[Paper Review] EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback

Peter Richtárik, Igor Sokolov+1 more|arXiv (Cornell University)|Jun 9, 2021

Stochastic Gradient Optimization Techniques37 references40 citations

TL;DR

EF21 introduces a Markov-based error feedback mechanism for biased gradient compression in distributed optimization, achieving faster convergence under standard assumptions and better empirical performance than prior EF methods.

ABSTRACT

Error feedback (EF), also known as error compensation, is an immensely popular convergence stabilization mechanism in the context of distributed training of supervised machine learning models enhanced by the use of contractive communication compression mechanisms, such as Top-$k$. First proposed by Seide et al (2014) as a heuristic, EF resisted any theoretical understanding until recently [Stich et al., 2018, Alistarh et al., 2018]. However, all existing analyses either i) apply to the single node setting only, ii) rely on very strong and often unreasonable assumptions, such global boundedness of the gradients, or iterate-dependent assumptions that cannot be checked a-priori and may not hold in practice, or iii) circumvent these issues via the introduction of additional unbiased compressors, which increase the communication cost. In this work we fix all these deficiencies by proposing and analyzing a new EF mechanism, which we call EF21, which consistently and substantially outperforms EF in practice. Our theoretical analysis relies on standard assumptions only, works in the distributed heterogeneous data setting, and leads to better and more meaningful rates. In particular, we prove that EF21 enjoys a fast $O(1/T)$ convergence rate for smooth nonconvex problems, beating the previous bound of $O(1/T^{2/3})$, which was shown a bounded gradients assumption. We further improve this to a fast linear rate for PL functions, which is the first linear convergence result for an EF-type method not relying on unbiased compressors. Since EF has a large number of applications where it reigns supreme, we believe that our 2021 variant, EF21, can a large impact on the practice of communication efficient distributed learning.

Motivation & Objective

• Motivate the need for effective communication-efficient optimization in distributed, over-parameterized models.
• Develop a new error feedback mechanism suitable for heterogeneous data without strong assumptions.
• Establish theoretical guarantees: improved convergence rates under standard smoothness and lower-boundedness assumptions, and linear convergence under PL condition.
• Extend EF21 with EF21+ to adaptively choose between compressors for improved performance.
• Demonstrate empirical superiority over classical EF in synthetic and deep learning experiments.

Proposed method

• Introduce Markov compressors to create estimators that become better over time, enabling stable biased compression.
• Define EF21 as a distributed optimization method where each node applies a biased compressor to local gradients and communicates compressed residuals.
• Prove O(1/T) convergence for smooth nonconvex objectives under standard assumptions (L-smoothness and lower-boundedness).
• Prove linear convergence under Polyak-Lojasiewicz (PL) condition for EF21.
• Propose EF21+ as a hybrid that lets nodes pick the better between the original compressor and the Markov compressor on a per-iteration basis.
• Provide extensions to stochastic gradients and discuss the relationship to the original EF method.

Experimental results

Research questions

• RQ1 Can EF21 achieve faster nonconvex convergence rates under standard smoothness and lower-boundedness assumptions without requiring unbiased compressors?
• RQ2 Does EF21 exhibit linear convergence under the Polyak-Lojasiewicz condition, without relying on extra unbiased compressors?
• RQ3 How does EF21 compare to classical error feedback (EF) in practice, especially on distributed heterogeneous data?
• RQ4 Does the EF21+ variant provide practical gains by adaptively selecting between compressors per node?
• RQ5 How can EF21 be extended to stochastic gradient settings and maintain theoretical guarantees?

Key findings

• EF21 achieves an O(1/T) convergence rate for smooth nonconvex problems under standard assumptions.
• EF21 attains a fast linear convergence rate for functions satisfying the Polyak-Lojasiewicz condition.
• EF21+ preserves the theoretical guarantees while often improving practical performance by choosing the better compressor per node.
• Empirical results show EF21 consistently and substantially outperforms the original EF method in synthetic and deep learning benchmarks, enabling larger learning rates.
• The analysis requires only standard assumptions (L-smoothness and lower bound) and does not rely on bounded gradients or iterate-dependent bounds.
• EF21 and EF21+ extend to stochastic gradient settings with discussed adaptations.