Decentralized Network Optimization Algorithms and Theories Qing Ling

Motivation: network data processing We are in a world of networks and in a

Outline p p p p Decentralized network optimization Background applications Problem statement and related

Decentralized network: our future? 100, 000 users connected by Fire. Chat, Hong. Kong, September

Example: multi-sensor target localization 6

More examples Network Machine Learning Magnus Egerstedt (Ga. Tech) www. youtube. com/watch? v=gs. NHJw.

Example: multi-sensor target localization 11

Tradeoff in designing decentralized algorithms How fast is the convergence? Convergence Speed Where do

Theories for algorithms & from theories to algorithms 15

Assumptions in designing and analyzing algorithms 18

ADMM: alternating direction method of multipliers 19

Simulation results of DADMM: linear convergence 24

Simulation results of DADMM: topology versus speed 25

Analyzing DGM through reinterpretation 30

Assumptions in designing and analyzing algorithms 37

Approximate Hessian inverse on decentralized network 45

Linear and quadratic convergence rates of NN 47

EXact firs. T-orde. R Algorithm (EXTRA) 51

Summarizing DLM, NN & EXTRA DADMM Convergence Speed Solution Accuracy NN DLM EXTRA DGM

Decentralized Optimization Summarizing the whole work Primal Domain Methods Network Newton Dual Domain Methods

Concluding remarks and future research directions Thank you 61

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Decentralized Network Optimization: Algorithms and Theories Qing Ling Department of Automation, University of Science and Technology of China (USTC) Joint work with Aryan Mokhtari (Penn), Alejandro Ribeiro (Penn), Wei Shi (USTC, now UIUC), Gang Wu (USTC), Wotao Yin (UCLA), Kun Yuan (USTC, now UCLA) Institute of System Science, Chinese Academy of Sciences (CAS) 2015/12/14 1

Motivation: network data processing We are in a world of networks and in a sea of networked data Autonomous agents - Collect data - Process data - Communicate RESERCH PROBLEM: how to efficiently accomplish in-network information processing tasks through collaboration of agents? OUR FOCUS: decentralized optimization, control, and decision-making

Outline p p p p Decentralized network optimization Background applications Problem statement and related works With particular focus on decentralized consensus optimization Analysis of existing algorithms Dual domain: decentralized ADMM (DADMM) Primal domain: decentralized gradient method (DGM) p Design and analysis of new algorithms Dual domain: decentralized linearized ADMM (DLM) Primal domain: network Newton (NN) Cross primal and dual domains: exact first-order algorithm (EXTRA) p Concluding remarks and future research directions p p p 3

Decentralized network: our future? 100, 000 users connected by Fire. Chat, Hong. Kong, September 2014 - No transmission to fusion center (bandwidth, latency, privacy, etc) - Decentralized processing via collaboration of neighboring agents

Example: multi-sensor target localization 6

Example: network flow optimization 7

More examples Network Machine Learning Magnus Egerstedt (Ga. Tech) www. youtube. com/watch? v=gs. NHJw. A 7 V-U Wireless sensing - Temperature - Humidity - Other factors Wireless actuating - Circulating fan - Wet curtain - Other actuators Wireless Precise Agriculture Vijay Kumar (Penn) www. youtube. com/watch? v=4 Er. EBkj_3 PY Robot & Drone Networks 8

Decentralized consensus optimization 10

Example: multi-sensor target localization 11

Example: network flow optimization 12

Tradeoff in designing decentralized algorithms How fast is the convergence? Convergence Speed Where do the iterates converge? Solution Accuracy Computation Cost Is the computation affordable? 13

Related works 14

Theories for algorithms & from theories to algorithms 15

References of DADMM and DGM 17

Assumptions in designing and analyzing algorithms 18

ADMM: alternating direction method of multipliers 19

Reformulating to use ADMM 20

Outline of DADMM 21

Linear convergence rate of DADMM 22

Simulation settings of DADMM 23

Simulation results of DADMM: linear convergence 24

Simulation results of DADMM: topology versus speed 25

Decentralized gradient method (DGM) 26

Mixing matrix 27

Existing convergence analysis 28

Reinterpreting DGM

Analyzing DGM through reinterpretation 30

Linear convergence rate of DGM 31

Simulation settings of DGM 32

Simulation results of DGM 33

Summarizing DADMM & DGM 34

References of DLM, NN, and EXTRA

Assumptions in designing and analyzing algorithms 37

Decentralized linearized ADMM (DLM) 38

From DADMM to DLM

Outline of DLM 40

Linear convergence rate of DLM 41

Simulation settings of DLM 42

Simulation results of DLM 43

From DGM to network Newton (NN) 44

Approximate Hessian inverse on decentralized network 45

Outline of NN-T 46

Linear and quadratic convergence rates of NN 47

Simulation settings of NN 48

Simulation results of NN 49

Revisiting DGM 50

EXact firs. T-orde. R Algorithm (EXTRA) 51

Mixing matrices 52

Explanations of EXTRA 53

Sublinear convergence of EXTRA 54

Linear convergence of EXTRA 55

Simulation settings of EXTRA 56

Simulation of EXTRA 57

Summarizing DLM, NN & EXTRA DADMM Convergence Speed Solution Accuracy NN DLM EXTRA DGM Computation Cost 58

Decentralized Optimization Summarizing the whole work Primal Domain Methods Network Newton Dual Domain Methods Gradient Method EXTRA DADMM Correction Penalty Methods DLM Dilemma Decaying Stepsize Slow Convergence Accurate Solution Fixed Stepsize Fast Convergence Inaccurate Solution Fixed Stepsize Fast Convergence Accurate Solution Linear Convergence 60

Concluding remarks and future research directions Thank you 61