Unraveling Test Time Compute on Graph Structures

Table of Contents

1. Introduction
2. The Test Time Compute Dream
3. Generalization Improvement Mechanisms
4. Efficiency Issues
5. The Shortest Path Task
6. Evaluating Test Time Performance
7. Graph Neural Networks for Sudoku
8. Exploring the Graph Refinement Equation
9. Deep Equilibrium Models
10. Training the Deep Equilibrium GNN
11. Learning Adjacencies from Scratch
12. Applying GNNs to Sequence Modeling
13. Problems where Test Time Compute Shines
14. Applying GNNs without Inductive Biases
15. Thresholding the Learned Adjacency Weights

Introduction

In this article, we will explore the concept of test time compute on graph structured problems. The author of the text has spent a significant amount of time investigating whether models can continuously improve their outputs with more compute given at test time. This concept, referred to as the "test time compute dream," aims to address the discrepancy between human and machine learning models when it comes to improving performance over time.

The Test Time Compute Dream

The author argues that humans tend to become better at answering questions the longer they are given to think, while machine learning models often struggle to exhibit this ability. They categorize the test time compute dream into two general categories: generalization improvement mechanisms and efficiency issues.

Generalization Improvement Mechanisms

The first category focuses on creating models that use test time compute to learn more general algorithms instead of simple statistical associations. The goal is for these models to use additional compute to resolve ambiguity, correct and refine their own answers. By doing so, they aim to achieve better overall generalization.

However, the author's investigations into this area, particularly in the Context of the shortest path task, have shown limited success. While recurrent models with additional compute showed some improvements, they did not reach the performance level of larger models trained without recurrence. It seems that recurrence alone is not enough to achieve significant generalization improvement.

Efficiency Issues

The Second category of the test time compute dream focuses on the efficiency of models. The aim is to decouple the number of parameters in a model from the computational cost at inference time. The idea is to construct larger models without incurring a proportional increase in computation. This would allow for more powerful models without sacrificing efficiency.

The author's work on the shortest path task involved investigating the relationship between the flop budget and test time performance. They found that recurrence alone was not sufficient to match the performance of larger models. This suggests that additional structure, such as graph neural networks, is needed to achieve the desired efficiency improvements.

The Shortest Path Task

A significant portion of the author's project focused on the shortest path task. In this task, a model is given a pair of tokens representing pairs of U.S. cities and is expected to output a sequence representing the shortest path between the destinations. The author explored how different models performed on this task with varying levels of test time compute.

The results of their experiments showed that recurrence alone was not enough to significantly improve the performance of models. The models that used recurrence did not reach the performance level of larger models trained without recurrence. This suggests that learning a general shortest path algorithm is a challenging task even with additional compute.

Evaluating Test Time Performance

To understand the impact of test time compute on model performance, the author conducted experiments with varying levels of recurrence. They trained recurrent models with a fixed number of time steps and evaluated them with more steps of recurrence at test time. The goal was to determine if the extra compute could allow the recurrent models to catch up to larger models trained without recurrence.

Unfortunately, the experiments largely showed that recurrence alone is not enough to bridge the performance gap. The recurrent models did not reach the performance level of the larger models, indicating that additional structure is needed to improve generalization and efficiency.

Graph Neural Networks for Sudoku

The author also explored the use of graph neural networks (GNNs) in the context of solving Sudoku puzzles. GNNs are networks that operate on graph-structured data. They consist of an input representation phase, where the graph's nodes and their relationships are processed, followed by a refinement phase where the model iteratively updates its internal representation of the nodes.

In the case of Sudoku, each cell on the Sudoku board corresponds to a node on the graph. The GNN processes the graph by iteratively refining the representations of these nodes using a graph refinement equation. The goal is to learn a general algorithm to solve Sudoku puzzles.

Exploring the Graph Refinement Equation

The graph refinement equation is a key feature of GNNs. It takes the Hidden state of a node, its embedding, and the embeddings of its neighbors as input. The equation then updates the hidden state of the node by processing the embeddings through a function and aggregating them using an aggregation function. This equation captures the iterative refinement process of the GNN.

The author noticed that this graph refinement equation shares similarities with a fixed point equation. Inspired by this observation, they experimented with applying the machinery of deep equilibrium models to GNNs. Deep equilibrium models represent standard neural networks as implicit functions and can converge to a fixed point. This allows for analytically backpropagating through the equilibrium point using the implicit function theorem.

Deep Equilibrium Models

The author introduced the concept of deep equilibrium models, which leverage the machinery of deep equilibrium models to evaluate GNNs at infinite depth. By doing so, they could determine if evaluating the GNNs at greater depths could lead to better performance.

