Network Newton-Part II: Convergence Rate and Implementation

Alejandro Ribeiro; Aryan Mokhtari; Qing Ling

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Network Newton-Part II: Convergence Rate and Implementation

Aryan Mokhtari, Qing Ling, Alejandro Ribeiro

Optimization and Control · Submitted 23 Apr 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The numerical experiments corroborate reductions in the number of iterations and the communication cost that are necessary to achieve convergence relative to distributed gradient descent.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.17 · human participants 0.02 · controlled comparison 0.77 · randomized assignment 0.01 · states quantitative result 0.34 · compares with prior results 0.94 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The use of network Newton methods for the decentralized optimization of a sum cost distributed through agents of a network is considered. Network Newton methods reinterpret distributed gradient descent as a penalty method, observe that the corresponding Hessian is sparse, and approximate the Newton step by truncating a Taylor expansion of the inverse Hessian. Truncating the series at $K$ terms yields the NN-$K$ that requires aggregating information from $K$ hops away. Network Newton is introduced and shown to converge to the solution of the penalized objective function at a rate that is at least linear in a companion paper [3]. The contributions of this work are: (i) To complement the convergence analysis by studying the methods' rate of convergence. (ii) To introduce adaptive formulations that converge to the optimal argument of the original objective. (iii) To perform numerical evaluations of NN-$K$ methods. The convergence analysis relates the behavior of NN-$K$ with the behavior of (regular) Newton's method and shows that the method goes through a quadratic convergence phase in a specific interval. The length of this quadratic phase grows with $K$ and can be made arbitrarily large. The numerical experiments corroborate reductions in the number of iterations and the communication cost that are necessary to achieve convergence relative to distributed gradient descent.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The use of network Newton methods for the decentralized optimization of a sum cost distributed through agents of a network is considered.	background 0.96	0.19	0.04	asserted 0.68
s1	Network Newton methods reinterpret distributed gradient descent as a penalty method, observe that the corresponding Hessian is sparse, and approximate the Newton step by truncating a Taylor expansion of the inverse Hessian.	method 0.65	0.35	0.04	asserted 1.00
s2	Truncating the series at $K$ terms yields the NN-$K$ that requires aggregating information from $K$ hops away.	method 0.89	0.54	0.04	asserted 1.00
s3	Network Newton is introduced and shown to converge to the solution of the penalized objective function at a rate that is at least linear in a companion paper [3].	background 0.99	0.20	0.08	asserted 1.00
s4	The contributions of this work are: (i) To complement the convergence analysis by studying the methods' rate of convergence.	contribution 1.00	0.83	0.06	asserted 0.71
s5	(ii) To introduce adaptive formulations that converge to the optimal argument of the original objective.	contribution 1.00	0.88	0.08	asserted 0.85
s6	(iii) To perform numerical evaluations of NN-$K$ methods.	contribution 1.00	0.92	0.05	asserted 0.74
s7	The convergence analysis relates the behavior of NN-$K$ with the behavior of (regular) Newton's method and shows that the method goes through a quadratic convergence phase in a specific interval.	result 0.89	0.88	0.27	asserted 1.00
s8	The length of this quadratic phase grows with $K$ and can be made arbitrarily large.	result 0.98	0.83	0.14	asserted 0.93
s9	The numerical experiments corroborate reductions in the number of iterations and the communication cost that are necessary to achieve convergence relative to distributed gradient descent.	result 0.99	0.93	0.37	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.68 · evidence from proof or derivation 0.77.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The use of network Newton methods for the decentralized optimization of a sum cost distributed through agents of a network is considered. Network Newton methods reinterpret distributed gradient descent as a penalty method, observe that the corresponding Hessian is sparse, and approximate the Newton step by truncating a Taylor expansion of the inverse Hessian. Truncating the series at $K$ terms yields the NN-$K$ that requires aggregating information from $K$ hops away. Network Newton is introduced and shown to converge to the solution of the penalized objective function at a rate that is at least linear in a companion paper [3]. The contributions of this work are: (i) To complement the convergence analysis by studying the methods' rate of convergence. (ii) To introduce adaptive formulations that converge to the optimal argument of the original objective. (iii) To perform numerical evaluations of NN-$K$ methods. The convergence analysis relates the behavior of NN-$K$ with the behavior of (regular) Newton's method and shows that the method goes through a quadratic convergence phase in a specific interval. The length of this quadratic phase grows with $K$ and can be made arbitrarily large. The numerical experiments corroborate reductions in the number of iterations and the communication cost that are necessary to achieve convergence relative to distributed gradient descent.

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11 arXiv papers cite this in 25 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 13 sentences without a confident label

Builds on it 3 papers

Our paper builds upon the network Newton algorithm presented in [ref] .

A Fast Distributed Asynchronous Newton-Based Optimization Algorithm · Our Contribution · 0.97
In this paper, we extend [ref] and propose a different family of distributed Newton-like methods for solving ([ref]).

Newton-like method with diagonal correction for distributed optimization · Introduction · 1.00
To develop a decentralized protocol to approximately execute [ref], we generalize a recently proposed technique to approximate the Hessian inverse $[∇_ F(_k; t_k)]^-1$ which operates by truncating its Taylor expansion [ref] .

Decentralized Prediction-Correction Methods for Networked Time-Varying Convex Optimization · Decentralized prediction step · 0.83

Uses its method or tool 1 paper

ESOM resolves this issue by using the Hessian inverse approximation technique introduced in [ref] .

A Decentralized Second-Order Method with Exact Linear Convergence Rate for Consensus Optimization · Introduction · 0.99

Compares with it 1 paper

The clustering of the performance of NN-$0$, NN-$1$, and NN-$2$ (although not so pronounced as in our examples) emerges also in Simulations in [ref] .

Newton-like method with diagonal correction for distributed optimization · Simulations · 0.92

Cites it as background 6 papers

The class of Netwrok Newton method, refereed to as NN, is extensively analyzed in [ref] .

Newton-like method with diagonal correction for distributed optimization · Introduction · 0.91
Network Newton (NN) is a more recent alternative that accelerates convergence of DGD by incorporating second order information of the penalized objective [ref] .

A Decentralized Second-Order Method with Exact Linear Convergence Rate for Consensus Optimization · Introduction · 0.84
These methods include decentralized gradient descent (DGD) ( [ref] ), network Newton ( [ref] ), decentralized dual averaging ( [ref] ), the exact first order algorithm (EXTRA) ( [ref] ), as well as the alternating direction method of multipliers (ADMM) ( [ref] ) and its linearized variants ( [ref] ).

DSA: Decentralized Double Stochastic Averaging Gradient Algorithm · Introduction · 0.99

What it cites · 1 sentence without a confident label

Compares with 1 paper

This result is identical to the convergence behavior of DGD as shown in, e.g., [ref] .

On the Convergence of Decentralized Gradient Descent · Convergence Rate · 1.00

Cites as background 2 papers

These include distributed gradient descent (DGD) [ref] as well as distributed implementations of the alternating direction method of multipliers [ref] and dual averaging [ref] .

On the Convergence of Decentralized Gradient Descent · Introduction · 0.97
These include distributed gradient descent (DGD) [ref] as well as distributed implementations of the alternating direction method of multipliers [ref] and dual averaging [ref] .

EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization · Introduction · 0.97

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