A Survey on Distance Metric Learning (Part 2)

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A Survey on Distance Metric Learning (Part 2)

• Gerry Tesauro
• IBM T.J.Watson Research Center

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Acknowledgement

• Lecture material shamelessly adapted from the
  following sources
• Kilian Weinberger
• Survey on Distance Metric Learning slides
• IBM summer intern talk slides (Aug. 2006)
• Sam Roweis slides (NIPS 2006 workshop on
  Learning to Compare Examples)
• Yann LeCun talk slides (CVPR 2005, 2006)

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Outline Part 2

• Neighbourhood Components Analysis (Golderberger
  et al.), Metric Learning by Collapsing Classes
  (Globerson Roweis)
• Metric Learning for Kernel Regression (Weinberger
  Tesauro)
• Metric learning for RL basis function
  construction (Keller et al.)
• Similarity learning for image processing (LeCun
  et al.)

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Neighborhood Component Analysis
Distance metric for visualization and kNN
(Goldberger et. al. 2004)
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Metric Learning for Kernel Regression
Weinberger Tesauro, AISTATS 2007
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Killing three birds with one stone
We construct a method for linear dimensionality
reduction
that generates a meaningful distance metric
optimally tuned for distance-based kernel
regression
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Kernel Regression

• Given training set (xj , yj), j1,,N where x
  is ?-dim vector and y is real-valued, estimate
  value of a test point xi by weighted avg. of
  samples
• where kij kD (xi, xj) is a distance-based
  kernel function using distance metric D

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Choice of Kernel

• Many functional forms for kij can be used in
  MLKR our empirical work uses the Gaussian
  kernel
• where s is a kernel width parameter (can set s1
  W.L.O.G. since we learn D)
• softmax regression estimate similar to Roweis
  softmax classifier

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Distance Metric for Nearest Neighbor Regression
Learn a linear transformation that allows to
estimate the value of a test point from its
nearest neighbors
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Mahalanobis Metric
Distance function is a pseudo Mahalanobis metric
(Generalizes Euclidean distance)
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General Metric Learning Objective

• Find parmaterized distance function D? that
  minimizes total leave-one-out cross-validation
  loss function
• e.g. params ? elements Aij of A matrix
• Since were solving for A not M, optimization is
  non-convex ? use gradient descent

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Gradient Computation

• where xij xi xj
• For fast implementation
• Dont sum over all i-j pairs, only go up to 1000
  nearest neighbors for each sample i
• Maintain nearest neighbors in a heap-tree
  structure, update heap tree every 15 gradient
  steps
• Ignore sufficiently small values of kij ( lt e-34
  )
• Even better data structures cover trees, k-d
  trees

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Learned Distance Metric example
orig. Euclidean D lt 1
learned D lt 1
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Twin Peaks test
Training
n8000
we added 3 dimensions with 1000 noise
we rotated 5 dimensions randomly
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Input Variance
Noise
Signal
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Test data
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Test data
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Output Variance
Signal
Noise
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DimReduction with MLKR

• FG-NET face data 82 persons, 984 face images
  w/age

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DimReduction with MLKR

• FG-NET face data 82 persons, 984 face images
  w/age

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DimReduction with MLKR
PowerManagement data (d21)

• Force A to be rectangular
• Project onto eigenvectors of A
• Allows visualization of data

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Robot arm results (8,32dim)
regression error
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Unity Data Center Prototype

• Objective Learn long-range resource value
  estimates for each application manager
• State Variables (48)
• Arrival rate
• ResponseTime
• QueueLength
• iatVariance
• rtVariance
• Action of servers allocated
• by Arbiter
• Reward SLA(Resp. Time)

Maximize Total SLA Revenue
5 sec
Demand (HTTP req/sec)
Demand (HTTP req/sec)
Value(srvrs)
Value(srvrs)
Value(srvrs)
SLA
SLA
SLA
Value(RT)
WebSphere 5.1
Value(srvrs)
WebSphere 5.1
Value(RT)
DB2
DB2
Trade3
Batch
Trade3
8 xSeries servers
(Tesauro, AAAI 2005 Tesauro et al., ICAC 2006)
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Power Performance Management

• Objective Managing systems to multi-discipline
  objectives minimize Resp. Time and minimize
  Power Usage
• State Variables (21)
• Power Cap
• Power Usage
• CPU Utilization
• Temperature
• of requests arrived
• Workload intensity ( Clients)
• Response Time
• Action Power Cap
• Reward SLA(Resp. Time) Power Usage

(Kephart et al., ICAC 2007)
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IBM Regression Results TEST ERROR
MLKR
14/47
3/5
10/22
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IBM Regression Results TRAINING ERROR
MLKR
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Metric Learning for RL basis function
construction (Keller et al. ICML 2006)

• RL Dataset of state-action-reward tuples (si,
  ai, ri) , i1,,N

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Value Iteration

• Define an iterative bootstrap calculation
• Each round of VI must iterate over all states in
  the state space
• Try to speed this up using state aggregation
  (Bertsekas Castanon, 1989)
• Idea Use NCA to aggregate states
• project states into lower-dim rep keep states
  with similar Bellman error close together
• use projected states to define a set of basis
  functions ?
• learn linear value function over basis functions
  V ? ?i ?i

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Chopra et. al. 2005
Similarity metric for image verification.
Problem Given a pair of face-images, decide if
they are from the same person.
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Chopra et. al. 2005
Similarity metric for image verification.
Problem Given a pair of face-images, decide if
they are from the same person.
Too difficult for linear mapping!
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