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Certified Roundoff Error Bounds using Semidefinite Programming and - - PowerPoint PPT Presentation

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Certified Roundoff Error Bounds using Semidefinite Programming and Formal Floating Point Arithmetic Victor Magron , CNRS VERIMAG Certification is joint work with G. Constantinides and A. Donaldson Formalization is joint work with T. Weisser and

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SLIDE 1

Certified Roundoff Error Bounds using Semidefinite Programming and Formal Floating Point Arithmetic

Victor Magron, CNRS VERIMAG

Certification is joint work with G. Constantinides and A. Donaldson Formalization is joint work with T. Weisser and B. Werner

Effective Analysis: Foundations, Implementations, Certification

CIRM, 13 January 2016

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 1 / 28

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SLIDE 2

Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 2 / 28

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SLIDE 3

Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

  • M. Lecat, Erreurs des Mathématiciens des origines à nos

jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, . . . )

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 2 / 28

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SLIDE 4

Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

  • M. Lecat, Erreurs des Mathématiciens des origines à nos

jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, . . . ) Ariane 5 launch failure, Pentium FDIV bug

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 2 / 28

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SLIDE 5

Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

  • M. Lecat, Erreurs des Mathématiciens des origines à nos

jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, . . . ) Ariane 5 launch failure, Pentium FDIV bug U.S. Patriot missile killed 28 soldiers from the U.S. Army’s Internal clock: 0.1 sec intervals Roundoff error on the binary constant “0.1”

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 2 / 28

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Errors and Proofs

GUARANTEED OPTIMIZATION Input : Linear problem

(LP), geometric, semidefinite (SDP)

Output : solution + certificate

numeric-symbolic ❀ formal

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 3 / 28

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SLIDE 7

Errors and Proofs

GUARANTEED OPTIMIZATION Input : Linear problem

(LP), geometric, semidefinite (SDP)

Output : solution + certificate

numeric-symbolic ❀ formal

VERIFICATION OF CRITICAL SYSTEMS

Reliable software/hardware embedded codes Aerospace control

molecular biology, robotics, code synthesis, . . .

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 3 / 28

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SLIDE 8

Errors and Proofs

GUARANTEED OPTIMIZATION Input : Linear problem

(LP), geometric, semidefinite (SDP)

Output : solution + certificate

numeric-symbolic ❀ formal

VERIFICATION OF CRITICAL SYSTEMS

Reliable software/hardware embedded codes Aerospace control

molecular biology, robotics, code synthesis, . . .

Efficient Verification of Nonlinear Systems

Automated precision tuning of systems/programs

analysis/synthesis

Efficiency sparsity correlation patterns Certified approximation algorithms

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 3 / 28

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SLIDE 9

Roundoff Error Bounds

Real : p(x) := x1 × x2 + x3 Floating-point : ˆ p(x, e) := [x1x2(1 + e1) + x3](1 + e2) Input variable constraints x ∈ S Finite precision ❀ bounds over e | ei | 2−m m = 24 (single) or 53 (double) Guarantees on absolute round-off error | ˆ p − p | ?

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 4 / 28

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SLIDE 10

Nonlinear Programs

Polynomials programs : +, −, × x2x5 + x3x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 5 / 28

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Nonlinear Programs

Polynomials programs : +, −, × x2x5 + x3x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) Semialgebraic programs: | · |, √, /, sup, inf 4x 1 + x 1.11

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 5 / 28

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Nonlinear Programs

Polynomials programs : +, −, × x2x5 + x3x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) Semialgebraic programs: | · |, √, /, sup, inf 4x 1 + x 1.11 Transcendental programs: arctan, exp, log, . . . log(1 + exp(x))

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 5 / 28

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Existing Frameworks

Classical methods: Abstract domains [Goubault-Putot 11] FLUCTUAT: intervals, octagons, zonotopes Interval arithmetic [Daumas-Melquiond 10] GAPPA: interface with COQ proof assistant

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 6 / 28

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Existing Frameworks

Recent progress: Affine arithmetic + SMT [Darulova 14] rosa: sound compiler for reals (in SCALA) Symbolic Taylor expansions [Solovyev 15] FPTaylor: certified optimization (in OCAML and HOL-LIGHT)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 6 / 28

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SLIDE 15

Contributions

Maximal Roundoff error of the program implementation of f: r⋆ := max |ˆ f(x, e) − f(x)| Decomposition: linear term l w.r.t. e + nonlinear term h r⋆ max |l(x, e)| + max |h(x, e)| Semidefinite programming (SDP) bounds for l Coarse bound of h with interval arithmetic

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 7 / 28

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Contributions

1 Comparison with SMT and linear/affine/Taylor

arithmetic: ❀ Efficient optimization + Tight upper bounds

2 Extensions to transcendental/conditional programs 3 Formal verification of SDP bounds 4 Open source tool Real2Float (in OCAML and COQ)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 7 / 28

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SLIDE 17

Introduction Semidefinite Programming for Polynomial Optimization Roundoff Error Bounds with Sparse SDP Formal Floating-Point Arithmetic Conclusion

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What is Semidefinite Programming?

Linear Programming (LP): min

z

c

⊤z

s.t. A z d .

Linear cost c Linear inequalities “∑i Aij zj di”

Polyhedron

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 8 / 28

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SLIDE 19

What is Semidefinite Programming?

Semidefinite Programming (SDP): min

z

c

⊤z

s.t.

∑

i

Fi zi F0 .

Linear cost c Symmetric matrices F0, Fi Linear matrix inequalities “F 0” (F has nonnegative eigenvalues)

Spectrahedron

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 9 / 28

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SLIDE 20

What is Semidefinite Programming?

Semidefinite Programming (SDP): min

z

c

⊤z

s.t.

∑

i

Fi zi F0 , A z = d .

