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Motivation – 1Drawing Visual Concepts
Dataset - Human Drawn Shape
Synthesize
Program..Goto(r1,0); draw(shape1);Goto(r3,10);draw(shape3);assert (contains 1 0);..
Execute
Goto(r1,0)
Routine
(given as primitives)
argument
Machine OutputProgram(I1)
Ii – Input representation
for the input.
1
2 3
Motivation – 2Learning Morphological Rules
Style, styledhatch,hatchedArticulate,articulatedPay,paidLay,laidNeed,needed
Program
if [ property1.value1 == True ] (stem + d)elseif [ property3.value2 > 5 ] (stem +ed)Elseif [property4.value5 == “y”] (stem + id)
Synthesize Execute
Style, styledhatch,hatchedArticulate,articulatedPay,paidLay,laidNeed,neededRun,ran
Noise
<stem, word in past tense>Program(snatch) = snatched
Can we quantify the length of the Program description?
Can we quantify the length of dataset encoded/represented in terms of the properties as required by the program ?
• Can we cast the problem as an optimization problem, so that we can find an optimal solution i.e. program & data encoding with the minimum description length
Optimization Problem
• Task is about compressing the data, and yet represent the same in terms of interpretable entities i.e. Logical Dimensionality reduction
Logical Dimensionality
Reduction
Introduction
Problem Framing
Description length priors over programs Pf (·), (eg, linguistic rules)
Priors over the inputs I, PI (·) to f,(eg, stems)
N observations, { xi }i = 1 to N , (eg, words)
Noise model: Px|z(· | ·) , where z I is defined as f(Ii)
Plate Diagram
Solution
• Manually provide a rough outline of the program to be induced.
• Also called as sketch
• probabilistic context-free grammar
• automatically translate sketches into Satisfiability Modulo Theories (SMT) problems.
• Intractable in general, but often solved efficiently in practice (Formal verification)
Solution
A context-free grammar (CFG) is a 4-tuple G = (N,Σ, R, S) where:• N – Non terminals set | Σ – Terminals set | R - is a finite set of rules of the form
X→Y1Y2. . . Yn , where X ∈ N, n ≥ 0 , and YI ∈ ( N ∪ Σ) for I = 1. . . N
CFG
A context-free grammar (CFG) is a 4-tuple G = (N,Σ, R, S) where:• N – Non terminals set | Σ – Terminals set | R - is a finite set of rules of the form
X→Y1Y2. . . Yn , where X ∈ N, n ≥ 0 , and YI ∈ ( N ∪ Σ) for I = 1. . . N
A PCFG is a CFG with a probability on production rules i.e. G = (N,Σ, R, S, q)• q - Probabilities on the production
PCFG
PCFG - Sketch
Define the program
primitives.
Constrain the program space with a
PCFG
Sketch AND/OR Graph
OR Node corresponnds to choice
AND node corresponds to descendant
Each program is a path through the AND/OR Graph
Recursiveness helps to have paths of any length.
Currently authors bound the length (arbitrary constant)
Cij – is a Boolean value 1 or 0, depending on
which production is being derived
All the edges in a path will have value 1. All others will be 0
OR
AND
OR
Constraints The SMT Solver can verify the correctness of the path over inputs
when the path is represented as a set of constraints.
Denotations
Mathematical Objects, that describe the meaning of entities in a
language
Every node in the selected path has a denotation
In denotation, each non-terminal is an expression (or a routine), which takes an input I and the range for
the output is known
The path will give the sequence of the routines with the appropriate
values for the arguments, which is obtained from the input
[Expression] (Input) = Output
The output is dependent on the input
The optimization algorithm iterates by finding numerous solutions. At each step along with constraints of the program a new constraint is
current length of the program
Initialize N inputs (unknown)
Find denotations and constraints for all paths and feed to a SMT
solver
Iteratively add the minimum length as constraint to find satisfiable
solutions of lesser length
Optimization loop
Tress rooted at each non-terminal
Descendants in the trees rooted at non
terminal
Encoding of the Input w.r.t. the program
Encoding the programs for SMT
Calculate length of the program
Denotation of the program
Form the constraints
•shapes, coordinates, distances, angles, scales
Program inputs:
•Image parseProgram output:
•control a turtle, but:
•Restricted to alternatingly moving and drawing
•No arithmetic on real variables
•No rotation of shapes
Constraints on program
space:
Experiments: Visual Concepts
• Comparing human performance on the SVRT with classification accuracy for machine learning approaches.
• Human accuracy is the fraction of humans that learned the concept: 0% is chance level.
• Machine accuracy is the fraction of correctly classified held out examples: 50% is chance level.
• Area of circles is proportional to the number of observations at that point.
• Dashed line is average accuracy. • Program synthesis: this work trained on 6
examples. ConvNet: A variant of LeNet5 trained on 2000 examples. Parse (Image) features: discriminative learners on features of parse (pixels) trained on 6 (10000) examples. Humans given an average of 6.27 examples and solve an average of 19.85 problems
Experiments: Results
Experiments: Morphology Learning
• The underlying stemsProgram inputs:
• Tuple of all inflections for a stemProgram output:
• Has form: tuple of expressions, one for each tense.
• Attend only to stem ending
• Consider only suffixes
Constraints on program space:
Experiments: Results
Thanks