Crash Course on Machine LearningPart V

Several slides from Derek Hoiem, Ben Taskar, and Andreas Krause

Structured Prediction

• Use local information • Exploit correlations

b r ea c

Min-max Formulation

LP duality

Before

QP duality

Exponentially many constraints/variables

After

By QP duality

Dual inherits structure from problem-specific inference LPVariables correspond to a decomposition of variables of the flat case

The Connection

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Duals and Kernels

Kernel trick works: Factored dual Local functions (log-potentials) can use kernels

3D Mapping

Laser Range Finder

GPS

IMU

Data provided by: Michael Montemerlo & Sebastian Thrun

Label: ground, building, tree, shrub Training: 30 thousand points Testing: 3 million points

• Simple iterative method

• Unstable for structured output: fewer instances, big updates– May not converge if non-separable– Noisy

• Voted / averaged perceptron [Freund & Schapire 99, Collins 02]– Regularize / reduce variance by aggregating over iterations

Alternatives: Perceptron

• Add most violated constraint

• Handles several more general loss functions• Need to re-solve QP many times• Theorem: Only polynomial # of constraints needed to achieve -

error [Tsochantaridis et al, 04]

• Worst case # of constraints larger than factored

Alternatives: Constraint Generation

[Collins 02; Altun et al, 03]

Integration

• Feature Passing

• Margin Based– Max margin Structure Learning

• Probabilistic– Graphical Models

Graphical Models

• Joint distribution– Factoring using independent variables

• Representation

• Inference

• Learning

Big Picture

• Two problems with using full joint distribution tables as our probabilistic models: – Unless there are only a few variables, the joint is WAY too

big to represent explicitly – Hard to learn (estimate) anything empirically about more

than a few variables at a time

• Describe complex joint distributions (models) using simple, local distributions – We describe how variables locally interact – Local interactions chain together to give global, indirect

interactions

Joint Distribution• For n variables with domain sizes d

– joint distribution table with dn -1 free parameters • Size of representation if we use the chain rule

Concretely, counting the number of free parameters accounting for that we know probabilities sum to one: (d-1) + d(d-1) + d2(d-1) + ... + dn-1 (d-1) = (dn-1)/(d-1) (d-1)= dn - 1

Conditional Independence• Two variables are conditionally independent:

• What about this domain? – Traffic– Umbrella– Raining

RepresentationExplicitly model uncertainty and dependency structure

a

b

c

a

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c

Directed Undirected Factor graph

d d

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c d

Key concept: Markov blanket

Bayes Net: Notation

• Nodes: variables– Can be assigned (observed)

or – unassigned (unobserved)

• Arcs: interactions – Indicate “direct influence”

between variables – Formally: encode conditional

independence

Cavity

Toothache Catch

Weather

Example: Flip Coins

• N independent flip coins

• No interactions between variables– Absolute independence

X1 X2 Xn

Example: Traffic

• Variables: – Traffic– Rain

• Model 1: absolute independence• Model 2: rain causes traffic• Which makes more sense?

Rain

Traffic

Semantics

• A set of nodes, one per variable X • A directed, acyclic graph • A conditional distribution for each

node – A collection of distributions over X,

one for each combination of parents’ values

– Conditional Probability Table (CPT)

A1 An

X

A2

Parents

A Bayes net = Topology (graph) + Local Conditional Probabilities

Example: Alarm

• Variables:– Alarm– Burglary– Earthquake– Radio– Calls John

Earthquake

Radio

Burglary

Alarm

Call

Example: AlarmEarthquake

Radio

Burglary

Alarm

Call

P(C|A)

P(R|E) P(A|E,B)

P(E) P(B)

P(E,B,R,A,C)=P(E)P(B)P(R|E)P(A|B,E)P(C|A)

Bayes Net Size

• How big is a joint distribution over N Boolean variables?

• How big is the size of CPT with k parents?

• How big is the size of BN with n node if nodes have up to k parents?

• BNs: – Compact representation– Use local properties to define CPTS– Answer queries more easily

2n

2k+1

n.2k+1

Independence in BN

• BNs present a compact representation for joint distributions– Take advantage of conditional independence

• Given a BN let’s answer independence questions:– Are two nodes independent given certain

evidence?

– What can we say about X, Z? (Example: Low pressure, Rain, Traffic}

X Y Z

Causal Chains

• Question: Is Z independent of X given Y?

