lirong xia planning tuesday, april 22, 2014. the kernel trick neuron network clustering (k-means) 1...
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Lirong Xia
Planning
Tuesday, April 22, 2014
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• The kernel trick
• Neuron network
• Clustering (k-means)
2
Last time
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Kernelized Perceptron
3
• If we had a black box (kernel) which told us the dot product of two examples x and y:– Could work entirely with the dual representation– No need to ever take dot products (“kernel trick”)
• Downside: slow if many examples get nonzero alpha
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Non-Linear Separators
4
• General idea: the original feature space can always be mapped to some higher-dimensional feature space where the training set is separable:
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Some Kernels
5
• Kernels implicitly map original vectors to higher dimensional spaces, take the dot product there, and hand the result back
• Linear kernel:
• Quadratic kernel:
• RBF: infinite dimensional representation
• Mercer’s theorem: if the matrix A with Ai,j=K(xi,xj) is positive definite, then K can be represented as the inner product of a higher dimensional space
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• Neuron = Perceptron with a non-linear
activation function
ai=g(ΣjWj,iaj)
6
Neuron
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• Feed-forward networks
– single-layer (perceptrons)
– multi-layer (perceptrons)
• can represent any boolean function
– Learning
• Minimize squared error of prediction
• Gradient descent: backpropagation algorithm
• Recurrent networks
– Hopfield networks have symmetric weights
Wi,j=Wj,i
– Boltzmann machines use stochastic activation
functions
7
Neural networks
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• a5=g5 (W3,5×a3+W4,5×a4)
=g5 (W3,5×g3 (W1,3×a1+W2,3×a2)+W4,5×g4 (W1,4×a1+W2,4×a2))
8
Feed-forward example
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K-Means
9
• An iterative clustering algorithm– Pick K random points
as cluster centers (means)
– Alternate:• Assign data instances
to closest mean
• Assign each mean to the average of its assigned points
– Stop when no points’ assignments change
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K-Means as Optimization
10
• Consider the total distance to the means:
• Each iteration reduces phi• Two states each iteration:
– Update assignments: fix means c, change assignments a
– Update means: fix assignments a, change means c
dis, = ,, tii i ik i ax c x ca
points assignments means
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Today: Planning
• A brief review of logic
• Planning problems
• Search algorithms
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• Atomic variables (atoms)
– can be true or false
– IsRaining
– Literal: an atomic variable or its negation
• Operators
– Not(x), And(x,y), Or(x,y)
– applied to atomic variables or formulas
• Semantics
– “meaning” w.r.t. true/false valuation
• Syntax
– purely symbolic inference rules
12
Language of propositional logic
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• Terms
– objects
– can be combined by functions
– e.g. NextInteger(x)
• Relation symbols
– functions that takes terms as inputs
– can take true or false
– E.g. IsRainyToday (0-ary), IsHuman(x) (1-ary), AtPosition(x,y) (2-ary)
– Literal: a relation symbol or its negation
• Operators
– Not(x), And(x,y), Or(x,y)
– Quantifiers: and ∀ ∃
– applied to relation symbols or formulas, but not to terms
• Semantics
– “meaning” w.r.t. true/false valuation of relation symbols
– Note: a term cannot be true or false
• Syntax
– purely symbolic inference rules13
Language in first-order logic
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• Caution: deliberately confusing semantics and syntax for illustration
• Syllogism
– All men are mortal
– Socrates is a man
– Therefore, Socrates is mortal
• Propositional logic
– SocratesIsMan→SocratesIsMortal
– SocratesIsMan = true
– Therefore, SocratesIsMortal = true
• First-order logic
– ∀ x (IsMan(x) →IsMortal(x))
– IsMan(Socrates) = true
– IsMortal(Socrates) = true14
Example of semantics
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Today: Planning
• A brief review of logic
• Planning problems
• Search algorithms
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Search Problems
16
• A search problem consists of:
– A state space ……
– A successor function
(with actions, costs)
– A start state and a goal test
