Network Latency Prediction for Personal Devices: Distance-Feature - - PowerPoint PPT Presentation

Network Latency Prediction for Personal Devices: Distance-Feature Decomposition from 3D Sampling

Presented by Di Niu

Assistant Professor University of Alberta, Canada

• An INFOCOM ‘15 paper by Bang Liu, Di Niu, Zongpeng Li, and H. Vicky Zhao

Internet Latency Prediction

The most common approach is via network tomography

Measure the end-to-end latency between some points (i, j) Use measured latencies to recover all end-to-end latencies Mij

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? ? ? ? ? ? 4 3 8 9

Euclidean Embedding

Given measured delays, embed nodes into a Euclidean space

Finds the co-ordinate xi of each node in the space Distance between two nodes i and j predicts their latency

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• F. Dabek, R. Cox, F. Kaashoek, and R. Morris, “Vivaldi: A decentralized network

coordinate system,” in ACM SIGCOMM, 2004.

ˆ Mij = kxi xjk Variations: 2D, 3D, 5D, 7D spaces, or with a Height parameter

Matrix Factorization

Factorize the latency matrix into 2 low-dimensional matrices

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M = MT

l Mr,

Ml, Mr ∈ Rr×n M = [Mij] ∈ Rn×n If M is complete, can use Singular Value Decomp to factorize With missing entries, usually use Gradient Descent

Guess the factors, find the error Compute error gradients w.r.t. factors update factors according to the gradients

Liao, Du, Geurts, Leduc, “DMFSGD: A decentralized matrix factorization algorithm for network distance prediction,” IEEE/ACM Transactions on Networking, 2013.

Are these good enough to handle personal devices?

Existing Approaches can’t Handle…

Violation of Triangle Inequality [Liao13 et al.]

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et- hurn sive w ut

Asymmetric latencies Unknown rank of the latency matrix M

But the rank must be known in matrix factorization

Unstable and time-varying latencies

All current methods are for mean/static latency prediction

Data Collected

Planetlab: donated computers/servers mostly in universities

490 nodes, 9 day period 18 frames in total 14.7 hour per frame

• Seattle: consists of donated personal devices (iPhone, Android,

laptops) 99 nodes, in a 3 hour-period 688 frames in total 15.7 seconds per frame

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Personal Devices vs. Desktops

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0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 RTT (second) CDF

Seattle PlanetLab

(a) RTT distributions

1 2 3 4 0.2 0.4 0.6 0.8 1 Max RTT (second) CDF Seattle PlanetLab

(b) Max RTT for each pair of nodes

Personal Devices vs. Desktops

Asymmetric Latencies

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0.5 1 1.5 2 0.25 0.5 0.75 1 RTT (Second) CDF |RTT(i,j) − RTT(j,i)| Seattle RTT

(a) |RTT(i, j) RTT(j, i)| (Seattle)

Personal Devices vs. Desktops

Low-rank structures: heat maps of latency matrices

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Node Index Node Index 1 20 40 60 80 99 1 20 40 60 80 99

(a) A Seattle RTT matrix

Node Index Node Index 1 100 200 300 400 490 1 100 200 300 400 490

(b) A PlanetLab RTT matrix

Personal Devices vs. Desktops

Low-rank structures: singular values of latency matrices

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1 5 10 15 20 20 40 60 80 Singular Value Magnitude

(c) A Seattle RTT matrix

1 5 10 15 20 20 40 60 80 Singular Value Magnitude

(d) A PlanetLab RTT matrix

Personal Devices vs. Desktops

Time-varying latencies between nodes

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1 30 60 90 120 0.5 1 1.5 Time Frame Index RTT (second) Seattle Pair 1 Seattle Pair 2 Seattle Pair 3 1 4 7 10 13 16 0.2 0.4 Time Frame Index RTT (second) PlanetLab Pair 1 PlanetLab Pair 2 PlanetLab Pair 3

Innovation 1: Distance-Feature Decomposition

Distance-Feature Decomposition

Assume there is a distance matrix D, and a network feature matrix F

• Dij models the geo distance (Euclidean part)

Fij models the network connectivity (non-Euclidean part)

Network connectivities are highly correlated Allow asymmetry and TIV (triangle inequality violation)

Combine the strengths of Euclidean embedding and low-rank matrix completion

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Mij = DijFij M = D F

Distance Feature Decomposition

Input: Incomplete RTT measurement matrix M Output: Compete RTT estimate matrix Algorithm Perform Euclidean Embedding on M to get D Get the incomplete remainder F = M/D Perform Rank Minimization to complete the matrix F:

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minimize

ˆ F∈Rm×n

rank(ˆ F) subject to | ˆ Fij − Fij| ≤ τ, (i, j) / ∈ Θ,

Distance-Feature Decomposition

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Algorithm 1 Iterative Distance-Feature Decomposition

1: D0 := M 2: for k = 1 to maxIter do 3:

Perform Euclidean Embedding on Dk−1 to get the complete matrix of distance estimates ˆ Dk

4:

F k

ij :=

( Mij

ˆ Dk

ij

8(i, j) / 2 Θ unknown 8(i, j) 2 Θ

5:

Perform Matrix Completion (4) on F k to get the complete matrix of network feature estimates ˆ F k

