An OpenCL implementation of a forward sampling algorithm for - - PowerPoint PPT Presentation

An OpenCL implementation of a forward sampling algorithm for CP-logic

Wiebe Van Ranst Joost Vennekens

• KU Leuven

Campus De Nayer

CP-logic

• PLP language in family of distribution semantics languages
• PRISM [Sato,Kameya], Independent Choice Logic

[Poole], CP-logic [Vennekens et al.], …

(E1 : 1) ∨ · · · ∨ (En : n) ← .

FOL formula Atom Probability
 (sum ≤ 1) Condition φ, when true, causes one of the effects Ei (or none if sum < 1). Each of the Ei is caused with probability ϵi.

Example

• Random walk
• Never turn back
• What is the probability?
• Of getting stuck
• Finding exit

Example (1/4)

(Go(Right,t) : ⅓) ∨ (Go(Straight,t) : ⅓) ∨ (Go(Left,t): ⅓) ← At(p,t) ∧ FourWay(p).

• (Go(Left,t) : ½) ∨ (Go(Right, t) : ½) ← At(p,t) ∧ Facing(d,t) ∧ TJunction(p,d).

(Go(Left,t) : ½) ∨ (Go(Straight, t) : ½) ← At(p,t) ∧ Facing(d,t) ∧ LeftTurn(p). (Go(Right,t) : ½) ∨ (Go(Straight, t) : ½) ← At(p,t) ∧ Facing(d,t) ∧ RightTurn(p).

• Go(Left,t) ← At(p,t) ∧ LeftBend(p).

Go(Right,t) ← At(p,t) ∧ RightBend(p). Go(Straight,t) ← At(p,t) ∧ Straight(p).

At(to, t +1) ← At(from, t) ∧ Go(dir,t) ∧ Road(from,dir,to).

• Facing(d2,t +1) ← Facing(d1,t) ∧ Go(Left,t) ∧ Next(d2,d1).

Facing(d2,t +1) ← Facing(d1,t) ∧ Go(Right,t) ∧ Next(d1,d2). Facing(d, t +1) ← Facing(d,t) ∧ Go(Straight,t).

• Next(N,E). Next(E,S). Next(S,W). Next(W,N).
• At(Start,0).

Heading(East,0).

Example (2/4)

Road(1,N,2). Road(1,E,3). Road(2,N,4). Road(2,E,5). …

• Road(x,S,y) ← Road(y,N,x).

Road(x,W,y) ← Road(y,W,x).

Example (3/4)

FourWay(p) ← Road(p,N,_) ∧ Road(p,E,_) ∧ Road(p,S,_) ∧ Road(p,W,_).

• TJunction(p,d) ← ¬FourWay(p) ∧ Next(left,d) ∧ Next(d,right) ∧ Road(p,left,_) ∧

Road(p,right,_). LeftTurn(p,d) ← ¬FourWay(p) ∧ Next(left,d) ∧ Road(p,left,_) ∧ Road(p,d,_). RightTurn(p,d) ← ¬FourWay(p) ∧ Next(d,right) ∧ Road(p,right,_) ∧ Road(p,d,_).

• ThreeJunction(p,d) ← TJunction(p,d) ∨ LeftTurn(p,d) ∨ RightTurn(p,d).
• LeftBend(p,d) ← ¬FourWay(p) ∧ ¬ThreeJunction(p) ∧ Next(left,d) ∧ Road(p,left,_).

RightBend(p,d) ← ¬FourWay(p) ∧ ¬ThreeJunction(p) ∧ Next(d,right) ∧ Road(p,right,_). Straight(p,d) ← ¬FourWay(p) ∧ ¬ThreeJunction(p) ∧ Road(p,d,_).

