The document proposes a Riemannian gossip algorithm for decentralized matrix completion. Each agent has its own data matrix and aims to complete the matrix while reaching consensus on the common factor matrix U with other agents. The optimization problem is formulated on a Grassmann manifold by minimizing a weighted combination of completion and consensus terms. A parallel variant of the gossip algorithm is also developed, which converges at the same rate as the original algorithm. Numerical tests on synthetic and Netflix data show the algorithm achieves good performance compared to benchmark methods.
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Maximum a posteriori (MAP) inference is a particularly complex type of probabilistic inference in Bayesian networks. It consists of finding the most probable configuration of a set of variables of interest given observations on a collection of other variables. In this paper we study scalable solutions to the MAP problem in hybrid Bayesian networks parameterized using conditional linear Gaussian distributions. We propose scalable solutions based on hill climbing and simulated anneal- ing, built on the Apache Flink framework for big data processing. We analyze the scalability of the solution through a series of experiments on large synthetic networks.
Full text paper: http://www.jmlr.org/proceedings/papers/v52/ramos-lopez16.pdf
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Scalable MAP inference in Bayesian networks based on a Map-Reduce approach (P...AMIDST Toolbox
Maximum a posteriori (MAP) inference is a particularly complex type of probabilistic inference in Bayesian networks. It consists of finding the most probable configuration of a set of variables of interest given observations on a collection of other variables. In this paper we study scalable solutions to the MAP problem in hybrid Bayesian networks parameterized using conditional linear Gaussian distributions. We propose scalable solutions based on hill climbing and simulated anneal- ing, built on the Apache Flink framework for big data processing. We analyze the scalability of the solution through a series of experiments on large synthetic networks.
Full text paper: http://www.jmlr.org/proceedings/papers/v52/ramos-lopez16.pdf
Solving Fuzzy Maximal Flow Problem Using Octagonal Fuzzy NumberIJERA Editor
In this paper a general fuzzy maximal flow problem is discussed . A crisp maximal flow problem can be solved
in two methods : linear programming modeling and maximal flow algorithm . Here I tried to fuzzify the
maximal flow algorithm using octagonal fuzzy numbers introduced by S.U Malini and Felbin .C. kennedy [26].
By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
it is solved using fuzzy octagonal number than when it is solved using trapezoidal fuzzy number . To illustrate
this a numerical example is solved and the obtained result is compared with the existing results . If there is no
uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp
maximal flow problems.
It is a short project on the Boston Housing dataset available in R. It shows the variables in the dataset and its interdependencies. A Regression Model is created taking some of the most dependent variables and adjusted to make a best possible fit. Lastly, the variances are analysed and adjusted.
Block-Wise Density Distribution of Primes Less Than A Trillion in Arithmetica...paperpublications3
Abstract: Primes in arithmetical progressions 11n + k are considered for their comparative abundance in the ten power blocks of 1-10n for 1 ≤ n ≤ 12. In each of this block, the first and the last primes of respective forms are given. For inspecting scarcity of primes, minimum number of primes of these forms in these blocks, the first and the last blocks of minimality and their frequency within the limit of one trillion are determined. This analysis is also carried out for the maximal prime container blocks.
Pastor Elio Marrocco's "Gossip" sermon at New Life Christian Church on February 15, 2013. You can learn more about New Life Christian Church here: http://www.newlifecc.ca
Pastor Elio Marrocco's "Gossip" sermon at New Life Christian Church on February 15, 2013. You can learn more about New Life Christian Church here: http://www.newlifecc.ca
Riemannian stochastic variance reduced gradient on Grassmann manifold (ICCOPT...Hiroyuki KASAI
Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance reduced gradient algorithm (R-SVRG) to a compact manifold search space. To this end, we show the developments on the Grassmann manifold. The key challenges of averaging, addition, and subtraction of multiple gradients are addressed with notions like logarithm mapping and parallel translation of vectors on the Grassmann manifold. We present a global convergence analysis of the proposed algorithm with a decay step-size and a local convergence rate analysis under a fixed step-size with under some natural assumptions. The proposed algorithm is applied on a number of problems on the Grassmann manifold like principal components analysis, low-rank matrix completion, and the Karcher mean computation. In all these cases, the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm.