6:

Dk

ij :=

( Mij

ˆ F k

ij

8(i, j) / 2 Θ unknown 8(i, j) 2 Θ

7: end for 8:

ˆ Mij := ˆ DmaxIter

ij

ˆ F maxIter

ij

, 1  i, j  n

The rank minimization part can be done using, e.g., singular value thresholding (SVT) by Candes’ group, Penalty Decomp,…

Distance-Feature Decomposition

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)

1 10 99 20 40 60 Singular Value Magnitude (Second) Distance Matrix D Network Feature Matrix F RTT Matrix M

(b) Singular values (Seattle)

1 10 100 490 20 40 60 80 100 Singular Value Magnitude (Second) Distance Matrix D Network Feature Matrix F RTT Matrix M

(c) Singular values (PlanetLab)

Singular values of decomposed matrices D and F Effect is more significant for Seattle

Results (D-F Decomposition)

Seattle low sample rate (most entries unknown)

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.4 0.6 0.8 1 Relative Error CDF Algorithm 1 (D−F Decomposition) DMFSGD Matrix Factorization PD Matrix Completion Vivaldi (7D) Vivaldi (3D) Vivaldi (3D + Height)

(a) Sample rate R = 0.3

Results (D-F Decomposition)

Seattle high sample rate (most entries are known)

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.4 0.6 0.8 1 Relative Error CDF Algorithm 1 (D−F Decomposition) DMFSGD Matrix Factorization PD Matrix Completion Vivaldi (7D) Vivaldi (3D) Vivaldi (3D + Height)

(b) Sample rate R = 0.7

Innovation 2: Dynamic Prediction based on 3D Data

Personal Devices vs. Desktops

Time-varying latencies between nodes

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1 30 60 90 120 0.5 1 1.5 Time Frame Index RTT (second) Seattle Pair 1 Seattle Pair 2 Seattle Pair 3 1 4 7 10 13 16 0.2 0.4 Time Frame Index RTT (second) PlanetLab Pair 1 PlanetLab Pair 2 PlanetLab Pair 3

Multi-Frame Completion Problem

Collect multiple frames of latency matrices

• Each frame represents data sampled within a time period t
• Estimate the current frame using past information

Removing distance component, becomes tensor completion:

• Computing the rank of a tensor is NP-hard

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Extraction from A Tensor

• r M = (Mijt) 2 Rn×n×T

minimize

ˆ X∈Rn×n×T

rank(X) subject to | ˆ Xijt Xijt|  τ, (i, j, t) 2 Ω,

Matrix Approximation to Tensor

Solve a simpler problem based on “stacking”or “unfolding”

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minimize

ˆ X∈Rn×n×T 3

X

l=1

αl · rank(X(l)) subject to | ˆ Xijt Xijt|  τ, (i, j, t) 2 Ω,

X X X X(1) X(2) X(3)

I1 I1 I1 I1 I1 I1 I2 I2 I2 I2 I2 I2 I3 I3 I3 I3 I3 I3 I1 I2 I3 I2 · I3 I3 · I1 I1 · I2

The question is— How to stack or unfold?

Differentiation in Treatment

For use column-stacking

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ΘA = {(i, j)|Mijt is known for at least one t ∈ {1, . . . , T − 1}}, ΘB = {(i, j)|Mijt is missing for all t ∈ {1, . . . , T − 1}}.

Original Matrices Frame-Stacked Matrix

? M12 ? ? M11 ? M12 ? ? M12 ? M21 M12 ? ? ? M12 ? M11 ? ? M21 M12 ? ? M12 ? M21 M11 M12 ? ?

?

M12

? ?

Column-Stacked Matrix

Time

Current Frame

ΘB ΘA

T = 3

ΘA ΘB For use frame-stacking, then apply D-F Decomposition

Rank of Column-Stacked Matrix

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Time Frame Index Node Pair Index 1 100 300 500 688 1 2000 4000 6000 8000 9801

(a) Column-Stacked Matrix Heat Map

1 20 40 60 80 99 200 400 600 800 Singular Value Magnitude

(b) Column-Stacked Matrix SVD

Recovery Results on the Current Frame of 3D Data

Seattle low sample rate (most entries are unknown)

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1 2 3 4 5 0.2 0.4 0.6 0.8 1 Relative Error CDF Vivaldi (3D) DMFSGD Matrix Factorization Algorithm 1 Algorithm 2

(a) Sample rate R = 0.3

Recovery Results on the Current Frame of 3D Data

Seattle high sample rate (most entries are known)

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1 2 3 4 5 0.2 0.4 0.6 0.8 1 Relative Error CDF Vivaldi (3D) DMFSGD Matrix Factorization Algorithm 1 Algorithm 2

(d) Sample rate R = 0.7

Conclusions and Future Work

Contribution Summary

Studied latency prediction between Personal Devices D-F Decomposition: combines the strengths of Euclidean embedding and matrix completion 3D sampled data: the first dynamic latency estimation scheme for Internet latency recovery

Challenges ahead: Speed and Scale!

How to make matrix completion faster and more scalable? How to make it a distributed/parallel algorithm? How to make tensor approximation fast for fast-changing latencies?

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