[Paper Review] On the Benefits of Multiple Gossip Steps in Communication-Constrained Decentralized Optimization

Abolfazl Hashemi, Anish Acharya+4 more|arXiv (Cornell University)|Nov 20, 2020

Distributed Control Multi-Agent Systems58 references5 citations

TL;DR

This paper proposes DeLi-CoCo, a communication-efficient decentralized optimization algorithm that accelerates convergence by performing multiple gossip steps between gradient updates under arbitrary communication compression. It proves linear convergence to within $\epsilon$ of the optimal solution for smooth non-convex objectives satisfying the Polyak-Łojasiewicz condition using $O(\log \frac{1}{\epsilon})$ gossip steps per gradient iteration, matching the rate of uncompressed decentralized gradient descent while enabling arbitrarily reduced sub-optimality via increased gossip steps.

ABSTRACT

In decentralized optimization, it is common algorithmic practice to have nodes interleave (local) gradient descent iterations with gossip (i.e. averaging over the network) steps. Motivated by the training of large-scale machine learning models, it is also increasingly common to require that messages be {\em lossy compressed} versions of the local parameters. In this paper, we show that, in such compressed decentralized optimization settings, there are benefits to having {\em multiple} gossip steps between subsequent gradient iterations, even when the cost of doing so is appropriately accounted for e.g. by means of reducing the precision of compressed information. In particular, we show that having $O(\log\frac{1}ε)$ gradient iterations {with constant step size} - and $O(\log\frac{1}ε)$ gossip steps between every pair of these iterations - enables convergence to within $ε$ of the optimal value for smooth non-convex objectives satisfying Polyak-Łojasiewicz condition. This result also holds for smooth strongly convex objectives. To our knowledge, this is the first work that derives convergence results for nonconvex optimization under arbitrary communication compression.

Motivation & Objective

• To address the challenge of high communication costs in decentralized machine learning, especially with compressed parameter exchanges.
• To investigate whether multiple gossip steps between gradient updates can improve convergence efficiency under communication constraints.
• To develop a theoretically grounded algorithm that maintains fast convergence despite aggressive compression.
• To establish convergence guarantees for non-convex objectives under arbitrary (biased or unbiased) compression operators.
• To demonstrate that increasing gossip steps while reducing compression precision can improve communication efficiency for a fixed budget.

Proposed method

• Proposes DeLi-CoCo, an iterative decentralized algorithm that performs $Q > 1$ gossip steps after each local gradient update.
• Integrates arbitrary communication compression (e.g., quantization, sparsification) into the gossip phase, allowing lossy transmission of local parameters.
• Uses a consensus mechanism over a network graph to average local estimates, with compression applied to messages exchanged between nodes.
• Employs a novel error feedback mechanism to track and correct for compression-induced errors across gossip iterations.
• Derives convergence bounds using the Polyak-Łojasiewicz (PL) condition, enabling linear convergence for non-convex objectives.
• Analyzes the trade-off between number of gossip steps $Q$ and compression precision, showing that increasing $Q$ reduces sub-optimality radius.

Experimental results

Research questions

• RQ1 Can multiple gossip steps between gradient updates improve convergence in communication-constrained decentralized optimization?
• RQ2 Does increasing the number of gossip steps allow for lower-precision compression while maintaining or improving convergence performance?
• RQ3 Can linear convergence be achieved for non-convex objectives under arbitrary communication compression, including biased operators?
• RQ4 How does the number of gossip steps $Q$ affect the sub-optimality radius of the solution?
• RQ5 Is there a theoretical trade-off between the number of gossip steps and compression precision that improves communication efficiency?

Key findings

• DeLi-CoCo achieves linear convergence to within $\epsilon$ of the optimal solution for smooth non-convex objectives satisfying the Polyak-Łojasiewicz condition using $O(\log \frac{1}{\epsilon})$ gradient iterations and $O(\log \frac{1}{\epsilon})$ gossip steps per iteration.
• The convergence rate matches that of decentralized gradient descent (DGD) with no compression, even under arbitrary compression operators.
• The sub-optimality radius can be arbitrarily reduced by increasing the number of gossip steps $Q$, improving upon standard DGD.
• For a fixed communication budget, increasing $Q$ and reducing compression precision leads to better convergence performance, as shown theoretically and verified numerically.
• The algorithm maintains convergence under both biased and unbiased compression operators, extending prior results to non-convex settings.
• Theoretical analysis confirms that the error from compression is bounded and diminishes over time, enabling stable and fast convergence.