additionally, natural language benchmarks often make it challenging to determine whether a model truly generalizes, as the differences between training and evaluation settings are not cleanly controlled.
Core Problem
The core problem of this study is to evaluate the systematic generalization capabilities of language models in composable optimization problems, particularly in shortest-path planning tasks. Shortest-path planning is a classic composable optimization problem that requires models to generate a complete solution path between given start and end nodes. The challenge lies in independently assessing the impact of training data, paradigms, and inference strategies on model generalization within a controlled environment.
Innovation
The core innovations of this study include constructing a synthetic environment based on shortest-path planning, allowing researchers to evaluate model performance along two orthogonal axes of generalization: spatial transfer and length scaling. β’ Spatial Transfer: Tests the model's ability to solve tasks on entirely unseen maps. β’ Length Scaling: Evaluates the model's ability to handle longer paths within the same map. β’ By controlling training data, paradigms, and inference strategies, the study independently assesses their impact on model generalization.
Methodology
The study employs the following methods: β’ Constructing a Synthetic Environment: Based on shortest-path planning, providing a globally verifiable objective and unambiguous optimal solutions. β’ Spatial Transfer Testing: Evaluating model generalization on entirely unseen maps. β’ Length Scaling Testing: Handling longer paths within the same map. β’ Controlling training data, paradigms, and inference strategies to independently assess their impact on model generalization.
Experiments
The experimental design includes: β’ Datasets: Randomly generated maps and paths. β’ Baselines: Comparison with existing shortest-path algorithms. β’ Evaluation Metrics: Success Rate (SR), the probability of the model generating valid shortest paths. β’ Key Hyperparameters: Coverage and diversity of training data, inference-time scaling strategies.
Results
Experimental results show: β’ Models exhibit strong generalization capabilities in spatial transfer tasks, achieving success rates above 90%. β’ In length scaling tasks, model performance significantly declines, indicating recursive instability as the main cause. β’ Increasing question diversity rather than solution diversity enhances spatial transfer capability under limited training budgets.
Applications
The study's findings have potential applications in the following scenarios: β’ Natural Language Processing: Enhancing model performance in complex reasoning tasks. β’ Robotics Navigation: Applying to path planning in dynamic environments. β’ Data Science: Optimizing strategies for solving composable problems.
Limitations & Outlook
The study's limitations include: β’ Models perform poorly in length scaling tasks primarily due to recursive instability. β’ Reinforcement learning improves training stability but does not extend the model's capability boundaries. β’ Future research needs to explore new algorithms to overcome these limitations.
Plain Language Accessible to non-experts
Imagine you're in a giant maze, and your task is to find the exit from the entrance. This maze has many different paths, some short and some long. You have a helper who can assist you in finding the shortest path. This helper is like our language model. The helper is very good at finding the exit in new mazes because he can quickly adapt to new environments. This is similar to how our model performs well in spatial transfer tasks. However, when the maze becomes very large, the helper starts making mistakes because he becomes unstable when dealing with longer paths. This is similar to how our model performs poorly in length scaling tasks. To help the helper perform better, we can give him more training to get used to handling longer paths. This is what we do in our study by adjusting training data and strategies to improve model performance.
ELI14 Explained like you're 14
Hey there, buddy! Imagine you're playing a super complex maze game. Your task is to find the shortest path from the start to the end. You have a super smart robot helper who can help you find the fastest route. This robot is like a language model, and he's really good at adapting to new mazes, just like our model does well in spatial transfer tasks. But when the maze gets really big, the robot gets a bit confused because he's not so good at handling longer paths. To make the robot even better, we can give him more training to get used to handling longer paths. That way, he can perform better in more complex mazes! Cool, right?
Glossary
Language Model
A model used in natural language processing that can generate and understand human language.
Used in this paper to solve the shortest path problem.
Shortest Path
The shortest path from a start to an end point in a graph, often used in optimization problems.
Serves as the core task of the study.
Spatial Transfer
The ability of a model to solve tasks in entirely unseen environments.
Used to evaluate model generalization capabilities.
Length Scaling
The ability of a model to handle longer paths within the same environment.
Used to test the model's recursive stability.
Recursive Instability
The instability of a model when recursively applying learned rules, leading to performance degradation.
Identified as the main cause of failure in length scaling tasks.
Supervised Fine-Tuning
A training paradigm that fine-tunes a model using labeled data.
Used to enhance model performance.
Reinforcement Learning
A training paradigm that guides model learning through reward signals.
Used to improve training stability.
Success Rate
The probability of a model generating valid shortest paths.
Serves as an evaluation metric.
Synthetic Environment
A controlled environment used to test model generalization capabilities.
Allows independent assessment of training data and strategy impacts.
Composable Optimization Problem
A problem that requires composing a sequence of local decisions to satisfy a global objective.
Shortest-path planning is a typical example.
Open Questions Unanswered questions from this research
- 1 How can model performance in length scaling tasks be improved? Current methods perform poorly when handling longer paths, primarily due to recursive instability. New algorithms are needed to overcome this limitation.