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A lambda calculus for density matrices with classical and probabilistic controlsAlejandro Díaz-Caro
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Motivated by a real-world financial dataset, we propose a distributed variational message passing scheme for learning conjugate exponential models. We show that the method can be seen as a projected natural gradient ascent algorithm, and it therefore has good convergence properties. This is supported experimentally, where we show that the approach is robust wrt. common problems like imbalanced data, heavy-tailed empirical distributions, and a high degree of missing values. The scheme is based on map-reduce operations, and utilizes the memory management of modern big data frameworks like Apache Flink to obtain a time-efficient and scalable implementation. The proposed algorithm compares favourably to stochastic variational inference both in terms of speed and quality of the learned models. For the scalability analysis, we evaluate our approach over a network with more than one billion nodes (and approx. 75% latent variables) using a computer cluster with 128 processing units.
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Riemannian gossip algorithms for decentralized matrix completion
1. Riemannian gossip algorithms for
decentralized matrix completion
Hiroyuki Kasai†
, Bamdev Mishra‡
, and Atul Saroop‡
†The University of Electro-Communications, Japan
‡Amazon Development Centre India, India
IEICE meeting 2016
2. Motivation
The matrix completion problem
? ? * ?
* * ? *
? * * ?
* ? * ?
≈
Low-rank prior
m Movies
n Users
m
nr
(n + m − r)r, r ≪ (m, n)
WT
U
X?
U and W factor matrices.
[Netflix Challenge, 2006]
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 2 / 25
3. Motivation
Our interest is to look at the decentralized scenario
? ? * ?
* * ? *
? * * ?
* ? * ?
m Movies
n1 Users n2 Users
X?
1 X?
2
U[WT
1 WT
2 ]≈
An agent i has access to its own data matrix Xi .
The matrix U is common across all the agents.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 3 / 25
4. Motivation
Contributions
We develop a nonlinear gossip algorithm with minimal
communication between agents.
The optimization formulation is based on a weighted combination of
matrix completion and consensus terms.
We develop a parallel variant of the proposed gossip algorithm.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 4 / 25
5. Motivation
Paper and codes available online
at
www.bamdevmishra.com.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 5 / 25
6. Motivation
Outline
Problem formulation on the Riemannian Grassmann manifold.
Proposed gossip algorithms.
Numerical comparisons on synthetic and Netflix data.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 6 / 25
7. Problem formulation
Outline
Problem formulation on the Riemannian Grassmann manifold.
Proposed gossip algorithms.
Numerical comparisons on synthetic and Netflix data.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 7 / 25
8. Problem formulation
Batch problem formulation
min
U∈St(r,m)
min
W∈Rn×r
PΩ(UWT
) − PΩ(X ) 2
F .
W ∈ Rn×r
and
U ∈ St(r, m), the set of m × r matrices with orthonormal columns.
PΩ is the sampling operator, a convenient way to denote known
entries.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 8 / 25
9. Problem formulation
Eliminate W
minU∈St(r,m) minW∈Rn×r PΩ(UWT
) − PΩ(X ) 2
F
≡
minU∈St(r,m) f (U, WU), a Grassmann optimization problem.
Solve blue problem in closed form to obtain WU.
Final optimization problem is on Grassmann manifold, i.e.,
variable is ‘column space’ of U.
[Boumal and Absil, LAA, 2015]
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 9 / 25
10. Problem formulation
Decentralized problem formulation
X = [X1, X2, . . . , XN].
i
min
U∈St(r,m),Wi ∈Rni ×r
1
2
PΩi
(UWi
T
) − PΩi
(Xi ) 2
F
= min
U∈St(r,m)
1
2 i
PΩi
(UWT
iU) − PΩi
(Xi ) 2
F ,
where WiU is computed by agent i independently.
Although the problem is distributed, we still need to learn a common
U.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 10 / 25
11. Problem formulation
We add a consensus term to our optimization
formulation
Key idea: introduce multiple copies of U among N agents, but allow
them to reach consensus.
min
U1,...,UN ∈St(r,m)
1
2 i
PΩi
(Ui WT
iUi
) − PΩi
(Xi ) 2
F
completion task handled by agent i
+
ρ
2
(d(U1, U2)2
+ d(U2, U3)2
+ . . . + d(UN−1, UN)2
)
consensus among agents
.
d is the Riemannian distance on the Grassmann manifold.
A large ρ trades-off completion with consensus.
Minimizing only consensus ⇒ U1 = U2 = . . . = UN−1 = UN.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 11 / 25
12. Proposed Riemannian gossip algorithms
Outline
Problem formulation on the Riemannian Grassmann manifold.
Proposed gossip algorithms.
Numerical comparisons on synthetic and Netflix data.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 12 / 25
13. Proposed Riemannian gossip algorithms
Riemannian online gossip on Grassmann
1 Agents i and i + 1 are neighbors for all i N − 1. (ordering of
agents)
2 At each time slot, say t, we pick an agent i N − 1 randomly
with uniform probability. (SGD updates)
Equivalently, we also pick agent i + 1 (the neighbor of agent i).
