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 M ij 3 4 ? 9 ? ? ? 8 ? ? 2

Euclidean Embedding Given measured delays, embed nodes into a Euclidean space Finds the co-ordinate x i of each node in the space Distance between two nodes i and j predicts their latency ˆ M ij = k x i � x j k Variations: 2D, 3D, 5D, 7D spaces, or with a Height parameter F. Dabek, R. Cox, F. Kaashoek, and R. Morris, “Vivaldi: A decentralized network coordinate system,” in ACM SIGCOMM, 2004. 3

Matrix Factorization Factorize the latency matrix into 2 low-dimensional matrices M = [ M ij ] ∈ R n × n M = M T M l , M r ∈ R r × n l M r , If M is complete, can use Singular Value Decomp to factorize With missing entries, usually use Gradient Descent Guess the factors, fi nd 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. 4

Are these good enough to handle personal devices?

Existing Approaches can’t Handle… Violation of Triangle Inequality [Liao13 et al.] 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 6

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 7

Personal Devices vs. Desktops 1 1 0.8 0.8 0.6 0.6 CDF CDF 0.4 0.4 0.2 0.2 Seattle Seattle PlanetLab PlanetLab 0 0 0 0.5 1 1.5 2 0 1 2 3 4 RTT (second) Max RTT (second) (a) RTT distributions (b) Max RTT for each pair of nodes 8

Personal Devices vs. Desktops Asymmetric Latencies 1 0.75 CDF 0.5 0.25 |RTT(i,j) − RTT(j,i)| Seattle RTT 0 0 0.5 1 1.5 2 RTT (Second) (a) | RTT( i, j ) � RTT( j, i ) | (Seattle) 9

Personal Devices vs. Desktops Low-rank structures: heat maps of latency matrices 1 1 20 100 Node Index Node Index 40 200 60 300 80 400 99 490 1 20 40 60 80 99 1 100 200 300 400 490 Node Index Node Index (a) A Seattle RTT matrix (b) A PlanetLab RTT matrix 10

Personal Devices vs. Desktops Low-rank structures: singular values of latency matrices 80 80 60 60 Magnitude Magnitude 40 40 20 20 0 0 1 5 10 15 20 1 5 10 15 20 Singular Value Singular Value (c) A Seattle RTT matrix (d) A PlanetLab RTT matrix 11

Personal Devices vs. Desktops Time-varying latencies between nodes 1.5 Seattle Pair 1 Seattle Pair 2 Seattle Pair 3 RTT (second) 1 0.5 0 1 30 60 90 120 Time Frame Index PlanetLab Pair 1 PlanetLab Pair 2 PlanetLab Pair 3 RTT (second) 0.4 0.2 0 1 4 7 10 13 16 Time Frame Index 12

Innovation 1: Distance-Feature Decomposition

Distance-Feature Decomposition Assume there is a distance matrix D , and a network feature matrix F M = D � F � M ij = D ij F ij � D ij models the geo distance (Euclidean part) F ij 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 14

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: rank(ˆ minimize F ) ˆ F ∈ R m × n | ˆ subject to F ij − F ij | ≤ τ , ( i, j ) / ∈ Θ , 15

Distance-Feature Decomposition Algorithm 1 Iterative Distance-Feature Decomposition 1: D 0 := M 2: for k = 1 to maxIter do Perform Euclidean Embedding on D k − 1 to get the 3: complete matrix of distance estimates ˆ D k ( M ij 8 ( i, j ) / 2 Θ ˆ D k F k ij := 4: ij 8 ( i, j ) 2 Θ unknown Perform Matrix Completion (4) on F k to get the 5: complete matrix of network feature estimates ˆ F k ( M ij 8 ( i, j ) / 2 Θ ˆ F k D k ij := 6: ij 8 ( i, j ) 2 Θ unknown 7: end for M ij := ˆ ˆ ˆ D maxIter F maxIter , 1  i, j  n 8: ij ij The rank minimization part can be done using, e.g., singular value thresholding (SVT) by Candes’ group, Penalty Decomp,… 16

Distance-Feature Decomposition 60 100 Distance Matrix D Distance Matrix D Magnitude (Second) Magnitude (Second) Network Feature Matrix F Network Feature Matrix F 80 RTT Matrix M RTT Matrix M 40 60 40 20 20 0 0 1 10 99 1 10 100 490 Singular Value Singular Value ) (b) Singular values (Seattle) (c) Singular values (PlanetLab) Singular values of decomposed matrices D and F E ff ect is more signi fi cant for Seattle 17

