News from Mahout

News From Mahout

©MapR Technologies - Confidential 1

whoami – Ted Dunning

 Chief Application Architect, MapR Technologies
 Committer, member, Apache Software Foundation
– particularly Mahout, Zookeeper and Drill

(we’re hiring)

 Contact me at
tdunning@maprtech.com
tdunning@apache.com
ted.dunning@gmail.com
@ted_dunning


 Slides and such (available late tonight):
– http://www.mapr.com/company/events/nyhug-03-05-2013

 Hash tags: #mapr #nyhug #mahout


New in Mahout

 0.8 is coming soon (1-2 months)
 gobs of fixes
 QR decomposition is 10x faster
– makes ALS 2-3 times faster
 May include Bayesian Bandits
 Super fast k-means
– fast
– online (!?!)


New in Mahout

 gobs of fixes
– fast
– online (!?!)
– fast
 Possible new edition of MiA coming
– Japanese and Korean editions released, Chinese coming


Real-time Learning


We have a product
to sell …
from a web-site


What tag-
What line?
picture?
Bogus Dog Food is the Best!
Now available in handy 1 ton
bags!

Buy 5!

What call to
action?


The Challenge

 Design decisions affect probability of success
– Cheesy web-sites don’t even sell cheese

 The best designers do better when allowed to fail
– Exploration juices creativity

 But failing is expensive
– If only because we could have succeeded
– But also because offending or disappointing customers is bad


More Challenges

 Too many designs
– 5 pictures
– 10 tag-lines
– 4 calls to action
– 3 back-ground colors
=> 5 x 10 x 4 x 3 = 600 designs

 It gets worse quickly
– What about changes on the back-end?
– Search engine variants?
– Checkout process variants?


Example – AB testing in real-time

 I have 15 versions of my landing page
 Each visitor is assigned to a version
– Which version?
 A conversion or sale or whatever can happen
– How long to wait?
 Some versions of the landing page are horrible
– Don’t want to give them traffic


A Quick Diversion

 You see a coin
– What is the probability of heads?
– Could it be larger or smaller than that?
 I flip the coin and while it is in the air ask again
 I catch the coin and ask again
 I look at the coin (and you don’t) and ask again
 Why does the answer change?
– And did it ever have a single value?


A Philosophical Conclusion

 Probability as expressed by humans is subjective and depends on
information and experience


I Dunno


5 heads out of 10 throws


2 heads out of 12 throws


So now you understand
Bayesian probability


Another Quick Diversion

 Let’s play a shell game
 This is a special shell game
 It costs you nothing to play
 The pea has constant probability of being under each shell
(trust me)
 How do you find the best shell?
 How do you find it while maximizing the number of wins?


Pause for short
con-game


Interim Thoughts

 Can you identify winners or losers without trying them out?

 Can you ever completely eliminate a shell with a bad streak?

 Should you keep trying apparent losers?


So now you understand
multi-armed bandits


Conclusions

 Can you identify winners or losers without trying them out?
No

 Can you ever completely eliminate a shell with a bad streak?
No

 Should you keep trying apparent losers?
Yes, but at a decreasing rate


Is there an optimum
strategy?


Bayesian Bandit

 Compute distributions based on data so far
 Sample p1, p2 and p2 from these distributions
 Pick shell i where i = argmaxi pi

 Lemma 1: The probability of picking shell i will match the
probability it is the best shell

 Lemma 2: This is as good as it gets


And it works!

0.12

0.11

0.1

0.09

0.08

0.07
regret

0.06
ε- greedy, ε = 0.05
0.05

0.04 Bayesian Bandit with Gam m a- Norm al
0.03

0.02

0.01

0
0 100 200 300 400 500 600 700 800 900 1000 1100

n


Video Demo


The Code

 Select an alternative
n = dim(k)[1]
p0 = rep(0, length.out=n)
for (i in 1:n) {
p0[i] = rbeta(1, k[i,2]+1, k[i,1]+1)
}
return (which(p0 == max(p0)))

 Select and learn
for (z in 1:steps) {
i = select(k)
j = test(i)
k[i,j] = k[i,j]+1
}
return (k)

 But we already know how to count!


The Basic Idea

 We can encode a distribution by sampling
 Sampling allows unification of exploration and exploitation

 Can be extended to more general response models


The Original Problem

x2
x1

bags!

Buy 5!

x3


Response Function

æ ö
p(win) = w çåqi xi ÷
è i ø
1

0.5
y

0
-6 -4 -2 0 2 4 6
x


Generalized Banditry

 Suppose we have an infinite number of bandits
– suppose they are each labeled by two real numbers x and y in [0,1]
– also that expected payoff is a parameterized function of x and y

E [ z ] = f (x, y | q )
– now assume a distribution for θ that we can learn online
 Selection works by sampling θ, then computing f
 Learning works by propagating updates back to θ
– If f is linear, this is very easy
– For special other kinds of f it isn’t too hard
 Don’t just have to have two labels, could have labels and context


Context Variables

x2
x1

bags!