• A solution is a sequence of actions (a plan) which transforms the start state to a goal state
• Algorithms– BFS, DFS, UCS
– Heuristics: best first, A*
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• State space
– compactly represented by logic
• Actions
– compactly represented by
• preconditions: activation rule
• effects: consequences
• A solution is a sequence of actions
17
Planning as a search problem
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State of the world (STRIPS language)
• Stanford Research Institute Problem Solver
• State of the world = conjunction of positive, ground, function-free literals– At(Home) AND IsAt (Umbrella, Home) AND
CanBeCarried(Umbrella) AND IsUmbrella(Umbrella) AND HandEmpty AND Dry
– Not OK as part of the state:• NOT(At(Home)) (negative)
• At(x) (not ground)
• At(Bedroom(Home)) (uses the function Bedroom)
• Any literal not mentioned is assumed false– Other languages make different assumptions, e.g.,
negative literals part of state, unmentioned literals unknown
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An action: TakeObject• TakeObject(location, x)
• Preconditions:– HandEmpty
– CanBeCarried(x)
– At(location)
– IsAt(x, location)
• Effects (“NOT something” means that that something should be removed from state):– Holding(x)
– NOT(HandEmpty)
– NOT(IsAt(x, location))
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Another action
• WalkWithUmbrella(location1, location2, umbr)
• Preconditions:–At(location1)
–Holding(umbr)
– IsUmbrella(umbr)
• Effects:– At(location2)
– NOT(At(location1))
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Yet another action
• WalkWithoutUmbrella(location1,
location2)
• Preconditions:
–At(location1)
• Effects:
– At(location2)
– NOT(At(location1))
– NOT(Dry)
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A goal and a plan• Goal: At(Work) AND Dry
• Recall initial state:
– At(Home) AND IsAt(Umbrella, Home) AND
CanBeCarried(Umbrella) AND IsUmbrella(Umbrella)
AND HandEmpty AND Dry
• TakeObject(Home, Umbrella)
– At(Home) AND CanBeCarried(Umbrella) AND
IsUmbrella(Umbrella) AND Dry AND Holding(Umbrella)
• WalkWithUmbrella(Home, Work, Umbrella)
– At(Work) AND CanBeCarried(Umbrella) AND
IsUmbrella(Umbrella) AND Dry AND Holding(Umbrella)
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Planning to write a paper• Suppose your goal is to be a co-author on an AI paper
with both theorems and experiments, within a year
LearnAbout(x,y)
Preconditions: HasTimeForStudy(x)
Effects: Knows(x,y), NOT(HasTimeForStudy(x))
HaveNewIdea(x)
Preconditions: Knows(x,AI), Creative(x)
Effects: Idea, Contributed(x)
ProveTheorems(x)
Preconditions: Knows(x,AI), Knows(x,Math), Idea
Effect: Theorems, Contributed(x)
PerformExperiments(x)
Preconditions: Knows(x,AI), Knows(x,Coding), Idea
Effect: Experiments, Contributed(x)
WritePaper(x)
Preconditions: Knows(x,AI), Knows(x,Writing), Idea, Theorems, Experiments
Effect: Paper, Contributed(x)
Goal: Paper AND Contributed(You)
FindExistingOpenProblem(x)
Preconditions: Knows(x,AI)
Effects: Idea
Name a few things that are missing/unrealistic…
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Some start states
Start1: HasTimeForStudy(You) AND Knows(You,Math) AND Knows(You,Coding) AND Knows(You,Writing)
Start2: HasTimeForStudy(You) AND Creative(You) AND Knows(Advisor,AI) AND Knows(Advisor,Math) AND Knows(Advisor,Coding) AND Knows(Advisor,Writing)
(Good luck with that plan…)
Start3: Knows(You,AI) AND Knows(You,Coding) AND Knows(OfficeMate,Math) AND HasTimeForStudy(OfficeMate) AND Knows(Advisor,AI) AND Knows(Advisor,Writing)
Start4: HasTimeForStudy(You) AND Knows(Advisor,AI) AND Knows(Advisor,Math) AND Knows(Advisor,Coding) AND Knows(Advisor,Writing)
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Today: Planning
• A brief review of logic
• Planning problems
• Search algorithms
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Forward state-space search (progression planning)
• Successors: all states that can be reached with an action whose preconditions are satisfied in current state
At(Home)
IsAt(Umbrella, Home)
CanBeCarried(Umbrella)
IsUmbrella(Umbrella)
HandEmpty
Dry
At(Home)
Holding(Umbrella)
CanBeCarried(Umbrella)
IsUmbrella(Umbrella)
Dry
TakeObject(Home, Umbrella)
At(Work)
IsAt(Umbrella, Home)
CanBeCarried(Umbrella)
IsUmbrella(Umbrella)
HandEmpty
WalkWithoutUm
brella(Home,
Work)
WalkWithUmbrella(
Home, Work,
Umbrella)
At(Work)
Holding(Umbrella)
CanBeCarried(Umbrella)
IsUmbrella(Umbrella)
Dry
WalkWithout
Umbrella(Wor
k, Home)
At(Home)
IsAt(Umbrella, Home)
CanBeCarried(Umbrella)
IsUmbrella(Umbrella)
HandEmpty
GOAL!
