Interactive Learning using
Manifold Geometry
Eric Eaton, Gary Holness, and Daniel McFarlane
Lockheed Martin Advanced Technology Laboratories
Artificial Intelligence Research Group
This work was supported by internal funding from Lockheed Martin and the
National Science Foundation under NSF ITR #0325329.
Introduction: Motivation
 Information monitoring
systems use a scoring
function f to focus user
attention
– f is customized to the
current situation
– Often, no data are
available to learn f
– Users require fine control over the
scoring function
Maritime Situational Awareness
 We propose an interactive
learning method that enables
the user to iteratively refine f
Network Security Monitoring
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Introduction: Interactive Refinement
 Uses a combination of manual input and machine learning:
1. The user manually selects and repositions a data point
2. The system relearns the model f, and updates the scatterplot
 Key idea: each adjustment should generalize naturally to the model
 We use least squares with Laplacian regularization to learn f,
based on the manifold underlying the data
Model View
Relevancy
User View
1D Projection of Data
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Related Work: Interactive Learning
 Crayons tool for interactive
object classification (Fails &
Olsen, 2003)
 Interactive decision tree
construction (Ware et al., 2001)
 Interactive visual clustering
(desJardins et al., 2008)
Crayons by Fails & Olsen
 Feature selection
(Figure used with permission)
(Dy & Brodley, 2000)
 Hierarchical clustering
(Wills, 1998)
Initial view
After 2
adjustments
After 14
adjustments
Interactive Visual Clustering by desJardins et al.
(Figure used with permission)
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Related Work: Interactive vs Active Learning
 Active learning – selects instances for labeling by an oracle
(Cohn et al., 1996; McCallum & Nigam, 1998; Tong, 2001)
Interactive ML
Active Learning
Starts with…
 Unlabeled data
 Incorrect model
 Unlabeled data
 No model
Selection of
instances
User determines
adjustments
System selects
instances for labeling
Goal
Collaborate with
the user to define
or adjust a model
Minimize number of
labels needed to
learn a model
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Mechanisms for User Interaction
 Data set
where
Relevancy
 The user supplies the initial
scoring function
Score:
55
Id:
dmaskes2
Event:
ACL-Monitor
System:
Julius-laptop
------------------------------Freq:
8 (1hr)
8 (24hr)
------------------------------DETAILS:
UID:
dmaskes2
Role:
App_Update
Policy:
finCloseLock
StartTime: 0 17 * * 5
EndTime: 0 8 * * 1
Res_type: triggerOverride
View_type: AcctClerk
DS_name: tbl_wklyTotals
Error:
unauth_update
Value:
(2 3 -2334 conf)
– We used a linear function for
 Current scoring function is given
by f (initially
)
1D Projection of Data
User View
 The user adjusts the score of individual data points to
change f until it matches the true (hidden) function F
– Details of each instance are available in a side panel
– User selects and drags an instance up or down to change its score
 Future work: similarity metric updates, qualitative feedback
Eric Eaton, Gary Holness, & Daniel McFarlane - Interactive Learning using Manifold Geometry
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Approach: Learning the Scoring Function
 Key Idea: each adjustment should generalize naturally to the model
– Adjustments should affect similar instances
– Generalizations should be based on the geometry underlying the data
 Our approach:
– Construct the manifold underlying the data
– Learn/update f using the manifold’s basis
v13
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v3
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v7
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v5
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v1
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Approach: Constructing the Manifold
 Represent data set X as an undirected graph G = (V,A),
with vertex vi representing instance xi
 Adjacency matrix A is given by:
– Weighting each edge (vi, vj) by a radial basis function of the distance
– Connecting each instance to its k nearest neighbors
 G is a discrete approximation of the continuous manifold
initial scoring
function
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Eric Eaton, Gary Holness, & Daniel McFarlane - Interactive Learning using Manifold Geometry
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Approach: Learning the Function on the Manifold
 Form the graph Laplacian of G (Chung, 1994)
where
 Take the eigendecomposition of
=
Q
Λ
λ1
λ2
λ3
λn
QT
λ1 = 0
 QThe
= [q
first
eigenvector
1…q
n] forms a complete
orthonormal
basis for G
is constant
q1
q2
q5
q10
q20
q50
Meshes provided by Gabriel Peyré
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Approach: Learning the Function on the Manifold
 The scoring function f : V → [0,1] is given by f = QW
 Fit W by least squares with Laplacian regularization:
– This is a special case of Belkin et al.’s (2006) Manifold Regularization
– Eigenvalues ¤ increasingly regularize the higher-order components
 A slider bar controls the weight
of adjustments
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Complete Algorithm for Interactive Refinement
Given: the data X, the user’s initial scoring function

Set

Construct the manifold underlying X, represented by G = (V,A)

Compute the graph Laplacian

Compute the eigenvectors Q and eigenvalues ¤ of

Repeat
of G
–
Display the scatterplot of X using the scores given by f
–
(Optional) The user adjusts the score of data instance xi
–
(Optional) The user updates the adjustment weight ! via a slider bar
–
If there were changes, update the scoring function as f = QW, where
W is given by
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Scaling to Large Volumes of Data
 A can be stored efficiently as a symmetric banded matrix

is also a symmetric banded matrix
– Use sparse eigensolvers (e.g., Lanczos methods) for efficiency
 Nyström method (Baker 1977) extends the eigenvectors to new
vertices for inductive learning
– Learn on a sample
, with Laplacian
– Extend eigenvectors to new instances by
– Score for a new instance x (represented by vertex v) is then given by
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Evaluation
 Simulate user by adjusting the current “most incorrect”
instance to the correct score
– Users are adept at identifying outliers, motivating our approach
–
is a linear model fit to X using ridge regression
 Compared against interactive learning using:
– SMO support vector regression with an RBF kernel
– Least squares regularized with a ridge parameter of 10E-8
Data Sets
Name
CPU
Heart Disease
Pharynx
Pyrimidines
Sleep
Wisconsin Breast Cancer
#Inst #Dim
209
6
303
13
195
10
74
27
62
7
194
32
Source
UCI repository
UCI repository
Kalbfleisch & Prentice (1980)
King et al. (1992)
StatLib archive
UCI repository
Eric Eaton, Gary Holness, & Daniel McFarlane - Interactive Learning using Manifold Geometry
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Evaluation: Adjusting the “most incorrect” instance
Eric Eaton, Gary Holness, & Daniel McFarlane - Interactive Learning using Manifold Geometry
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Evaluation: Adjusting a random instance (100 trials)
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Related Work: Manifold Learning
 Belkin et al.’s (2006) Manifold Regularization
– We use a special case regularizing only the solution’s smoothness
 Semi-supervised learning using Gaussian random fields
(Zhu et al., 2003; Cai et al., 2006)
 Zhou et al.’s (2004; 2005) “Distribution Regularization”
– Uses a regularized form of the graph Laplacian as the basis
– Learns a function
 Spectral Graph Transduction (Joachims, 2003)
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Conclusion and Future Work
 We presented a method for interactive learning based on
least squares with Laplacian regularization
 Manifold-based interactive learning continuously improves
with each correction
 In practice, the technique shows an interactive response
time for hundreds of data instances
 Future Work:
– User adjustment of the similarity metric
between data instances
– Incorporate passive observation of the user
– Handling drifting or recurring concepts
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Thank You!
Questions?
Eric Eaton
eeaton@atl.lmco.com
This work was supported by internal funding from Lockheed Martin and the
National Science Foundation under NSF ITR #0325329.
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