- 2 What are the root causes of recursive instability? While known as the main cause of failure in length scaling tasks, its fundamental mechanisms remain unclear.
- 3 How can more effective training data and inference strategies be designed? Existing strategies fail to address failures in length scaling tasks.
- 4 How can systematic generalization capabilities of language models in complex reasoning tasks be further enhanced? Existing studies indicate that model reasoning abilities are influenced by multiple factors.
- 5 How can model generalization capabilities be improved without increasing training data? Existing studies indicate that data coverage sets the capability limits of models.
- 6 What is the potential of reinforcement learning in enhancing model generalization capabilities? While it improves training stability, it does not extend the model's capability boundaries.
- 7 How can model inference capabilities be improved without affecting training stability? Existing studies indicate that inference-time scaling strategies, while enhancing performance, fail to address failures in length scaling tasks.
Applications
Immediate Applications
Natural Language Processing
Enhancing model performance in complex reasoning tasks, particularly in composable optimization problems.
Robotics Navigation
Applying to path planning in dynamic environments, improving robot navigation capabilities in unknown environments.
Data Science
Optimizing strategies for solving composable problems, improving efficiency and accuracy in data analysis.
Long-term Vision
Intelligent Transportation Systems
Improving efficiency and safety of intelligent transportation systems through improved path planning algorithms.
Autonomous Driving
Enhancing decision-making capabilities of autonomous driving systems in complex environments, improving driving safety.
Abstract
Whether language models can systematically generalize remains actively debated. Yet empirical performance is jointly shaped by multiple factors such as training data, training paradigms, and inference-time strategies, making failures difficult to interpret. We introduce a controlled synthetic environment based on shortest-path planning, a canonical composable sequential optimization problem. The setup enables clean separation of these factors and supports two orthogonal axes of generalization: spatial transfer to unseen maps and length scaling to longer-horizon problems. We find that models exhibit strong spatial transfer but consistently fail under length scaling due to recursive instability. We further analyze how distinct stages of the learning pipeline influence systematic problem-solving: for example, data coverage sets capability limits; reinforcement learning improves training stability but does not expand those limits; and inference-time scaling enhances performance but cannot rescue length-scaling failures.
References (20)
Understanding R1-Zero-Like Training: A Critical Perspective
Zi-Yan Liu, Changyu Chen, Wenjun Li et al.
Self-Consistency Improves Chain of Thought Reasoning in Language Models
Xuezhi Wang, Jason Wei, D. Schuurmans et al.
Generalization without Systematicity: On the Compositional Skills of Sequence-to-Sequence Recurrent Networks
B. Lake, Marco Baroni
On the generalization of language models from in-context learning and finetuning: a controlled study
Andrew Kyle Lampinen, Arslan Chaudhry, Stephanie C Y Chan et al.
Instruction Tuning with GPT-4
Baolin Peng, Chunyuan Li, Pengcheng He et al.
Contrastive Decoding: Open-ended Text Generation as Optimization
Xiang Lisa Li, Ari Holtzman, Daniel Fried et al.
SWE-bench: Can Language Models Resolve Real-World GitHub Issues?
Carlos E. Jimenez, John Yang, Alexander Wettig et al.
Vision-R1: Incentivizing Reasoning Capability in Multimodal Large Language Models
Wenxuan Huang, Bohan Jia, Zijie Zhai et al.
Task Generalization With AutoRegressive Compositional Structure: Can Learning From D Tasks Generalize to DT Tasks?
Amirhesam Abedsoltan, Huaqing Zhang, Kaiyue Wen et al.
Compositional Generalization from First Principles
ThaddΓ€us Wiedemer, P. Mayilvahanan, M. Bethge et al.
On Provable Length and Compositional Generalization
Kartik Ahuja, Amin Mansouri
SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training
Tianzhe Chu, Yuexiang Zhai, Jihan Yang et al.
LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code
Naman Jain, King Han, Alex Gu et al.
CTL++: Evaluating Generalization on Never-Seen Compositional Patterns of Known Functions, and Compatibility of Neural Representations
R'obert Csord'as, K. Irie, J. Schmidhuber
COGS: A Compositional Generalization Challenge Based on Semantic Interpretation
Najoung Kim, Tal Linzen
The Paradox of the Compositionality of Natural Language: A Neural Machine Translation Case Study
Verna Dankers, Elia Bruni, Dieuwke Hupkes
All Roads Lead to Likelihood: The Value of Reinforcement Learning in Fine-Tuning
Gokul Swamy, Sanjiban Choudhury, Wen Sun et al.
Understanding Addition in Transformers
Philip Quirke, Fazl Barez
Compositional Abilities Emerge Multiplicatively: Exploring Diffusion Models on a Synthetic Task
Maya Okawa, E. Lubana, Robert P. Dick et al.
DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning
Adam Suma, Sam Dauncey