Transcript Machine Learning and Statistical Analysis

Jong Youl Choi
Computer Science Department
([email protected])
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Motivations
 Social indexing or collaborative annotation
 Collect knowledge from people
 Extract information
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Challenges
 Vast amount of data
 Very dynamic
 Unsupervised data
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Principles of Machine Learning
 Bayes’ theorem and maximum likelihood
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Machine Learning Algorithms
 Clustering analysis
 Dimension reduction
 Classification
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Parallel Computing
 General parallel computing architecture
 Parallel algorithms
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Definition
Algorithms or techniques that enable computer (machine) to
“learn” from data. Related with many areas such as data
mining, statistics, information theory, etc.
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Algorithm Types
 Unsupervised learning
 Supervised learning
 Reinforcement learning
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Topics
 Models
▪ Artificial Neural Network (ANN)
▪ Support Vector Machine (SVM)
 Optimization
▪ Expectation-Maximization (EM)
▪ Deterministic Annealing (DA)
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Posterior probability of i, given X
 i 2  : Parameter
 X : Observations
 P(i) : Prior (or marginal) probability
 P(X|i) : likelihood
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Maximum Likelihood (ML)
 Used to find the most plausible i 2 , given X
 Computing maximum likelihood (ML) or log-likelihood
 Optimization problem
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Problem
Estimate hidden parameters (={, })
from the given data extracted from
k Gaussian distributions
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Gaussian distribution
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Maximum Likelihood
(Mitchell , 1997)
 With Gaussian (P = N),
 Solve either brute-force or numeric method
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Problems in ML estimation
 Observation X is often not complete
 Latent (hidden) variable Z exists
 Hard to explore whole parameter space
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Expectation-Maximization algorithm
 Object : To find ML, over latent distribution P(Z |X,)
 Steps
0. Init – Choose a random old
1. E-step – Expectation P(Z |X, old)
2. M-step – Find new which maximize likelihood.
3. Go to step 1 after updating old à new
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Definition
Grouping unlabeled data into clusters, for the purpose of
inference of hidden structures or information
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Dissimilarity measurement
 Distance : Euclidean(L2), Manhattan(L1), …
 Angle : Inner product, …
 Non-metric : Rank, Intensity, …
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Types of Clustering
 Hierarchical
▪ Agglomerative or divisive
 Partitioning
▪ K-means, VQ, MDS, …
(Matlab helppage)
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Find K partitions with the total
intra-cluster variance minimized
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Iterative method
 Initialization : Randomized yi
 Assignment of x (yi fixed)
 Update of yi (x fixed)
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Problem?
 Trap in local minima
(MacKay, 2003)
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Deterministically avoid local minima
 No stochastic process (random walk)
 Tracing the global solution by changing
level of randomness
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Statistical Mechanics
(Maxima and Minima, Wikipedia)
 Gibbs distribution
 Helmholtz free energy F = D – TS
▪ Average Energy D = < Ex>
▪ Entropy S = - P(Ex) ln P(Ex)
▪ F = – T ln Z
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In DA, we make F minimized
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Analogy to physical annealing process
 Control energy (randomness) by temperature (high  low)
 Starting with high temperature (T = 1)
▪ Soft (or fuzzy) association probability
▪ Smooth cost function with one global minimum
 Lowering the temperature (T ! 0)
▪ Hard association
▪ Revealing full complexity, clusters are emerged
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Minimization of F, using E(x, yj) = ||x-yj||2
Iteratively,
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Definition
Process to transform high-dimensional data into lowdimensional ones for improving accuracy, understanding, or
removing noises.
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Curse of dimensionality
 Complexity grows exponentially
in volume by adding extra
dimensions
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Types
(Koppen, 2000)
 Feature selection : Choose representatives (e.g., filter,…)
 Feature extraction : Map to lower dim. (e.g., PCA, MDS, … )
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Finding a map of principle components
(PCs) of data into an orthogonal space,
such that
y = W x where W 2 Rd£h (hÀd)
x2
x1
PCs – Variables with the largest variances
 Orthogonality
 Linearity – Optimal least mean-square error
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Limitations?
