Hallmarks of Optimization Trajectories in Neural Networks: Directional Exploration and Redundancy (2403.07379v2)

Abstract: We propose a fresh take on understanding the mechanisms of neural networks by analyzing the rich directional structure of optimization trajectories, represented by their pointwise parameters. Towards this end, we introduce some natural notions of the complexity of optimization trajectories, both qualitative and quantitative, which hallmark the directional nature of optimization in neural networks: when is there redundancy, and when exploration. We use them to reveal the inherent nuance and interplay involved between various optimization choices, such as momentum and weight decay. Further, the trajectory perspective helps us see the effect of scale on regularizing the directional nature of trajectories, and as a by-product, we also observe an intriguing heterogeneity of Q,K,V dynamics in the middle attention layers in LLMs and which is homogenized by scale. Importantly, we put the significant directional redundancy observed to the test by demonstrating that training only scalar batchnorm parameters some while into training matches the performance of training the entire network, which thus exhibits the potential of hybrid optimization schemes that are geared towards efficiency.

• The paper introduces Trajectory Map and Mean Directional Similarity metrics to capture the structure and directional exploration of optimization paths.
• The paper demonstrates the impact of hyperparameters such as momentum, weight decay, and learning rate on the regularity and complexity of training trajectories.
• The paper shows that increased model size leads to more regular optimization paths, enhancing parameter alignment and potential generalization.

Insights into Optimization Trajectories of Neural Networks and LLMs

Recent work explores the optimization trajectories of neural networks with a focus on understanding the underlying mechanisms and implications for both small and large-scale networks, including LLMs. The paper specifically explores the structure and dynamics of parameter updates during training, asserting that the paths taken during optimization can exhibit various complex patterns such as length, bends, and potential dead ends. By investigating these trajectories, the authors provide valuable insights into the optimization process, characterized by the implicit bias and regularity within the sequence of steps.

Key Methodological Contributions

1. Trajectory Map (TM): A qualitative tool used to visualize and evaluate the directional similarity and the complexity of optimization paths, providing an overview of how parameters evolve across different stages of the training process.
2. Mean Directional Similarity (MDS): A quantitative measure that captures the average cosine similarity between pairs of points in the trajectory, facilitating a clearer understanding of parameter alignment and movement patterns during optimization.
3. Angular and Norm-based Measures: These focus on the angles between consecutive updates and the norms, offering further granularity to the analysis of how optimization paths explore the parameter space.

Empirical Observations

The research advocates a systematic approach to relate various hyperparameter settings such as momentum, weight decay, learning rate, and batch size to the qualitative nature of optimization trajectories. An illustrative set of experimental findings includes:

• Impact of Momentum and Weight Decay: The paper reveals an intricate interplay between momentum and weight decay, leading to enhanced directional exploration within the loss landscape. Contrarily, the absence of these hyperparameters leads to a simplification of the trajectory, effectively limiting the exploration of possible solutions and potentially locking the optimization into suboptimal paths.
• Model Size and Trajectory Structure: Extensive experiments with GPT-NeoX models on different scales reveal an increase in trajectory regularity as model size grows. This observation has theoretical backing; at large widths, parameter updates align with their initialization, necessitating stability in feature updates and implicitly enhancing directional similarities.

Theoretical Implications

The analysis of trajectories lends credence to the hypothesis that understanding the optimization path is crucial beyond merely evaluating the final parameters at convergence. The introduction of MDS as a potential correlator of optimization quality may offer a new dimension in predicting generalization performance, suggesting that the nature of the optimization trajectory itself can reveal deep insights into the learned representations and model efficacy.

At a broader theoretical level, these methods promise advances in understanding implicit biases inherent in various optimization routines. These biases are not only dictated by hyperparameters but are also critically influenced by the underlying geometry of the optimization path.

Practical Implications and Future Directions

The practical implications of this work revolve primarily around more efficient training regimens. By optimizing under the lens of trajectory geometry, it is conceivable to develop optimization strategies that leverage the observed directional Explorations, potentially leading to reductions in computation and resource requirements during training.

This work also opens numerous avenues for future work, such as exploring the trajectory perspective for different architectures and tasks, extending studies to non-convex landscapes beyond vision and LLMs, and integrating these insights to improve existing optimization algorithms.

In summation, this research underscores the complexity and richness of optimization paths in neural networks and LLMs. It marks advancements in both theoretical understanding and practical application, setting the stage for enhanced training methodologies informed by a deeper appreciation of the dynamic behavior inherent in the optimization trajectories of modern neural networks.