Linearly Converging Error Compensated SGD (2010.12292v1)

Abstract: In this paper, we propose a unified analysis of variants of distributed SGD with arbitrary compressions and delayed updates. Our framework is general enough to cover different variants of quantized SGD, Error-Compensated SGD (EC-SGD) and SGD with delayed updates (D-SGD). Via a single theorem, we derive the complexity results for all the methods that fit our framework. For the existing methods, this theorem gives the best-known complexity results. Moreover, using our general scheme, we develop new variants of SGD that combine variance reduction or arbitrary sampling with error feedback and quantization and derive the convergence rates for these methods beating the state-of-the-art results. In order to illustrate the strength of our framework, we develop 16 new methods that fit this. In particular, we propose the first method called EC-SGD-DIANA that is based on error-feedback for biased compression operator and quantization of gradient differences and prove the convergence guarantees showing that EC-SGD-DIANA converges to the exact optimum asymptotically in expectation with constant learning rate for both convex and strongly convex objectives when workers compute full gradients of their loss functions. Moreover, for the case when the loss function of the worker has the form of finite sum, we modified the method and got a new one called EC-LSVRG-DIANA which is the first distributed stochastic method with error feedback and variance reduction that converges to the exact optimum asymptotically in expectation with a constant learning rate.

• The paper introduces a general framework for analyzing distributed SGD methods with error feedback, unifying existing approaches and improving convergence rates.
• The paper develops linearly converging methods like EC-GD-DIANA and EC-LSVRG-DIANA, overcoming the challenges posed by biased compression.
• The paper empirically demonstrates that the proposed methods reduce communication costs while maintaining convergence speed and accuracy on real-world datasets.

Linearly Converging Error Compensated SGD: An Overview

This paper presents a comprehensive analysis and development of variants of distributed Stochastic Gradient Descent (SGD) with specific emphasis on error compensation and communication efficiency. The authors introduce a unified framework that encapsulates various existing methods like quantized SGD, error-compensated SGD (EC-SGD), and SGD with delayed updates (D-SGD), and furthermore, propose new variants that integrate additional techniques such as variance reduction, arbitrary sampling, error feedback, and quantization.

The paper outlines a general theoretical framework to analyze and derive convergence complexities of methods that fit within its proposed structure. Notably, this framework recovers and often improves upon the best-known results for existing methods. The authors establish a significant result that showcases linearly converging EC-SGD methods despite employing biased communication compression, which was a challenging theoretical problem unresolved by previous work.

Contributions and Implications

1. General Framework and Analysis: The paper introduces a robust framework leveraging parametric assumptions for analyzing a broad class of methods expressed through an error-feedback update mechanism. This framework allows for the development and analysis of 16 novel methods and captures various existing approaches as special cases, often deriving sharper rates of convergence.
2. Linearly Converging Methods: Among the key contributions is the development of linearly converging EC-SGD methods, unexpected in the context of biased compression operators. Specifically, methods such as EC-GD-DIANA and EC-LSVRG-DIANA demonstrate linear convergence rates while incorporating both compression and variance reduction. This breakthrough result addresses a significant gap in distributed optimization literature, where ensuring convergence to an exact solution with biased compression was a persistent challenge.
3. Numerical Results: The paper provides empirical evaluations showing that the proposed methods reduce communication costs without sacrificing convergence speed or accuracy compared to standard SGD approaches. Experiments conducted on logistic regression problems using real-world datasets illustrate the practical efficacy of these methods.

Theoretical Insights and Future Directions

The authors highlight an essential trade-off in distributed optimization: while compression can drastically reduce communication overhead, without careful analysis and methodical compensation, it may degrade convergence properties. Through rigorous theoretical work, they have shown how to maintain strong convergence guarantees even when using sophisticated compression techniques.

The establishment of variance reduction as a tool to mitigate the noise introduced by both stochastic gradients and compression underscores its importance in developing communication-efficient distributed methods. Similarly, error-feedback mechanisms are solidified as a viable approach to addressing issues introduced by biased compressors.

Looking forward, the research opens avenues for further exploration into more sophisticated interaction models and adaptive strategies for choosing among multiple variance reduction or compression techniques in real-time. Effectively deploying these techniques in dynamically varying network conditions, as encountered in federated learning or large-scale distributed systems, promises substantial performance enhancements.

Conclusion

This paper significantly advances the state of distributed SGD by addressing critical communication bottlenecks through theoretically grounded and experimentally validated methods. The blend of rigorous analysis with practical algorithm development underlines its relevance, catering to the increasing demand for efficient distributed learning frameworks. It sets a precedent for future research in distributed machine learning, emphasizing the importance of integrating robust theoretical insights with real-world applicability.