 Strict linearity
 specific distribution
 Large variance assumption
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Like PCA, reduction of dimension by y = R x where R is a
random matrix with i.i.d columns and R 2 Rd£p (pÀd)
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Johnson-Lindenstrauss lemma
 When projecting to a randomly selected subspace, the distance
are approximately preserved
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Generating R
 Hard to obtain orthogonalized R
 Gaussian R
 Simple approach
choose rij = {+31/2,0,-31/2} with probability 1/6, 4/6, 1/6
respectively
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Dimension reduction preserving distance proximities
observed in original data set
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Loss functions
 Inner product
 Distance
 Squared distance
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Classical MDS: minimizing STRAIN, given 
 From , find inner product matrix B (Double centering)
 From B, recover the coordinates X’ (i.e., B=X’X’T )
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SMACOF : minimizing STRESS
 Majorization – for complex f(x),
find auxiliary simple g(x,y) s.t.:
 Majorization for STRESS
(Cox, 2001)
 Minimize tr(XT B(Y) Y), known as Guttman transform
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Competitive and unsupervised learning process for
clustering and visualization
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Learning
 Choose the best similar model
Input
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Model
vector mj with xi
 Update the winner and its
neighbors by
mk = mk + (t) (t)(xi – mk)
(t) : learning rate
(t) : neighborhood size
Result : similar data getting closer in the model space
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Definition
A procedure dividing data into the given set of categories
based on the training set in a supervised way
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Generalization Vs. Specification
 Hard to achieve both
Underfitting
Overfitting
 Avoid overfitting(overtraining)
▪
▪
▪
▪
Early stopping
Holdout validation
K-fold cross validation
Leave-one-out cross-validation
Validation Error
Training Error
(Overfitting, Wikipedia)
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Perceptron : A computational unit with binary threshold
Weighted Sum
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Activation Function
Abilities
 Linear separable decision surface
 Represent boolean functions (AND, OR, NO)
 Network (Multilayer) of perceptrons
 Various network architectures and capabilities
(Jain, 1996)
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Learning weights – random initialization and updating
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Error-correction training rules
 Difference between training data and output: E(t,o)
 Gradient descent (Batch learning)
▪ With E =  Ei ,
 Stochastic approach (On-line learning)
▪ Update gradient for each result
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Various error functions
 Adding weight regularization term ( wi2) to avoid overfitting
 Adding momentum (wi(n-1)) to expedite convergence
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Q: How to draw the optimal linear
separating hyperplane?
 A: Maximizing margin
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Margin maximization
 The distance between H+1 and H-1:
 Thus, ||w|| should be minimized
Margin
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Constraint optimization problem
 Given training set {xi, yi} (yi 2 {+1, -1}):
 Minimize :
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Lagrangian equation with saddle points
 Minimized w.r.t the primal variable w and b:
 Maximized w.r.t the dual variables i (all i ¸ 0)
 xi with i > 0 (not i = 0) is called support vector (SV)
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Soft Margin (Non-separable case)
 Slack variables i < C
 Optimization with additional constraint
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Non-linear SVM
 Map non-linear input to feature space
 Kernel function k(x,y) = h(x), (y)i
 Kernel classifier with support vectors si
Input Space
Feature Space
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Memory Architecture
Shared Memory
 Symmetric Multiprocessor (SMP)
 OpenMP, POSIX, pthread, MPI
 Easy to manage but expensive
Distributed Memory
 Commodity, off-the-shelf processors
 MPI
 Cost effective but hard to maintain
(Barney, 2007)
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Decomposition Strategy
 Task – E.g., Word, IE, …
 Data – scientific problem
 Pipelining – Task + Data
(Barney, 2007)
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Shrinking
 Recall : Only support vectors (i>0) are
used in SVM optimization
 Predict if data is either SV or non-SV
 Remove non-SVs from problem space
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Parallel SVM
 Partition the problem
 Merge data hierarchically
 Each unit finds support vectors
 Loop until converge
(Graf, 2005)
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