[Paper Review] Distributed second order methods with variable number of working nodes.

Dragana Bajović, Dušan Jakovetić+2 more|arXiv (Cornell University)|Sep 5, 2017

Sparse and Compressive Sensing Techniques3 citations

TL;DR

This paper introduces an idling mechanism into distributed second-order optimization by adapting it to the Distributed Quasi-Newton (DQN) method, enabling nodes to probabilistically activate based on increasing probabilities over iterations. The approach maintains R-linear convergence rates while significantly reducing communication and computational costs through cheaper, adaptive updates.

ABSTRACT

Recently, an idling mechanism has been introduced in the context of distributed first order methods for minimization of a sum of nodes' local convex costs over a generic, connected network. With the idling mechanism, each node i, at each iteration k, is active -- updates its solution estimate and exchanges messages with its network neighborhood -- with probability pk (pk increasing to one as k grows large), and it stays idle with probability 1-pk, while the activations are independent both across nodes and across iterations. The idling mechanism involves an increasing number of nodes in the algorithm (on average) as the iteration counter k grows, thus avoiding unnecessarily expensive exact updates at the initial iterations while performing beneficial close-to-exact updates near the solution. In this paper, we demonstrate that the idling mechanism can be successfully incorporated in distributed second order methods also. Specifically, we apply the idling mechanism to the recently proposed Distributed Quasi Newton method (DQN). We first show theoretically that DQN with idling exhibits very similar theoretical convergence and convergence rates properties as the standard DQN method, thus achieving the same order of convergence rate (R-linear) as the standard DQN, but with significantly cheaper updates. Further, we demonstrate by simulation examples significant communication and computational savings gained through incorporation of the idling mechanism.

Motivation & Objective

• Address the high computational and communication cost of exact second-order updates in distributed optimization, especially in early iterations.
• Overcome inefficiencies in traditional distributed methods that perform full updates regardless of progress, leading to unnecessary resource usage.
• Introduce a dynamic activation strategy that increases the number of active nodes over time, improving efficiency without sacrificing convergence quality.
• Extend the idling mechanism—previously used in first-order methods—to second-order distributed optimization, specifically to the DQN framework.
• Demonstrate that the idling-enhanced DQN maintains the same convergence rate as standard DQN while reducing computational and communication overhead.

Proposed method

• Adapt the idling mechanism from first-order methods to the Distributed Quasi-Newton (DQN) algorithm, where each node activates with probability pk that increases toward one over iterations.
• At each iteration k, each node i independently decides to update its solution estimate and exchange messages with neighbors with probability pk, otherwise remaining idle.
• Maintain the DQN update rule for active nodes, using local Hessian approximations and gradient information to compute search directions.
• Ensure that the Hessian approximation is updated only when nodes are active, reducing the frequency of expensive Hessian computations.
• Design the activation probability pk to grow over time, ensuring that more nodes participate as the algorithm approaches convergence.
• Theoretical analysis shows that the idling DQN maintains the same R-linear convergence rate as standard DQN, despite reduced update frequency.

Experimental results

Research questions

• RQ1 Can the idling mechanism, effective in first-order distributed optimization, be successfully extended to second-order methods like DQN?
• RQ2 Does the idling DQN method preserve the R-linear convergence rate of the standard DQN method under the same assumptions?
• RQ3 What are the communication and computational savings achieved by the idling mechanism in DQN compared to standard DQN?
• RQ4 How does the dynamic activation probability pk affect convergence behavior and efficiency in distributed second-order optimization?
• RQ5 Can the idling mechanism reduce resource usage in early iterations without compromising final solution accuracy or convergence speed?

Key findings

• The idling DQN method achieves the same R-linear convergence rate as the standard DQN method, confirming theoretical equivalence in convergence quality.
• The method significantly reduces computational and communication costs by avoiding exact updates in early iterations when progress is less critical.
• Simulation results demonstrate substantial savings in communication and computation, particularly in the initial phase of optimization.
• The increasing activation probability pk ensures that the number of active nodes grows over time, enabling high-accuracy updates near convergence.
• The idling mechanism enables a smooth transition from low-cost early iterations to high-accuracy final iterations, improving overall efficiency.
• Theoretical analysis confirms that the idling mechanism does not degrade convergence properties, maintaining the same order of convergence as standard DQN.