Learning Iterative Reasoning through Energy Diffusion (2406.11179v1)

Abstract: We introduce iterative reasoning through energy diffusion (IRED), a novel framework for learning to reason for a variety of tasks by formulating reasoning and decision-making problems with energy-based optimization. IRED learns energy functions to represent the constraints between input conditions and desired outputs. After training, IRED adapts the number of optimization steps during inference based on problem difficulty, enabling it to solve problems outside its training distribution -- such as more complex Sudoku puzzles, matrix completion with large value magnitudes, and pathfinding in larger graphs. Key to our method's success is two novel techniques: learning a sequence of annealed energy landscapes for easier inference and a combination of score function and energy landscape supervision for faster and more stable training. Our experiments show that IRED outperforms existing methods in continuous-space reasoning, discrete-space reasoning, and planning tasks, particularly in more challenging scenarios. Code and visualizations at https://energy-based-model.github.io/ired/

• The paper presents IRED, an energy-based framework that formulates iterative reasoning tasks as optimization problems using annealed energy landscapes.
• It employs supervised energy shaping and contrastive loss to guide the energy function and enhance stability during training.
• IRED adapts its optimization steps based on problem difficulty, outperforming specialized models on tasks like Sudoku, matrix completion, and path planning.

Iterative Reasoning through Energy Diffusion

The research paper "Learning Iterative Reasoning through Energy Diffusion" by Du, Mao, and Tenenbaum presents the framework "Iterative Reasoning through Energy Diffusion" (IRED) for complex reasoning and decision-making tasks across diverse domains. The primary contribution of this work lies in the efficiency and versatility of the proposed energy-based optimization method, which forms a robust computational model capable of generalizing well beyond the training distribution.

Framework Overview

IRED formulates reasoning tasks as energy-based optimization problems. An energy function is learned to represent constraints between input conditions and desired outputs. The method employs annealed energy landscapes and combines score function supervision with energy landscape supervision for efficient and stable training.

During inference, IRED adapts the number of optimization steps based on problem difficulty, allowing it to tackle more complex instances than those seen during training. This adaptability is crucial for solving harder reasoning tasks not encountered during the model's training phase, such as more challenging Sudoku puzzles, matrix completion tasks, and pathfinding in larger graphs.

Key Techniques

1. Annealed Energy Landscapes: IRED leverages multiple annealed energy landscapes, gradually refining solutions from smooth to complex energy landscapes. This approach mitigates the complexities involved in directly optimizing high-dimensional, sharply varying landscapes, thereby enhancing stability during the optimization process.
2. Supervised Energy Shaping: The framework utilizes denoising supervision to guide the energy function's gradient towards the ground truth label, addressing potential instability and slow training speeds observed in earlier EBM methodologies. Additionally, a contrastive loss supervises the energy landscape, ensuring that the minimal energy corresponds to the correct solutions.

Experimental Evaluation

IRED was rigorously tested across three categories: continuous-space reasoning, discrete-space reasoning, and planning tasks.

Continuous-Space Reasoning

Tasks such as matrix addition, matrix completion, and matrix inversion demonstrated IRED's superior generalization capabilities. The method outperformed existing models on both test sets and more difficult instances with larger magnitudes or poorly-conditioned matrices.

Discrete-Space Reasoning

For discrete reasoning tasks like Sudoku solving and graph connectivity, IRED exhibited notable success. On the Sudoku task, for example, it significantly surpassed the performance of domain-specific models like SAT-Net and RNN-based approaches, especially on harder Sudoku puzzles with fewer initial clues. In graph connectivity prediction, IRED achieved higher accuracy on both training distribution and more challenging graphs.

Planning Tasks

The shortest path finding task illustrated IRED's planning abilities within discrete spaces. Compared to alternatives, IRED provided better accuracy in predicting actions that shorten the distance to the goal. Moreover, its approach does not rely on pre-specified task formulations, distinguishing it from specialized polynomial algorithms.

Implications and Future Directions

The implications of IRED span practical and theoretical domains. Practically, its ability to generalize and adapt computational load based on problem difficulty makes it suitable for real-world applications where task complexity can vary significantly. Theoretically, the framework offers a robust foundation for further research into energy-based models, particularly in areas necessitating dynamic, iterative reasoning processes.

Future developments may focus on accelerating the inference-time optimization procedure using guided optimizers or neural network generators. Additionally, adapting IRED to hybrid discrete-continuous decision-making spaces and improving the learning sequence of energy landscapes for enhanced adaptability could present valuable advancements.

Conclusion

This paper presents a comprehensive and versatile framework for iterative reasoning utilizing energy diffusion. By integrating annealed energy landscapes and supervision techniques, IRED sets a new benchmark in solving complex reasoning tasks. This work illustrates a significant step forward in energy-based optimization methodologies and opens avenues for future research and applications in AI reasoning and planning domains.