1) The document proposes GASGD, a distributed asynchronous stochastic gradient descent algorithm for matrix completion via graph partitioning. 2) GASGD aims to efficiently exploit computer cluster architectures by mitigating load imbalance, minimizing communication, and tuning synchronization frequency among computing nodes. 3) The key contributions of GASGD are a graph partitioning-based input slicing solution for load balancing, reducing shared data usage by leveraging characteristics of bipartite power-law datasets, and introducing a synchronization parameter to trade off communication cost and convergence rate.

1. GASGD: Stochastic Gradient Descent for Distributed
Asynchronous Matrix Completion via Graph Partitioning
Fabio Petroni
Cyber Intelligence and information Security
CIS Sapienza
Department of Computer Control and
Management Engineering Antonio Ruberti,
Sapienza University of Rome -
petroni|querzoni@dis.uniroma1.it
Foster City, Silicon Valley
6th-10th October 2014
and Leonardo Querzoni

2. Matrix Completion & SGD
R
Q
x
P
?
item vector
user vector
LOSS(P, Q) =
X
(Rij Pi Qj )2
+ ...
I Stochastic Gradient Descent works by taking steps
proportional to the negative of the gradient of the LOSS.
I stochastic = P and Q are updated for each given training
case by a small step, toward the average gradient descent.
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3. Scalability
7 Lengthy training stages;
7 high computational costs;
7 especially on large data sets;
7 input data may not fit in main memory.
I goal = e ciently exploit computer cluster architectures.
...
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4. Distributed Asynchronous SGD
R
Q3
P3
Q4
P4
Q2
P2
Q1
P1
1 2 3 4
R1 R4
R3R2
I R is splitted;
I vectors are
replicated;
I replicas
concurrently
updated;
I replicas deviate
inconsistently;
I synchronization.
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5. Bulk Synchronous Processing Model
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6. Challenges 1/2
R
1 2 3 4
R1 R4
R3R2
I 1. Load balance
B ensure that computing nodes are
fed with the same load.
I 2. Minimize communication
B minimize vector replicas.
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7. Challenges 2/2
...
I 3. Tune synchronization frequency
among computing nodes.
I Current implementations synchronize
vector copies:
B continuously during the epoch
(waste of resources);
B once after every epoch
(slow convergence).
I epoch = a single iteration over the ratings.
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8. Contributions
3 We mitigate the load imbalance by proposing an input
slicing solution based on graph partitioning algorithms;
3 we show how to reduce the number of shared data by
properly leveraging known characteristics of the input dataset
(bipartite power-law nature);
3 we show how to leverage the tradeo↵ between
communication cost and algorithm convergence rate by
tuning the frequency of the bulk synchronization phase
during the computation.
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9. Graph representation
I The rating matrix describes a bipartite graph.
x
x x
x
x
x
xx
x
x
x
x
R
I Real data: skewed power-law degree distribution.
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10. Input partitioner
I vertex-cut performs better than edge-cut in power-law graphs.
I Assumption: the input
data doesn’t fit in
main memory.
I Streaming algorithm.
I Balanced k-way
vertex-cut graph
partitioning:
B minimize replicas;
B balance edge load.
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11. Balanced Vertex-Cut Streaming Algorithms
I Hashing: pseudo-random edge assignment;
I Grid: shu✏e and split the rating matrix in identical blocks;
I Greedy: [Gonzalez et al. 2012] and [Ahmed et al. 2013].
Bipartite Aware Greedy Algorithm
I Real word bipartite graphs are often significantly skewed: one
of the two sets is much bigger than the other.
I By perfectly splitting the bigger set it is possible to achieve a
smaller replication factor.
I GIP (Greedy - Item Partitioned)
I GUP (Greedy - User Partitioned)
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12. Evaluation: The Data Sets
Degree distributions:
100
101
102
10
3
10
4
100
101
102
103
104
105
numberofvertices
degree
items
users
MovieLens 10M
100
101
102
10
3
10
4
100
101
102
103
104
105
numberofvertices
degree
items
users
Netflix
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13. Experiments: Partitioning quality
MovieLens
1
10
100
8 16 32 64 128 256
replicationfactor
partitions
hashing
grid
greedy
GIP
GUP
1
10
100
8 16 32 64 128 256
loadstandarddeviation(%)
partitions
hashing
grid
greedy
GIP
GUP
Netflix
1
10
100
8 16 32 64 128 256
replicationfactor
partitions
hashing
grid
greedy
GIP
GUP
1
10
100
8 16 32 64 128 256
loadstandarddeviation(%)
partitions
hashing
grid
greedy
GIP
GUP
I RF
Replication
Factor
I RSD
Relative
Standard
Deviation
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14. Synchronization frequency
f = 1
f = 100
Netflix
5
6
7
10
20
30
40
50
60
0 5 10 15 20 25 30
loss(×10
7
)
epoch
grid
greedy
GUP
centralized
5
6
7
8
9
10
20
30
40
50
60
0 5 10 15 20 25 30
loss(×10
7
)
epoch
grid
greedy
GUP
centralized
I f = synchronization
frequency parameter
B number of
synchronization steps
during an epoch.
I tradeo↵ between
communication cost and
convergence rate.
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15. Evaluation: SSE and Communication cost
MovieLens
1011
1012
10
13
10
14
10
15
100
101
102
103
SSE
frequency
grid, |C|=32
greedy, |C|=32
GUP, |C|=32
grid, |C|=8
greedy, |C|=8
GUP, |C|=8
1
2
3
4
5
6
7
8
9
10
100
101
102
103
104
105
communacationcost(×107
)
frequency
hashing
grid
greedy
GUP
0.01
0.05
0.1
0.15
0.2
0.25
101
Zoom
Netflix
1013
1014
1015
10
16
1017
100
101
102
103
SSE
frequency
grid, |C|=32
greedy, |C|=32
GUP, |C|=32
grid, |C|=8
greedy, |C|=8
GUP, |C|=8
1
2
3
4
5
6
7
8
9
10
100
101
102
103
104
105
106
communacationcost(×10
9
)
frequency
hashing
grid
greedy
GUP
0.01
0.05
0.1
0.15
0.2
10
1
Zoom
I SSE
between
ASGD
variants and
SGD curves
I CC
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16. Communication cost
I T = the training set
I U = users set
I I = items set
I V = U [ I
I C = processing nodes
I RF = replication factor
I RFU = users’ RF
I RFI = items’ RF
RF =
|U|RFU + |I|RFI
|V |
f = 1 ! CC ⇡ 2(|U|RFU + |I|RFI ) = 2|V |RF
f =
|T|
|C|
! CC ⇡ |T|(RFU + RFI )
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17. Conclusions
I three distinct contributions aimed at improving the e ciency
and scalability of ASGD:
1. we proposed an input slicing solution based on graph
partitioning approach that mitigates the load imbalance
among SGD instances (i.e. better scalability);
2. we further reduce the amount of shared data by exploiting
specific characteristics of the training dataset. This provides
lower communication costs during the algorithm execution
(i.e. better e ciency);
3. we introduced a synchronization frequency parameter driving
a tradeo↵ that can be accurately leveraged to further improve
the algorithm e ciency.
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18. Thank you!
Questions?
Fabio Petroni
Rome, Italy
Current position:
PhD Student in Computer Engineering, Sapienza University of Rome
Research Areas:
Recommendation Systems, Collaborative Filtering, Distributed Systems
petroni@dis.uniroma1.it
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