Agents i and i + 1 update Ui and Ui+1, respectively, by taking
a gradient descent step with stepsize γt on Grassmann manifold.
γ2
t < ∞ and γt = +∞.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 13 / 25
14. Proposed Riemannian gossip algorithms
A graphical illustration
Agent 1 Agent 2 Agent 3 Agent N-1 Agent N
Universal clock
Each pair is chosen
. . .
with probability 1=(N − 1)
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 14 / 25
15. Proposed Riemannian gossip algorithms
Convergence of Riemannian online gossip
Asymptotic convergence follows standard SGD analysis on manifold.
The proposed algorithm is readily implementable, e.g., with the
toolbox Manopt.
[Bonnabel, IEEE TAC, 2013; Absil, Mahoney, and Sepulchre,
Princeton Press, 2008; Boumal et al., JMLR, 2014]
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 15 / 25
16. Proposed Riemannian gossip algorithms
Parallelizing Riemannian gossip with particular
sampling
Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
Universal clock
Solid" pairs chosen at same time.
. . .
Dotted' pairs chosen at same time.
“Dotted’ and “solid” groups are chosen with probability 1/2.
Convergence guarantees remain the same.
(N − 1)/2 times faster.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 16 / 25
17. Numerical comparisons
Outline
Problem formulation on the Riemannian Grassmann manifold.
Proposed gossip algorithms.
Numerical comparisons on synthetic and Netflix data.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 17 / 25
18. Numerical comparisons
Effect of ρ
10 000×100 000 matrix with N = 6.
0 10 20 30 40
Every 10th update
10-10
10-8
10
-6
10-4
10
-2
100
Meansquareerrorontrainingset
Agent 1 rho=103
Agent 2, rho=103
Distance 1-2, rho=103
Agent 1, rho=10
10
Agent 2, rho=1010
Distance 1-2, rho=1010
A very large ρ minimizes only achieves consensus among agents.
A tuned ρ achieves both completion and consensus.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 18 / 25
19. Numerical comparisons
Performance of online and parallel variants
10 000×100 000 matrix with N = 6 and ρ = 103
.
0 10 20 30 40
Every 10th update
10-10
10-8
10
-6
10-4
10
-2
100
Meansquareerrorontrainingset
Onl. agent 1
Onl. agent 2
Onl. distance 1 - 2
Para. agent 1
Para. agent 2
Para. distance 1 - 2
There is no loss of performance in parallelizing the updates.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 19 / 25
20. Numerical comparisons
Comparison with D-LMaFit
500 × 12000 and N = 6.
0 10 20 30 40
Every 10th update
10-12
10-10
10-8
10-6
10
-4
10-2
100
102Meansquareerrorontrainingset
Online agent 1
Online agent 2
Online distance 1 - 2
D-LMaFit agent 1
D-LMaFit agent 2
D-LMaFit distance 1 - 2
D-LMaFit code is not scalable to large data.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 20 / 25
21. Numerical comparisons
Netflix data: different number of agents
Rank = 10
N = {2, 5, 10, 15, 20} agents
10 random 80/20 - train/test - 80/20 million split
Online gossip:
N = 2 N = 5 N = 10 N = 15 N = 20
Test
RMSE
0.877 0.885 0.891 0.894 0.900
Batch gradient descent algorithm RTRMC benchmark: 0.873.
[Boumal and Absil, LAA, 2015]
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 21 / 25
23. Numerical comparisons
Netflix data: test RMSE with updates
0 20 40 60 80
0.85
0.9
0.95
1
1.05
1.1
TestRMSE
Agent 1
Agent 2
Agent 3
RTRMC−1
Every 10th update
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 23 / 25
24. Summary and future work
Summary and future work
We proposed a Riemannian gossip approach to the decentralized
matrix completion problem.
We minimize weighted sum of completion and consensus terms
on Grassmann manifold.
Numerical comparisons show good performance of the proposed
algorithms, e.g., on Netflix dataset.
Currently, we intend to explore asynchronous updating of agents
on Grassmann manifold.
Kasai, Mishra, and Saroop Riemannian gossip 20 September 2016 24 / 25
25. Riemannian gossip algorithms for
decentralized matrix completion
Hiroyuki Kasai†
, Bamdev Mishra‡
, and Atul Saroop‡
†The University of Electro-Communications, Japan
‡Amazon Development Centre India, India
IEICE meeting 2016