Results (D-F Decomposition) 1 0.8 0.6 CDF Algorithm 1 (D − F Decomposition) DMFSGD Matrix Factorization 0.4 PD Matrix Completion Vivaldi (7D) 0.2 Vivaldi (3D) Vivaldi (3D + Height) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Relative Error (a) Sample rate R = 0 . 3 Seattle low sample rate (most entries unknown) 18

Results (D-F Decomposition) 1 0.8 0.6 CDF Algorithm 1 (D − F Decomposition) DMFSGD Matrix Factorization 0.4 PD Matrix Completion Vivaldi (7D) 0.2 Vivaldi (3D) Vivaldi (3D + Height) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Relative Error (b) Sample rate R = 0 . 7 Seattle high sample rate (most entries are known) 19

Innovation 2: Dynamic Prediction based on 3D Data

Personal Devices vs. Desktops Time-varying latencies between nodes 1.5 Seattle Pair 1 Seattle Pair 2 Seattle Pair 3 RTT (second) 1 0.5 0 1 30 60 90 120 Time Frame Index PlanetLab Pair 1 PlanetLab Pair 2 PlanetLab Pair 3 RTT (second) 0.4 0.2 0 1 4 7 10 13 16 Time Frame Index 21

Multi-Frame Completion Problem Extraction from A Tensor Collect multiple frames of latency matrices or M = ( M ijt ) 2 R n × n × T � Each frame represents data sampled within a time period t � Estimate the current frame using past information Removing distance component , becomes tensor completion: � minimize rank( X ) ˆ X ∈ R n × n × T � | ˆ subject to X ijt � X ijt |  τ , ( i, j, t ) 2 Ω , � � Computing the rank of a tensor is NP-hard 22

Matrix Approximation to Tensor Solve a simpler problem based on “stacking”or “unfolding” 3 X minimize α l · rank( X ( l ) ) ˆ X ∈ R n × n × T l =1 | ˆ subject to X ijt � X ijt |  τ , ( i, j, t ) 2 Ω , X (1) I 2 · I 3 X I 1 I 1 I 3 I 3 I 3 I 3 I 2 X (2) I 3 · I 1 X I 2 I 1 I 3 I 1 I 1 I 1 I 2 X (3) I 1 · I 2 X I 3 I 1 I 3 I 2 I 2 I 2 I 2 23

The question is— How to stack or unfold?

Di ff erentiation in Treatment Θ A = { ( i, j ) |M ijt is known for at least one t ∈ { 1 , . . . , T − 1 }} , Θ B = { ( i, j ) |M ijt is missing for all t ∈ { 1 , . . . , T − 1 }} . Current Frame Θ A M 12 ? Time M 12 ? M 11 ? ? ? ? ? ? M 11 M 12 M 11 M 12 M 21 ? ? M 12 M 12 M 12 ? ? ? ? M 12 ? T = 3 M 12 ? ? ? ? M 21 ? M 21 ? Θ B Frame-Stacked Matrix Original Matrices Column-Stacked Matrix For use column-stacking Θ A For use frame-stacking, then apply D-F Decomposition Θ B 25

Rank of Column-Stacked Matrix 800 1 Node Pair Index 2000 600 Magnitude 4000 400 6000 200 8000 9801 0 1 100 300 500 688 1 20 40 60 80 99 Time Frame Index Singular Value (a) Column-Stacked Matrix Heat Map (b) Column-Stacked Matrix SVD 26

Recovery Results on the Current Frame of 3D Data 1 0.8 0.6 CDF 0.4 Vivaldi (3D) DMFSGD Matrix Factorization 0.2 Algorithm 1 Algorithm 2 0 0 1 2 3 4 5 Relative Error (a) Sample rate R = 0 . 3 Seattle low sample rate (most entries are unknown) 27

Recovery Results on the Current Frame of 3D Data 1 0.8 0.6 CDF 0.4 Vivaldi (3D) DMFSGD Matrix Factorization 0.2 Algorithm 1 Algorithm 2 0 0 1 2 3 4 5 Relative Error (d) Sample rate R = 0 . 7 Seattle high sample rate (most entries are known) 28

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 fi rst 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? 29