On Biased Compression for Distributed Learning (2002.12410v4)

Abstract: In the last few years, various communication compression techniques have emerged as an indispensable tool helping to alleviate the communication bottleneck in distributed learning. However, despite the fact biased compressors often show superior performance in practice when compared to the much more studied and understood unbiased compressors, very little is known about them. In this work we study three classes of biased compression operators, two of which are new, and their performance when applied to (stochastic) gradient descent and distributed (stochastic) gradient descent. We show for the first time that biased compressors can lead to linear convergence rates both in the single node and distributed settings. We prove that distributed compressed SGD method, employed with error feedback mechanism, enjoys the ergodic rate $O\left( \delta L \exp \left[-\frac{\mu K}{\delta L}\right] + \frac{(C + \delta D)}{K\mu}\right)$, where $\delta\ge 1$ is a compression parameter which grows when more compression is applied, $L$ and $\mu$ are the smoothness and strong convexity constants, $C$ captures stochastic gradient noise ($C=0$ if full gradients are computed on each node) and $D$ captures the variance of the gradients at the optimum ($D=0$ for over-parameterized models). Further, via a theoretical study of several synthetic and empirical distributions of communicated gradients, we shed light on why and by how much biased compressors outperform their unbiased variants. Finally, we propose several new biased compressors with promising theoretical guarantees and practical performance.

• The paper introduces three classes of biased compression operators and demonstrates their effectiveness in achieving linear convergence rates.
• It provides a detailed convergence analysis in both single-node and distributed gradient descent settings, highlighting reduced variance in communication.
• The study compares biased and unbiased compressors, offering theoretical guarantees and empirical evidence of efficiency gains in distributed learning.

Analysis of Biased Compression for Distributed Learning

The paper "On Biased Compression for Distributed Learning" explores the role of biased compressors in alleviating communication bottlenecks in distributed machine learning settings. Despite biased compressors performing well in real-world scenarios, their theoretical understanding has been limited. This paper provides a comprehensive examination of three classes of biased compression operators and demonstrates their effectiveness in achieving linear convergence rates.

Key Contributions

1. Definition of Biased Compression Operators:
   • The paper introduces three classes of biased compression operators: $\mathbb{B}^1(\alpha,\beta)$, $\mathbb{B}^2(\gamma,\beta)$, and $\mathbb{B}^3(\delta)$. These classes provide a systematic way to analyze biased compression techniques and relate them to existing unbiased methodologies.
2. Convergence Analysis:
   • The analysis of biased compression operators applied to single-node and distributed gradient descent settings reveals that biased compressors can achieve linear convergence rates. This is particularly significant given the challenges of communication in distributed systems.
3. Comparison with Unbiased Compressors:
   • The paper investigates the circumstances under which biased compressors outperform their unbiased counterparts. It leverages synthetic and empirical data to quantify these differences. The observations suggest substantial benefits in employing biased compressors, particularly in terms of reduced variance during compression.
4. Development of New Biased Compressors:
   • It proposes new biased compressor methods with motivating theoretical guarantees and empirical performance. These advancements could potentially lead to more efficient distributed learning systems.

Numerical Results and Implications

The paper’s theoretical insights are backed by numerical experiments which reflect the reduced empirical variance and communication costs associated with biased compressors. The efficiency gains observed in these experiments could lead to more practical implementations in large-scale distributed learning systems.

The implications of this research are twofold:

• Practical: The findings can be used to optimize communication strategies in distributed machine learning frameworks, potentially reducing computational overheads and increasing the feasibility of larger models.
• Theoretical: The comprehensive examination of biased compression operators enriches the theoretical understanding of these techniques, paving the way for future studies to build upon these foundations.

Future Developments

By establishing a correlation between compression techniques and convergence rates, this paper opens avenues for further exploration into adaptive compression strategies tailored to specific distributed learning tasks. Additionally, the proposed methodologies hold promise for improving federated learning environments where data heterogeneity is a significant concern.

Through its detailed analysis and insightful results, the paper emphasizes the critical role that biased compression operators play in enhancing distributed learning systems. It sets the stage for ongoing research to deepen their exploration and enhance the efficiency of machine learning algorithms deployed in networked environments.