The document discusses the concept of algorithmic data science, which combines theoretical and practical approaches in the field of data science. It emphasizes the necessity of domain expertise, mathematical modeling, and algorithms to solve various scientific questions while proposing methods for efficient data representation and approximations. Additionally, it highlights the importance of sampling techniques for achieving probabilistic approximations in graph analysis.

1. Algorithmic Data Science
=
Theory + Practice
Matteo Riondato – Labs, Two Sigma Investments
@teorionda – http://matteo.rionda.to
IEEE MIT URTC – November 5, 2016
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2. Matteo Riondato
Ph.D. in CS
Working at
Labs, Two Sigma Investments (Research Scientist);
CS Dept., Brown U. (Visiting Asst. Prof.);
Doing research on algorithmic data science;
Tweeting @teorionda;
Reading matteo@twosigma.com;
“Living” at http://matteo.rionda.to.
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3. Conjecture
Let X be a scientific discipline. Then
21st
-century X = datascience (X) + ε .
Partial evidence: “Computational X” exists for many X.
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4. data science : 21st
century = statistics : 20th
century
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5. data science for 21st
century society



questions
data
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6. data science
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7. data science
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8. data science =



1/4 data representation and management
1/4 mathematical and statistical modeling
1/4 computational thinking and algorithms
1/4 domain expertise
Shake well, and strain into a cocktail glass.
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9. domain expertise modeling
management
algorithms
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10. domain expertise modeling
management
algorithms
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11. domain expertise modeling
management
algorithms
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12. algorithmic data science:
=
algorithms for/with:



approximation guarantees
data streams
Spark/MapReduce
sampling
statistical testing
graph analysis
. . . 9 / 24

