Composing Global Optimizers to Reasoning Tasks via Algebraic Objects in Neural Nets (2410.01779v3)

Abstract: We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and $L_2$ loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables analytical construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity. We coin the framework as CoGO (Composing Global Optimizers). Specifically, we show that the weight space over different numbers of hidden nodes of the 2-layer network is equipped with a semi-ring algebraic structure, and the loss function to be optimized consists of monomial potentials, which are ring homomorphism, allowing partial solutions to be composed into global ones by ring addition and multiplication. Our experiments show that around $95\%$ of the solutions obtained by gradient descent match exactly our theoretical constructions. Although the global optimizers constructed only required a small number of hidden nodes, our analysis on gradient dynamics shows that over-parameterization asymptotically decouples training dynamics and is beneficial. We further show that training dynamics favors simpler solutions under weight decay, and thus high-order global optimizers such as perfect memorization are unfavorable. Code can be found at https://github.com/facebookresearch/luckmatters/tree/yuandong3/ssl/real-dataset.

• The paper introduces the CoGO framework that composes global optimizers from partial solutions using algebraic structures in neural nets.
• It unveils a semi-ring structure and ring homomorphism properties of monomial potentials in 2-layer neural networks with quadratic activations.
• Empirical studies show that about 95% of gradient descent solutions align with the theoretical predictions of the CoGO framework.

Overview of "Composing Global Optimizers to Reasoning Tasks via Algebraic Objects in Neural Nets"

This paper presents a theoretical framework called CoGO (Composing Global Optimizers) to model the algebraic structures within trained neural networks for reasoning tasks in Abelian groups. Focusing on modular addition, the work investigates 2-layer neural networks with quadratic activation functions and $L_2$ loss, uncovering a rich algebraic structure that can construct global optimizers from partial solutions.

Key Concepts and Theoretical Contributions

1. Algebraic Structure in Neural Networks: The research identifies a semi-ring structure over the solution space of weights for neural networks with varying hidden nodes. This structure facilitates the composition of global solutions using fundamental operations such as ring addition and multiplication.
2. Monomial Potentials (MPs) and Loss Function: The loss function is analyzed through MPs, revealing that they are ring homomorphisms. This allows partial solutions satisfying certain constraints to be composed into global solutions.
3. The Composition of Partial Solutions: The crux of the framework is the ability to utilize partial solutions that satisfy parts of the loss function and combine them to construct complete global optimizers. This is achieved by leveraging the semi-ring structure of the weight space and the homomorphic properties of MPs.
4. Training Dynamics and Overparameterization: The analysis on training dynamics demonstrates that gradient dynamics favor simpler solutions due to overparameterization, which asymptotically decouples the training dynamics. This preference for low-order solutions leads to better performance and generalization.
5. Empirical Validation: Through empirical studies, the paper validates that approximately 95% of solutions found by gradient descent align with the theoretical constructions proposed by CoGO. This strong alignment supports the framework's predictive power in modeling network training outcomes.

Implications and Future Directions

• Theorizing Neural Network Training: CoGO offers insights into the intrinsic mathematical structures of neural networks during training, potentially guiding the development of novel, efficient training algorithms that leverage compositional methods rather than traditional gradient descent.
• Designing Loss Functions: Since CoGO applies to any loss function dependent on monomial potentials, it opens possibilities for designing new types of loss functions aligned with specific algebraic structures to drive learning dynamics toward desired configurations.
• Expanding Theoretical Frameworks: Extending the framework to other activation functions and broader classes of reasoning tasks could deepen the understanding of learning dynamics and solution spaces across neural network architectures.

This work presents a structured, mathematical perspective on neural network optimization, with significant implications for theory and practice. It provides a roadmap for exploiting algebraic properties in neural network training, which could lead to innovations in both methodology and application.