Distributed Gradient Descent with Many Local Steps in Overparameterized Models (2412.07971v1)

Abstract: In distributed training of machine learning models, gradient descent with local iterative steps is a very popular method, variants of which are commonly known as Local-SGD or the Federated Averaging (FedAvg). In this method, gradient steps based on local datasets are taken independently in distributed compute nodes to update the local models, which are then aggregated intermittently. Although the existing convergence analysis suggests that with heterogeneous data, FedAvg encounters quick performance degradation as the number of local steps increases, it is shown to work quite well in practice, especially in the distributed training of LLMs. In this work we try to explain this good performance from a viewpoint of implicit bias in Local Gradient Descent (Local-GD) with a large number of local steps. In overparameterized regime, the gradient descent at each compute node would lead the model to a specific direction locally. We characterize the dynamics of the aggregated global model and compare it to the centralized model trained with all of the data in one place. In particular, we analyze the implicit bias of gradient descent on linear models, for both regression and classification tasks. Our analysis shows that the aggregated global model converges exactly to the centralized model for regression tasks, and converges (in direction) to the same feasible set as centralized model for classification tasks. We further propose a Modified Local-GD with a refined aggregation and theoretically show it converges to the centralized model in direction for linear classification. We empirically verified our theoretical findings in linear models and also conducted experiments on distributed fine-tuning of pretrained neural networks to further apply our theory.

Analyzing Distributed Gradient Descent with Local Steps in Overparameterized Models

The paper "Distributed Gradient Descent with Many Local Steps in Overparameterized Models" by Heng Zhu et al. addresses the effectiveness of Local-Gradient Descent (Local-GD) methodologies such as Local-SGD and FedAvg in the context of overparameterized models. In distributed machine learning, Local-GD is widely used due to its ability to reduce communication overhead by performing multiple local updates on the edge nodes before aggregating the models at a central server. Although theoretical analyses have suggested that increasing the number of local steps can degrade performance on heterogeneous data, empirical evidence shows that these methods are surprisingly efficient.

Key Findings and Contributions

1. Local-GD Convergence in Linear Regression: The authors start by analyzing Local-GD on linear regression problems in the overparameterized regime. They prove that, contrary to previous assumptions, the aggregated global model can converge exactly to the centralized one as the number of communication rounds increases. Theoretical insights are derived by considering the implicit bias of gradient descent, which in overparameterized settings drives solutions towards the minimum norm solution compatible with all constraints.
2. Characterization of Linear Classification via Parallel Projection Method: Extending the analysis to linear classification with overparameterized models, the paper ties the behavior of Local-GD with many local steps to the parallel projection method (PPM). PPM is a classical algorithm for finding a common point in intersecting convex sets, and it is used here to describe the iterative behavior of aggregating local models. Through this connection, the paper proves that the global model converges towards a feasible global set, which is analogous to the centralized model's convergence.
3. Proposition of Modified Local-GD: Recognizing the potential discrepancy between vanilla Local-GD and centralized solutions for classification tasks, the authors introduce a modified version of Local-GD. This variation implements a refined aggregation mechanism that provably aligns the direction of convergence of the global model with that of the centralized model.
4. Empirical Validation: The experimental section underscores the theoretical findings, especially noting the convergence of Local-GD and Modified Local-GD to centralized solutions in both linear regression and classification tasks. Additionally, experiments on fine-tuning the final layer of neural networks reinforce that the insights gained from studying linear models can be extrapolated to more complex neural architectures.

Implications and Future Directions

The analysis provided in this paper challenges previous conceptions about the limitations of Local-GD with numerous local steps. By rigorously demonstrating convergence to centralized solutions under overparameterization, the authors provide compelling evidence that these methods are well-suited for distributed training of large models, even with heterogeneous data distributions. This opens several pathways for future work:

• Theoretical Extensions to Nonlinear Models: While the paper focuses on linear models, extending the analysis to nonlinear scenarios, especially within neural networks, could yield valuable insights into the general applicability of these results.
• Generalization and Performance Guarantees: Further research might explore the impact of different data distributions beyond the uniformly considered settings to understand how general these findings are in diverse real-world configurations.
• Adapting Local Models in Federated Learning: The proposed insights and modifications might inform adaptive federated learning algorithms that dynamically adjust local step sizes according to model overparameterization and data heterogeneity.

Overall, this work enriches the theoretical understanding of federated optimization techniques in overparameterized settings and paves the way for more robust and efficient distributed training systems.