[Paper Review] Decentralized Composite Optimization with Compression

Yao Li, Xiaorui Liu+3 more|arXiv (Cornell University)|Aug 10, 2021

Stochastic Gradient Optimization Techniques69 references4 citations

TL;DR

This paper proposes Prox-LEAD, a novel decentralized proximal gradient algorithm with communication compression for stochastic composite optimization. It achieves linear convergence in both general stochastic and finite-sum settings, even with arbitrary compression precision, significantly reducing communication costs without sacrificing convergence rates, making it highly efficient for distributed machine learning with non-smooth regularizers.

ABSTRACT

Decentralized optimization and communication compression have exhibited their great potential in accelerating distributed machine learning by mitigating the communication bottleneck in practice. While existing decentralized algorithms with communication compression mostly focus on the problems with only smooth components, we study the decentralized stochastic composite optimization problem with a potentially non-smooth component. A \underline{Prox}imal gradient \underline{L}in\underline{EA}r convergent \underline{D}ecentralized algorithm with compression, Prox-LEAD, is proposed with rigorous theoretical analyses in the general stochastic setting and the finite-sum setting. Our theorems indicate that Prox-LEAD works with arbitrary compression precision, and it tremendously reduces the communication cost almost for free. The superiorities of the proposed algorithms are demonstrated through the comparison with state-of-the-art algorithms in terms of convergence complexities and numerical experiments. Our algorithmic framework also generally enlightens the compressed communication on other primal-dual algorithms by reducing the impact of inexact iterations, which might be of independent interest.

Motivation & Objective

• Address the gap in decentralized optimization with communication compression for problems involving non-smooth regularizers.
• Develop a unified algorithmic framework that integrates compression with proximal gradient methods in decentralized settings.
• Achieve linear convergence rates comparable to uncompressed counterparts under arbitrary compression precision.
• Demonstrate superior communication efficiency and convergence performance in both stochastic and finite-sum settings.
• Bridge connections between known decentralized algorithms and enable compressed communication for primal-dual methods.

Proposed method

• Propose Prox-LEAD, a decentralized proximal gradient algorithm with compressed communication using arbitrary compression operators.
• Integrate variance reduction techniques (SAGA, LSVRG) into the compressed decentralized framework to accelerate convergence.
• Use a local update mechanism with compressed communication over a connected network topology to minimize data transfer.
• Apply a general compression operator with bounded error to maintain convergence despite quantization or sparsification.
• Employ a diminishing stepsize strategy to achieve O(1/k) convergence to the exact solution in the stochastic setting.
• Theoretical analysis accounts for non-smooth regularizers and heterogeneous data distributions without bounded gradient assumptions.

Experimental results

Research questions

• RQ1 Can communication compression be effectively integrated into decentralized proximal gradient methods for composite optimization?
• RQ2 Does the proposed algorithm maintain linear convergence under arbitrary compression precision?
• RQ3 How does Prox-LEAD compare to uncompressed and state-of-the-art compressed algorithms in terms of communication cost and convergence speed?
• RQ4 Can variance reduction techniques be successfully combined with compressed communication in decentralized non-smooth optimization?
• RQ5 What is the impact of compression on convergence when data heterogeneity and non-smooth regularizers are present?

Key findings

• Prox-LEAD achieves linear convergence to the optimal solution in the finite-sum setting with variance reduction variants (LSVRG and SAGA), even under compression.
• In the general stochastic setting, Prox-LEAD converges linearly up to a neighborhood of the optimal solution, and with diminishing stepsize, it achieves O(1/k) convergence to the exact solution.
• The algorithm maintains convergence rates comparable to uncompressed counterparts, even without bounded gradient or data heterogeneity assumptions.
• Communication compression reduces communication cost by up to 16x (e.g., 2-bit vs. 32-bit) with minimal performance degradation, as shown in experiments.
• Prox-LEAD (2-bit) matches the convergence speed of 32-bit versions in terms of iterations and gradient evaluations, while using significantly fewer communication bits.
• The framework generalizes to other primal-dual algorithms and reduces the impact of inexact iterations through compressed communication.