Buy 5!

x3

user.geo env.time env.day_of_week env.weekend


Caveats

 Original Bayesian Bandit only requires real-time

 Generalized Bandit may require access to long history for learning
– Pseudo online learning may be easier than true online

 Bandit variables can include content, time of day, day of week

 Context variables can include user id, user features

 Bandit × context variables provide the real power


You can do this
yourself!


Super-fast k-means Clustering


Rationale


What is Quality?

 Robust clustering not a goal
– we don’t care if the same clustering is replicated
 Generalization is critical
 Agreement to “gold standard” is a non-issue


An Example


Diagonalized Cluster Proximity


Clusters as Distribution Surrogate


Theory


For Example

1
D (X) >
2
D (X)
2

s
4 2 5

Grouping these
two clusters
seriously hurts
squared distance


Algorithms


Typical k-means Failure

Selecting two seeds
here cannot be
fixed with Lloyds

Result is that these two
clusters get glued
together


Ball k-means

 Provably better for highly clusterable data
 Tries to find initial centroids in each “core” of each real clusters
 Avoids outliers in centroid computation

initialize centroids randomly with distance maximizing tendency
for each of a very few iterations:
for each data point:
assign point to nearest cluster
recompute centroids using only points much closer than closest cluster


Still Not a Win

 Ball k-means is nearly guaranteed with k = 2
 Probability of successful seeding drops exponentially with k
 Alternative strategy has high probability of success, but takes
O(nkd + k3d) time


Still Not a Win

 Ball k-means is nearly guaranteed with k = 2
 Probability of successful seeding drops exponentially with k
 Alternative strategy has high probability of success, but takes O(
nkd + k3d ) time

 But for big data, k gets large


Surrogate Method

 Start with sloppy clustering into lots of clusters
κ = k log n clusters
 Use this sketch as a weighted surrogate for the data
 Results are provably good for highly clusterable data


Algorithm Costs

 Surrogate methods
– fast, sloppy single pass clustering with κ = k log n
– fast sloppy search for nearest cluster,
O(d log κ) = O(d (log k + log log n)) per point
– fast, in-memory, high-quality clustering of κ weighted centroids
O(κ k d + k3 d) = O(k2 d log n + k3 d) for small k, high quality
O(κ d log k) or O(d log κ log k) for larger k, looser quality
– result is k high-quality centroids
• Even the sloppy surrogate may suffice


Algorithm Costs

 Surrogate methods
– fast, sloppy single pass clustering with κ = k log n
– fast sloppy search for nearest cluster,
O(d log κ) = O(d ( log k + log log n )) per point
– fast, in-memory, high-quality clustering of κ weighted centroids
O(κ k d + k3 d) = O(k2 d log n + k3 d) for small k, high quality
O(κ d log k) or O( d log k ( log k + log log n ) ) for larger k, looser quality
– result is k high-quality centroids
• For many purposes, even the sloppy surrogate may suffice


Algorithm Costs

 How much faster for the sketch phase?
– take k = 2000, d = 10, n = 100,000
– k d log n = 2000 x 10 x 26 = 500,000
– d (log k + log log n) = 10(11 + 5) = 170
– 3,000 times faster is a bona fide big deal


How It Works

 For each point
– Find approximately nearest centroid (distance = d)
– If (d > threshold) new centroid
– Else if (u > d/threshold) new cluster
– Else add to nearest centroid
 If centroids > κ ≈ C log N
– Recursively cluster centroids with higher threshold


Implementation


But Wait, …

 Finding nearest centroid is inner loop

 This could take O( d κ ) per point and κ can be big

 Happily, approximate nearest centroid works fine


Projection Search

total ordering!


LSH Bit-match Versus Cosine
1

0.8

0.6

0.4

0.2
Y Ax is

0
0 8 16 24 32 40 48 56 64

- 0.2

- 0.4

- 0.6

- 0.8

-1

X Ax is


Results


Parallel Speedup?

200

Non- threaded

✓
100
2
Tim e per point (μs)

Threaded version
3

50
4
40 6
5

8
30
10 14
12
20 Perfect Scaling 16

10
1 2 3 4 5 20

Threads

Quality

 Ball k-means implementation appears significantly better than
simple k-means

 Streaming k-means + ball k-means appears to be about as good as
ball k-means alone

 All evaluations on 20 newsgroups with held-out data

 Figure of merit is mean and median squared distance to nearest
cluster


Contact Me!

 We’re hiring at MapR in US and Europe

 MapR software available for research use

 Get the code as part of Mahout trunk (or 0.8 very soon)

 Contact me at tdunning@maprtech.com or @ted_dunning

 Share news with @apachemahout


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