However, the author noted that there are challenges with deep equilibrium GNNs. They observed a collapse during training, where the models performed well initially but then encountered difficulties. The cause of this collapse is still under investigation, but early stopping has proven to be a viable solution. Despite the challenges, the potential of deep equilibrium GNNs remains promising.

Learning Adjacencies from Scratch

The author also explored the idea of learning adjacencies from scratch. Traditionally, GNNs require explicit specifications of the graph's structure, meaning the relationships between nodes must be defined beforehand. However, the author proposed using a tension head from a standard transformer to extract an adjacency matrix, which could then be fed into the GNN.

This approach involves using a small transformer to extract Relevant pairs of tokens from the input. These extracted pairs are treated as neighborhoods and passed into the GNN. While the initial results showed slower training and worse performance compared to standard GNNs, it demonstrated the concept of learning adjacencies from scratch.

Applying GNNs to Sequence Modeling

The author discussed the potential application of GNNs to sequence modeling tasks, such as language modeling. They Mentioned the possibility of using GNNs in an autoregressive manner, where the model evaluates the state of the entire graph at each point and outputs an output sequence. This could potentially improve the modeling of sequences by leveraging the relational reasoning capabilities of GNNs.

While the specific methods discussed in the presentation may appear contrived, the author is optimistic about the potential of test time compute mechanisms. They believe that significant breakthroughs will be made in the coming years, leveraging the Core ideas of test time compute to achieve improved generalization and efficiency in machine learning models.

Problems where Test Time Compute Shines

The author highlighted the potential value of test time compute in problems that involve relational reasoning tasks. These are tasks where the model needs to relate previous outputs to the Current information being processed. The ability to condition the amount of compute on the complexity of inputs can lead to more efficient and effective solutions.

Applying GNNs without Inductive Biases

GNNs are often praised for their ability to incorporate inductive biases by explicitly defining the graph's structure. However, the author discussed the possibility of learning adjacencies from scratch, effectively applying GNNs without HAND-baking inductive biases. While initial results were not as promising compared to explicitly defined structures, it demonstrates the potential for models to learn the adjacencies from raw, unstructured data.

Thresholding the Learned Adjacency Weights

In response to a question, the author discussed the possibility of thresholding the learned adjacency weights to produce a discrete graph structure. The discrete nature of the adjacency weights allows for the direct sampling of relevant tokens, which serves as the input to the GNN. This thresholding approach could provide a way to learn adjacencies from scratch and enable the modeling of structured data with GNNs.

In conclusion, the concept of test time compute holds much promise in the field of machine learning. The exploration of generalization improvement mechanisms, efficiency issues, and the application of GNNs in various tasks demonstrate the potential for improved performance and more efficient models. While specific methods may require further refinement, the core idea of test time compute is expected to lead to critical breakthroughs in the coming years.

Highlights

• Test time compute aims to improve model performance with additional compute given at inference time.
• Generalization improvement mechanisms and efficiency issues are two main categories of the test time compute dream.
• The author investigated the performance of recurrent models with additional compute on the shortest path task and found limited success.
• Graph neural networks (GNNs) can improve performance on relational reasoning tasks such as Sudoku.
• Deep equilibrium models Show promise in evaluating GNNs at infinite depth but face challenges such as a collapse during training.
• Learning adjacencies from scratch is a possibility with GNNs, although initial results show slower training and worse performance.
• Test time compute shines in problems that require relational reasoning and the ability to condition compute on input complexity.
• GNNs can potentially be applied without explicit inductive biases by learning adjacencies from raw, unstructured data.
• Thresholding the learned adjacency weights enables the creation of a discrete graph structure for GNNs to model structured data effectively.

FAQ

Q: What is the test time compute dream? A: The test time compute dream refers to the idea that models can continuously improve their outputs with more compute given at test time.

Q: How can test time compute improve generalization? A: Test time compute can allow models to use additional compute to resolve ambiguity, correct and refine their answers, leading to better generalization.

Q: What are some challenges encountered in applying deep equilibrium models to GNNs? A: Deep equilibrium models may experience a collapse during training, and the growth of the spectral norm of operators inside the GNN can pose difficulties. Early stopping has proven effective in mitigating these challenges.

Q: Can GNNs be applied to tasks without explicit inductive biases? A: Yes, there is potential to learn adjacencies from scratch, effectively applying GNNs without hand-baking inductive biases. However, the performance may not match that of explicitly defined structures.

Q: How can GNNs be used for sequence modeling? A: GNNs can be applied in an autoregressive manner, where the model evaluates the entire graph and outputs an output sequence at each point. This leverages the relational reasoning capabilities of GNNs and may improve sequence modeling.

Q: What problems are suitable for test time compute mechanisms? A: Test time compute shines in problems that involve relational reasoning tasks, where previous outputs need to be related to current information. It is also beneficial in problems where the compute can be conditioned on input complexity.