Example (4/4)

Simplifying assumptions

• All theories have been grounded
• No negation in bodies
• Two-valued well-founded model semantics
• Makes both implementation and semantics bit more

difficult

• Probabilities in the head sum to 1

Distribution semantics [Sato]

CP-theory Instance 1 Instance n

(a : α) ∨ (b : β) ← φ. (c : γ) ∨ (d : δ) ← ψ. (a : α) ∨ (b : β) ← φ. (c : γ) ∨ (d : δ) ← ψ. (a : α) ∨ (b : β) ← φ. (c : γ) ∨ (d : δ) ← ψ.

…

α · δ

β · γ

Distribution semantics [Sato]

CP-theory Instance 1 Instance n

(a : α) ∨ (b : β) ← φ. (c : γ) ∨ (d : δ) ← ψ.

…

α · δ

β · γ

b ← φ. c ← ψ.

a ← φ. d ← ψ.

Distribution semantics [Sato]

CP-theory Instance 1 Instance n

(a : α) ∨ (b : β) ← φ. (c : γ) ∨ (d : δ) ← ψ.

…

α · δ

β · γ

World 1 World m

Σ Σ

| = | =

…

Query yes no yes … … … no

b ← φ. c ← ψ.

a ← φ. d ← ψ.

Distribution semantics

… …

Inference

• Problog system 


[De Raedt,Kimmig,Fierens,…]

• PITA in XSB [Riguzzi, Swift]
• CVE [Meert, Blockeel]
• MCINTYRE [Riguzzi]

Exact Approximative All query-based Query
 (+ evidence) All/some 
 proofs Disjoint sum

This talk: Sampling with forward chaining

• Sometimes no query
• Debugging: prob = 0
• Diagnosis: MPE
• Different search strategy
• Maybe more appropriate for

certain theories

• See experiments

Efficient implementation: GPGPU

http://michaelgalloy.com/2013/06/11/cpu-vs-gpu-performance.html

Writing efficient GPGPU code

• Massive number of simple cores in SIMD
• Each core executes dumb code
• Be smart about
• Memory transfer
• Memory access
• Good fit

Approach

CPU GPU

Ground CP-theory

1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample

Results

1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample

Approach

CPU GPU

Ground CP-theory Results

1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample 1 sample

I = {} Repeat: Find applicable rule Sample atom from head Add atom to I Until no more rules

Approach

CPU GPU

Ground CP-theory Results

Probability tree semantics

… 0.5 0.5 Equivalent [TPLP2009]
 Correctness: sample = branch … …

Optimisations

• [MEM] Ensure coalesced memory access
• [INI] Initialise data structures on GPU
• [PRI] Use private instead of local memory
• [RAN] Generate random numbers on GPU
• [RED] Compute results on GPU
• [CHU] Start m ≪ n samples on GPU, copy back, repeat


+ stopping criterion

Benchmarks

• Taken from: W. Meert, J. Struyf, and H. Blockeel. CP-logic

theory inference with contextual variable elimination and comparison to BDD based inference methods. In Proc. ILP, 2009.

• Bloodtype, GrowingBody, GrowingHead
• Linux machine with similarly priced GPU,CPU:
• Intel Core i7 965 CPU (3.20GHz)
• NVIDIA GeForce GTX 295 (GT200)

Optimisations

GrowingHead

best: MCINTYRE, ok: OpenCL

GrowingBody

best: CVE, ok: OpenCL, PITA, ProbLog

Bloodtype (non ground)

best: PITA, ok: (C)VE, OpenCL(?)

Bloodtype (non ground)

best: PITA, ok: (C)VE, OpenCL(?) $1000

Experimental results

best

• k

PITA 1 1 CVE 1 1 Problog 1 Problog MC Problog 
 k-best MCINTYRE 1 OpenCL 2 or 3

Conclusion

• Naive forward chaining sampling algorithm
• Implemented in a smart way in OpenCL
• Complements existing approaches
• Not query-based
• Different computation strategy
• Experimental results
• Optimisations really help
• Performance is always ok, though never best
• Close to formal semantics, maybe easier for extensions

Thank you