WalkWithoutUmbrella(
Home, Umbrella) (!)
WalkWithoutUm
brella(Home,
Work)
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Backward state-space search (regression planning)
• Predecessors: for every action that accomplishes one of the literals (and does not undo another literal), remove that literal and add all the preconditions
At(location1)
At(location2)
IsAt(umbr, location2)
CanBeCarried(umbr)
IsUmbrella(umbr)
HandEmpty
Dry
At(location1)
Holding(umbr)
IsUmbrella(umbr)
Dry
TakeObject(location2, umbr)
This is accomplished in the
start state, by substituting
location1=location2=Home,
umbr=Umbrella
WalkWithUmbrella(
location1, Work,
umbr)At(Work)
Dry
GOAL
WalkWithUmbrella(location2, location1)
WalkWithoutUmbrella can never be used, because it undoes Dry
(this is good)
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Heuristics for state-space search
• Cost of a plan: (usually) number of actions
• Heuristic 1: plan for each subgoal (literal) separately, sum costs of plans– Does this ever underestimate? Overestimate?
• Heuristic 2: solve a relaxed planning problem in which actions never delete literals (empty-delete-list heuristic)– Does this ever underestimate? Overestimate?
– Very effective, even though requires solution to (easy) planning problem
• Progression planners with empty-delete-list heuristic perform well
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Blocks world
• On(B, A), On(A, Table), On(D, C), On(C,
Table), Clear(B), Clear(D)
A
B
C
D
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Blocks world: Move action
• Move(x,y,z)
• Preconditions:
– On(x,y), Clear(x), Clear(z)
• Effects:
– On(x,z), Clear(y), NOT(On(x,y)), NOT(Clear(z))
A
B
C
D
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Blocks world: MoveToTable action
• MoveToTable(x,y)
• Preconditions:
– On(x,y), Clear(x)
• Effects:
– On(x,Table), Clear(y), NOT(On(x,y))
A
B
C
D
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Blocks world example
• Goal: On(A,B) AND Clear(A) AND On(C,D)
AND Clear(C)
• A plan: MoveToTable(B, A), MoveToTable(D,
C), Move(C, Table, D), Move(A, Table, B)
• Really two separate problem instances
A
B
C
D
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A partial-order plan
Goal: On(A,B) AND
Clear(A) AND
On(C,D) AND
Clear(C)
A
B
C
DStart
MoveToTable(
B,A)
MoveToTable(
D,C)
Move(A,
Table, B)
Move(C,
Table, D)
FinishAny total order on the actions consistent with
this partial order will work
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A partial-order plan (with more detail)
Start
MoveToTable(B,
A)
MoveToTable(D,
C)
Move(A,T
able, B)
Move(C,T
able, D)
Finish
On(B,A) Clear(B) On(D,C)Clear(D)
Clear(A)Clear(B) On(A, Table) Clear(D) Clear(C)On(C, Table)
On(A, B) On(C, D)Clear(A) Clear(C)
On(B,A) Clear(B) On(D,C)Clear(D)On(A, Table) On(C, Table)
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Not everything decomposes into multiple problems: Sussman
Anomaly
• Goal: On(A,B) AND On(B,C)
• Sequentially pursuing goals does not work
• Optimal plan: MoveToTable(C,A),
Move(B,Table,C), Move(A,Table,B)
A B
C
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An incorrect partial order plan for the Sussman Anomaly
Start
Finish
On(A, B)
On(A, Table)
MoveToTable(C,
A) Move(B,T
able,C)
Move(A,
Table,B)
On(B, C)
On(B, Table) Clear(B)On(C, A)Clear(C)
On(C, A)Clear(C)
Clear(A) On(A, Table) Clear(B)
Clear(B)Clear(C) On(B, Table)
Move(B,Table,C) must be
after MoveToTable(C,A),
otherwise it will ruin
Clear(C)
Move(A,Table,B) must be
after Move(B,Table,C),
otherwise it will ruin
Clear(B)
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Corrected partial order plan for the Sussman Anomaly
Start
Finish
On(A, B)
On(A, Table)
MoveToTable(C,
A) Move(B,T
able, C)
Move(A,
Table, B)
On(B, C)
On(B, Table) Clear(B)On(C, A)Clear(C)
On(C, A)Clear(C)
Clear(A) On(A, Table) Clear(B)
Clear(B)Clear(C) On(B, Table)
No more flexibility in the
order due to protection of
causal links