13. algorithmic data science
=
theory
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14. algorithmic data science
≈
theory + practice
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15. algorithmic data science
=
(theory × practice)(theory×practice)
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16. Example
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17. Scientific question: Find relevant webpages on the web, influential participants in
a email chain, key proteins in a network, . . .
Data representation: represent the data as a graph G = (V , E).
a
h
b
g f e
c d
Modeling question: What are the important nodes in a graph G = (V , E)?
We need f : V → R+ to express the importance of a node.
The higher is f (x), the more important is x ∈ V .
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18. Domain Knowledge / Modeling: Assume that
1) every node wants to communicate with every node; and
2) communication progresses along Shortest Paths (SPs).
Then, the higher the no. of SPs that a node v belongs to, the more important v is.
Definition
For each node x ∈ V , the betweeness b(x) of x is:
b(x) =
1
n(n − 1) u=x=v∈V
σuv (x)
σuv
∈ [0, 1]
• σuv : number of SPs from u to v, u, v ∈ V ;
• σuv (x): number of SPs from u to v that go through x.
I.e., b(x) is weighted fraction of SPs that go through x, among all SPs in G.
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19. a
h
b
g f e
c d
Node x a b c d e f g h
b(x) 0 0.250 0.125 0.036 0.054 0.080 0.268 0
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20. Algorithmic question: How to compute all b(x)?
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21. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
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22. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
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23. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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24. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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25. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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26. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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27. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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28. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
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29. Algorithmic question: How to compute all b(x)?
Brandes’ Algorithm
Intuition: For each vertex s ∈ V :
1) Build the SP DAG from s via Dijkstra/BFS;
2) Traverse the SP DAG from the most distant node towards s, in reverse order of
distance. During the walk, appropriately increment b(v) of each non-leaf node v
traversed.
Source s: 1
1
234
567
89
(update to b(v) not shown)
Time complexity: O(nm + n2 log n)
n Dijkstra’s, plus n backward walks,
taking at most n each
Too much even with just 104 nodes.
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30. Modeling / Domain knowledge:
High-quality approximations of all BCs are sufficient.
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31. Modeling / Domain knowledge:
High-quality approximations of all BCs are sufficient.
Let ε ∈ (0, 1), and δ ∈ (0, 1) be user-specified parameters;
An (ε, δ)-approximation is a set {b(x), x ∈ V } of n values s.t.
Pr(∃x ∈ V s.t. |b(x) − b(x)| > ε) ≤ δ
i.e., with prob. ≥ 1 − δ, for all x ∈ V , b(x) is within ε of b(x):
a uniform probabilistic guarantee over all the estimations.
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32. Algorithmic question:
How to obtain an (ε, δ)-approximation quickly?
Answer:
Sampling
Instead of computing all the SPs from each node x ∈ V , compute them only from
some randomly chosen nodes (samples).
Theory question:
How many samples do we need to obtain an (ε, δ)-approximation?
The more the better, but really, how many?
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33. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Hoeffding Bound + Union Bound
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34. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Hoeffding Bound + Union Bound
Need O
1
ε2
log |V | + log
1
δ
samples
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35. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Hoeffding Bound + Union Bound
Need O
1
ε2
log |V | + log
1
δ
samples
Comments
Practice:
Fewer samples than the above are sufficient for (ε, δ)-approx.
Theory:
Dependency on |V | and not on edge structure seems wrong.
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36. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Vapnik-Chervonenkis (VC) Dimension
Developed to evaluate supervised learning classifiers.
We twisted it to work in a non-supervised graph mining problem.
“The most practical theory ever” – Me, right now
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37. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Vapnik-Chervonenkis (VC) Dimension
Developed to evaluate supervised learning classifiers.
We twisted it to work in a non-supervised graph mining problem.
“The most practical theory ever” – Me, right now
Need O
1
ε2
log diam(G) + log
1
δ
samples
Decreased sample size exponentially on small-world networks.
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38. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Vapnik-Chervonenkis (VC) Dimension
Developed to evaluate supervised learning classifiers.
We twisted it to work in a non-supervised graph mining problem.
“The most practical theory ever” – Me, right now
Need O
1
ε2
log diam(G) + log
1
δ
samples
Decreased sample size exponentially on small-world networks.
Comments
Practice: Great improvement but still too many samples.
Theory: Graphs with the same diameter are not equally “hard”.
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39. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Progressive sampling + Rademacher Averages
Let’s start sampling, use the sample to decide when to stop.
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40. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Progressive sampling + Rademacher Averages
Let’s start sampling, use the sample to decide when to stop.
Stop when ηi ≤ ε, where ηi is. . .
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41. How many samples do we need to obtain an (ε, δ)-approximation?
Theory: Progressive sampling + Rademacher Averages
Let’s start sampling, use the sample to decide when to stop.
Stop when ηi ≤ ε, where ηi is. . .
ηi = 2 min
t∈R+
1
t
ln
(r,C)∈T
et2
r2
/(2S2
i )
+ 3
(i + 1) ln(2/δ)
2Si
Comments
Practice: Getting closer to the empirical bound
Theory: Proving stuff is getting complicated (isn’t that good?)
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42. Theory + Practice:
Get rid of “theoretical elegance” while maintaining correctness.
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43. Theory + Practice:
Get rid of “theoretical elegance” while maintaining correctness.
Let
gS(x, y) = 2 exp −2 x2
(y − 2RF (S))2
+ exp − ((1 − x)y + 2xRF (S))
φ
2RF (S)
(1 − x)y + 2xRF (S)
− 1 .
Then compute
min
x,ξ
ξ
s.t. gS(x, ξ) ≤ η
ξ ∈ (2RF (S), 1]
x ∈ (0, 1)
and check if ξ < ε.
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44. To be a data scientist, you need to get your hands dirty in data.
To be an algorithmic data scientist,
you need to get your hands dirty in



data
theory
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45. Other examples



pattern mining
(Rademacher Averages)
selectivity of database queries
(VC-dimension)
triangle counting from data streams
(non-i.i.d. sampling)
graph summarization
(Szemerédi Regularity)
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46. 1) Embrace data science
2) Combine theory and practice
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47. 1) Embrace data science
2) Combine theory and practice
Thank you!
EML: matteo@twosigma.com TWTR: @teorionda
WWW: http://matteo.rionda.to
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