X Y Z• X: low pressure• Y: Rain• Z: Traffic

Common Cause

• Are X, Z independent?

X

y

Z

• Are X, Z independent given Y?

• Observing Y blocks the influence between X,Z

• Y: low pressure• X: Rain• Z: Cold

Common Effect

• Are X, Z independent?

X

Y

Z

• X: Rain• Y: Traffic• Z: Ball Game

• Are X, Z independent given Y?

• Observing Y activates influence between X, Z

Independence in BNs

• Any complex BN structure can be analyzed using these three cases

Earthquake

Radio

Burglary

Alarm

Call

Directed acyclical graph (Bayes net)

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c d

P(a,b,c,d) = P(c|b)P(d|b)P(b|a)P(a)

• Can model causality• Parameter learning

– Decomposes: learn each term separately (ML)

• Inference– Simple exact inference if tree-

shaped (belief propagation)

Directed acyclical graph (Bayes net)

a

b

c d

• Can model causality• Parameter learning

– Decomposes: learn each term separately (ML)

• Inference– Simple exact inference if tree-

shaped (belief propagation)– Loops require approximation

• Loopy BP• Tree-reweighted BP• Sampling

P(a,b,c,d) = P(c|b)P(d|a,b)P(b|a)P(a)

• Example: Places and scenes

Directed graph

Place: office, kitchen, street, etc.

Car Person Toaster MicrowaveFire

Hydrant

Objects Present

P(place, car, person, toaster, micro, hydrant) = P(place) P(car | place) P(person | place) … P(hydrant | place)

Undirected graph (Markov Networks)

• Does not model causality• Often pairwise• Parameter learning difficult• Inference usually approximate

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x2

x3 x4

edgesji

jii

iZ dataxxdataxdataP,

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Markov Networks• Example: “label smoothing” grid

Binary nodes

0 10 0 K1 K 0

Pairwise Potential

Image De-Noising

Original Image Noisy Image

Image De-Noising

Image De-Noising

Noisy Image Restored Image (ICM)

Factor graphs• A general representation

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Bayes Net

Factor Graph

d

a

b

c d

Factor graphs• A general representation

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b

c

Markov Net

d

Factor Graph

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c d

Factor graphs

),()(),,(),,,( 321 dafdfcbafdcbaP

Write as a factor graph

Inference in Graphical Models

• Joint• Marginal• Max

• Exact inference is HARD

Approximate Inference

Approximation

Sampling a Multinomial Distribution

Sampling from a BN

- Compute Marginals- Compute Conditionals

Belief Propagation

• Very general • Approximate, except for tree-shaped graphs

– Generalizing variants BP can have better convergence for graphs with many loops or high potentials

• Standard packages available (BNT toolbox)

• To learn more:– Yedidia, J.S.; Freeman, W.T.; Weiss, Y., "Understanding Belief Propagation and Its

Generalizations”, Technical Report, 2001: http://www.merl.com/publications/TR2001-022/

Belief Propagation

i

)(

)()(iNa

iiaii xmxb

“beliefs” “messages”

a

)( \)(

)()()(aNi aiNb

iibaaaa xmXfXb

The “belief” is the BP approximation of the marginal probability.

BP Message-update Rules

Using ,)()(\

ia xX

aaii Xbxb we get

ai

ia xX iaNj ajNb

jjbaaiia xmXfxm\ \)( \)(

)()()(

i a=

Inference: Graph Cuts

• Associative: edge potentials penalize different labels• Associative binary networks can be solved optimally

(and quickly) using graph cuts

• Multilabel associative networks can be handled by alpha-expansion or alpha-beta swaps

• To learn more:– http://www.cs.cornell.edu/~rdz/graphcuts.html– Classic paper: What Energy Functions can be Minimized via Graph Cuts? (Kolmogorov

and Zabih, ECCV '02/PAMI '04)

Graph Cuts: Binary MRF

Unary terms (compatability of data with label y)

Pairwise terms (compatability of neighboring labels)

Graph cuts used to optimise this cost function:

Summary of approach

• Associate each possible solution with a minimum cut on a graph• Set capacities on graph, so cost of cut matches the cost function• Use augmenting paths to find minimum cut• This minimizes the cost function and finds the MAP solution

Denoising Results

Original Pairwise costs increasing

Pairwise costs increasing