How to guess a gradient (2312.04709v1)

Abstract: How much can you say about the gradient of a neural network without computing a loss or knowing the label? This may sound like a strange question: surely the answer is "very little." However, in this paper, we show that gradients are more structured than previously thought. Gradients lie in a predictable low-dimensional subspace which depends on the network architecture and incoming features. Exploiting this structure can significantly improve gradient-free optimization schemes based on directional derivatives, which have struggled to scale beyond small networks trained on toy datasets. We study how to narrow the gap in optimization performance between methods that calculate exact gradients and those that use directional derivatives. Furthermore, we highlight new challenges in overcoming the large gap between optimizing with exact gradients and guessing the gradients.

• The paper introduces architecture-aware methods that constrain gradient guessing to low-dimensional subspaces, reducing estimator variance.
• It develops feature-aware techniques that leverage activation subspaces to boost cosine similarity with the true gradients.
• Empirical evaluations on MNIST and CIFAR10 demonstrate improved performance over traditional directional methods, though results still lag behind backpropagation.

Insights into "How to Guess a Gradient"

The paper "How to Guess a Gradient," authored by Singhal et al., explores whether neural networks can be optimized without backpropagation—a fundamental technique in deep learning. The work addresses the challenge of estimating gradients using alternative methods, focusing on gradient-free optimization strategies that leverage directional derivatives.

Background and Context

The traditional reliance on backpropagation enables efficient gradient computation but poses scalability challenges due to memory and synchronization constraints. The paper revisits older ideas of using directional derivatives for optimization, such as those proposed by Polyak and Spall. Despite the theoretical interest, these methods typically struggle with high-dimensional tasks, resulting in impractical convergence rates.

Singhal et al. question whether it's possible to predict gradients more effectively by exploiting the neural network's structure and incoming features. They propose several methods to enhance gradient-free optimization techniques by narrowing the guess space and reducing estimator variance.

Core Contributions

1. Architecture-Aware Gradient Guessing: The authors introduce methods that limit the guessing space for gradients by utilizing known structures in neural networks. They suggest that gradients lie within a constrained, low-dimensional subspace determined by network architecture and activations. The proposed $W^\top$ approach notably reduces the dimensionality of the guessing space and achieves lower variance in gradient estimation.
2. Feature-Aware Gradient Guessing: The paper highlights that activations and gradients tend to reside in the same subspace. The "Activation Mixing" method leverages this observation by constructing gradient guesses within the activation subspace, thereby increasing the cosine similarity with the true gradient.
3. Empirical Evaluation: To assess the methods' effectiveness, experiments are conducted using MLPs on various datasets, including MNIST and CIFAR10. The results demonstrate significant improvements in cosine similarity and optimization performance over existing directional descent methods. However, there's an observable gap when compared to standard backpropagation, particularly for more complex datasets.
4. Self-Sharpening Phenomenon: An intriguing aspect of the paper is the "self-sharpening" phenomenon. This refers to a feedback loop where training dynamics inherently enhance the quality of gradient guesses, improving the cosine similarity over time. Yet, this phenomenon also introduces challenges, such as decreased generalization performance.
5. Theoretical and Practical Implications: The research underscores the potential of structural insights into gradient spaces to advance gradient-free optimization. The authors acknowledge that while these methods don't match backpropagation's efficiency on large-scale problems, they unveil new pathways for improving memory and computational efficiency in neural network training.

Numerical Outcomes and Observations

• On CIFAR10, methods like $W^\top$ achieve a train accuracy of 62.4% and a test accuracy of 48.0%, showing marked improvements over directional descent. However, the results still lag behind backpropagation, which nearly achieves perfect training accuracy.
• Activation subspace methodologies reported significant increases in cosine similarity, illustrating their capability to approximate true gradients more effectively than baseline approaches.

Future Directions

This research opens several avenues for future exploration. Reducing bias remains a significant hurdle, and addressing this could further narrow the performance gap with backpropagation. Additionally, integrating these gradient guessing techniques with contemporary training paradigms, such as fine-tuning and transfer learning, could offer novel insights and applications.

There's also potential for leveraging these methodologies in scenarios necessitating biologically plausible algorithms. By exploring combinatory approaches with local loss functions or auxiliary neural networks, as seen in recent literature, further advancements in scalable neural network training could be achieved.

In summary, this paper provides a detailed examination of alternative gradient estimation methods, yielding valuable theoretical contributions and empirical insights. It advances the discourse on non-backpropagation-based optimization and lays the groundwork for impactful future research to address the current limitations.