Universal Neural Functionals (2402.05232v1)

Abstract: A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and LLMs. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

• The paper introduces a novel algorithm that constructs permutation equivariant models for arbitrary weight tensors in complex neural networks.
• It extends prior frameworks by stacking linear layers with pointwise non-linearities to handle residual, recurrent, and convolutional architectures.
• Empirical evaluations demonstrate improved optimizer performance and higher predictive accuracy, with enhanced rank correlation over baseline methods.

An In-Depth Analysis of Universal Neural Functionals

The paper "Universal Neural Functionals" addresses an intricate challenge in modern machine learning: the processing and transformation of weight-space features of neural networks. This scope includes weights, gradients, and sparsity masks among other parameters critical in neural network architectures. One pressing issue is preserving equivalence when neural weights undergo permutations, more complex when dealing with architectures beyond simple feedforward networks.

Core Contributions and Methodology

The principal contribution of this paper is a novel algorithm capable of constructing permutation equivariant models for an arbitrary collection of weight tensors. These models, named Universal Neural Functionals (UNFs), address permutation symmetries that appear complicated due to the presence of residual or recurrent connections. This algorithm facilitates the creation of the most generalized linear layers while ensuring equivariance to the specified permutation symmetries. Additionally, the algorithm supports the automatic construction of deep permutation equivariant models through stacking multiple such layers with pointwise non-linearities.

In formalizing their approach, the authors extend previous frameworks for permutation equivariance. For instance, Navon et al. (2023) and Zhou et al. (2023a) had introduced permutation equivariant neural functionals primarily applicable to simple multilayer perceptrons (MLPs) and feedforward convolutional networks (CNNs). By developing a more generalized framework, this paper captures the permutation symmetries relevant for complex networks containing residual connections, recurrence, and layers of normalization.

Empirical Evaluation

The authors conducted rigorous evaluations demonstrating the efficacy and flexibility of UNFs. A standout application was integrating UNFs into learned optimizers. When tested on small image classifiers and LLMs, UNFs outperformed previous methods in optimization tasks, achieving a consistent and promising improvement in performance metrics. These findings suggest that learned optimizers must consider the symmetry structure inherent in the weight space they manipulate.

A compelling set of experiments involved predicting the generalization performance of recurrent sequence-to-sequence models from their weights, using a dataset coined as the Tiny RNN Zoo. The results showcased that UNF-based predictors achieved superior performance by delivering a higher rank correlation (Kendall’s τ) between the predicted and actual success rates compared to baselines such as STAT NN, which calculates statistics invariant to weight permutations.

Theoretical and Practical Implications

Theoretically, the paper contributes to a unified and principled framework for designing neural functionals operating on weight spaces by leveraging permutation equivariance. This advancement opens avenues for more complex weight-space processing architectures, potentially facilitating the discovery of more expressive and robust learned optimizers.

Practically, the implications are wide-reaching. The ability to process complex weight spaces with automatic symmetry handling can lead to more effective optimization strategies. Consequently, this impacts numerous applied domains, from computer vision to natural language processing, where neural networks are pivotal. Implementations open-sourced by this paper (available at https://github.com/AllanYangZhou/universal_neural_functional) further provide the machine learning community with immediate tools to experiment with and build upon this new class of neural functionals.

Comparisons and Future Directions

Comparative analysis with existing methods revealed that UNFs not only generalize better but also handle a broader spectrum of neural architectures more effectively. For example, while frameworks like NFNs (Zhou et al., 2023a) work well for MLPs due to similar underlying principles, they fall short when extended to architectures with complex interaction patterns like RNNs and Transformers.

Future research will likely explore heterogeneity in weight-space inputs, potentially combining diverse architectures within a single UNF framework. Moreover, further assessment of UNF-based learned optimizers across varied tasks and their generalization capacity will be crucial. Computationally efficient meta-gradient estimation methods may also be necessary to handle the higher parameter dimensions intrinsic to UNFs and enhance meta-optimization scalability.

In conclusion, the development of Universal Neural Functionals marks a significant step in the evolution of weight-space processing methods for neural networks. By constructing maximally expressive, permutation-equivariant models applicable to complex weight spaces, this paper lays a foundation for the next generation of advanced, symmetry-aware neural network optimizers and predictors.