Fine-Grained Semantics for Probabilistic Programs Benjamin Timon - - PowerPoint PPT Presentation

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fine grained semantics for probabilistic programs
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Fine-Grained Semantics for Probabilistic Programs Benjamin Timon - - PowerPoint PPT Presentation

Oct 17, 2022 257 likes •624 views

Fine-Grained Semantics for Probabilistic Programs Benjamin Timon Martin Bichsel Gehr Vechev Probabilistic models Security Networking Machine Learning Robotics Randomized Algorithms Probabilistic programming Probability density

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Fine-Grained Semantics for Probabilistic Programs

Benjamin Bichsel Timon Gehr Martin Vechev

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Probabilistic models

Networking Security Machine Learning Robotics Randomized Algorithms

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Probabilistic programming

x := uniform(0,1); y := uniform(0,1); return x+y;

Probability density function Cumulative distribution function

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Observations

x := uniform(0,1); y := uniform(0,1);

  • bserve(x+y>=1);

return x+y;

Probability density function Cumulative distribution function

0.5

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Partial functions

x := uniform(-1,1); x = sqrt(x); return x;

Probability density function Cumulative distribution function

0.5

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Loops

x := 2; while x>0 { if x<4 { if flip(0.5) { x++; } else { x--; } } } return x;

2 3 4 1

1/2 1/2 1/2 1/2 1/2 1/2 1 return 0 start

0.5

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Interaction of exceptions

x := 0; while 1 { x = x/x; } return x; Better:

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Interaction of exceptions II

x := 0; while x==0 { x = flip(0.5);

  • bserve(x==0);

} return x; Better:

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Interaction of exceptions III

x := 10; d := gauss(0,1);

  • bserve(d>=-0.5);

while x>=0 { x-=d; } return x; Better:

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Existing denotational semantics

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Our work – Denotational semantics

Explicitly distinguish exceptions:

  • non-termination ( )
  • errors ( )
  • observation failure ( )

Our semantics also support

  • Mixing continuous and discrete distributions
  • Arrays
  • The score primitive
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Benefits

  • Deeper understanding of

probabilistic semantics

  • Establish correctness of probabilistic

solvers (e.g. PSI)

  • More efficient normalization in

probabilistic solvers

  • Generalize to arbitrary number of

exceptions

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Measure theory

1

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Measure theory

1

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Denotational semantics

x := uniform(-1,1); x = sqrt(x); return x;

lift lift lift lift

sqrt(uniform(-1,1))

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sqrt(uniform(-1,1))

Denotational semantics - Constants

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sqrt(uniform(-1,1))

Denotational semantics - Constants

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sqrt(uniform(-1,1))

Denotational semantics - Constants

Dirac delta

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Denotational semantics - Constants

sqrt(uniform(-1,1))

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Denotational semantics - Functions

sqrt(uniform(-1,1))

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Denotational semantics - Functions

sqrt(uniform(-1,1))

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Denotational semantics - Functions

sqrt(uniform(-1,1))

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Denotational semantics - Functions

sqrt(uniform(-1,1))

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Denotational semantics - Functions

sqrt(uniform(-1,1))

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Denotational semantics - Composition

sqrt(uniform(-1,1))

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Denotational semantics - Loops

n := 0; while !flip(0.5) { n++; }

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Denotational semantics - Loops

n := 0; while !flip(0.5) { n++; }

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Denotational semantics - Loops

n := 0; while !flip(0.5) { n++; }

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Denotational semantics - Loops

n := 0; while !flip(0.5) { n++; }

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Probability kernel

Theorem: The semantics of each expression and each statement is a probability kernel.

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Properties of Semantics - Commutativity

F() { while 1 { skip; } return 0; }

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Properties of Semantics - Associativity

Proof: Relies on associativity of the product of kernels and composition of kernels.

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Score primitive

x := gauss(0,1); score( ); return x;

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Score primitive vs non-termination

i := 0; while 1 { if i==0 { score(2); } else { score(0.5); } i = 1-i; }

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Score primitive – s-finite kernels

Theorem: After adding the score primitive and abolishing non- termination, the semantics of each expression and each statement is an s-finite kernel.

Informally:

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Features of probabilistic programming languages

  • bservations

errors mix discrete and continuous higher-order score loops recursion non-determinism