Linear cost c Symmetric matrices F0, Fi Linear matrix inequalities “F 0” (F has nonnegative eigenvalues)

Spectrahedron

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 10 / 28

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SLIDE 21

Applications of SDP

Combinatorial optimization Control theory Matrix completion Unique Games Conjecture (Khot ’02) : “A single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems.” (Barak and Steurer survey at ICM’14) Solving polynomial optimization (Lasserre ’01)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 11 / 28

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SLIDE 22

SDP for Polynomial Optimization

Prove polynomial inequalities with SDP: p(a, b) := a2 − 2ab + b2 0 . Find z s.t. p(a, b) =

  • a

b z1 z2 z2 z3

  • a

b

  • .

Find z s.t. a2 − 2ab + b2 = z1a2 + 2z2ab + z3b2 (A z = d)

z1 z2 z2 z3

  • =

1

  • F1

z1 + 1 1

  • F2

z2 + 1

  • F3

z3

  • F0

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 12 / 28

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SLIDE 23

SDP for Polynomial Optimization

Choose a cost c e.g. (1, 0, 1) and solve: min

z

c

⊤z

s.t.

∑

i

Fi zi F0 , A z = d . Solution

z1 z2 z2 z3

  • =

1 −1 −1 1

  • (eigenvalues 0 and 2)

a2 − 2ab + b2 =

  • a

b 1 −1 −1 1

  • a

b

  • = (a − b)2 .

Solving SDP = ⇒ Finding SUMS OF SQUARES certificates

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 13 / 28

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SLIDE 24

SDP for Polynomial Optimization

General case: Semialgebraic set S := {x ∈ Rn : g1(x) 0, . . . , gm(x) 0} p∗ := min

x∈S p(x): NP hard

Sums of squares (SOS) Σ[x] (e.g. (x1 − x2)2) Q(S) :=

  • σ0(x) + ∑m

j=1 σj(x)gj(x), with σj ∈ Σ[x]

  • Fix the degree 2k of products:

Qk(S) :=

  • σ0(x) +

m

∑

j=1

σj(x)gj(x), with deg σj gj 2k

  • Victor Magron

Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 14 / 28

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SDP for Polynomial Optimization

Hierarchy of SDP relaxations: λk := sup

λ

  • λ : p − λ ∈ Qk(S)
  • Convergence guarantees λk ↑ p∗ [Lasserre 01]

Can be computed with SDP solvers (CSDP, SDPA) “No Free Lunch” Rule: (n+2k

n ) SDP variables

Extension to semialgebraic functions r(x) = p(x)/

  • q(x)

[Lasserre-Putinar 10]

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 15 / 28

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SLIDE 26

Sparse SDP Optimization [Waki, Lasserre 06]

Correlative sparsity pattern (csp) of variables x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6)

6 4 5 1 2 3

1 Maximal cliques C1, . . . , Cl 2 Average size κ ❀ (κ+2k κ )

variables C1 := {1, 4} C2 := {1, 2, 3, 5} C3 := {1, 3, 5, 6} Dense SDP: 210 variables Sparse SDP: 115 variables

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 16 / 28

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SLIDE 27

Introduction Semidefinite Programming for Polynomial Optimization Roundoff Error Bounds with Sparse SDP Formal Floating-Point Arithmetic Conclusion

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SLIDE 28

Polynomial Programs

Input: exact f(x), floating-point ˆ f(x, e), x ∈ S, | ei | 2−m Output: Bound for f − ˆ f

1: Error r(x, e) := f(x) − ˆ

f(x, e) = ∑

α

rα(e)xα

2: Decompose r(x, e) = l(x, e) + h(x, e), l linear in e 3: l(x, e) = ∑n′

i=0 si(x)ei

4: Maximal cliques correspond to {x, e1}, . . . , {x, en′} 5: Bound l(x, e) with sparse SDP relaxations (and h with IA)

Dense relaxation: (n+n′+2k

n+n′ )

SDP variables Sparse relaxation: n′(n+1+2k

n+1 )

SDP variables

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 17 / 28

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Preliminary Comparisons

f(x) := x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) x ∈ [4.00, 6.36]6 , e ∈ [−ǫ, ǫ]15 , ǫ = 2−24 Dense SDP: (6+15+4

6+15 ) = 12650 variables ❀ Out of memory

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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Preliminary Comparisons

f(x) := x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) x ∈ [4.00, 6.36]6 , e ∈ [−ǫ, ǫ]15 , ǫ = 2−24 Dense SDP: (6+15+4

6+15 ) = 12650 variables ❀ Out of memory

Sparse SDP Real2Float tool: 15(6+1+4

6+1 ) = 4950 ❀ 759ǫ

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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SLIDE 31

Preliminary Comparisons

f(x) := x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) x ∈ [4.00, 6.36]6 , e ∈ [−ǫ, ǫ]15 , ǫ = 2−24 Dense SDP: (6+15+4

6+15 ) = 12650 variables ❀ Out of memory

Sparse SDP Real2Float tool: 15(6+1+4

6+1 ) = 4950 ❀ 759ǫ

Interval arithmetic: 922ǫ (10 × less CPU)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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Preliminary Comparisons

f(x) := x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) x ∈ [4.00, 6.36]6 , e ∈ [−ǫ, ǫ]15 , ǫ = 2−24 Dense SDP: (6+15+4

6+15 ) = 12650 variables ❀ Out of memory

Sparse SDP Real2Float tool: 15(6+1+4

6+1 ) = 4950 ❀ 759ǫ

Interval arithmetic: 922ǫ (10 × less CPU) Symbolic Taylor FPTaylor tool: 721ǫ (21 × more CPU)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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SLIDE 33

Preliminary Comparisons

f(x) := x2x5 + x3x6 − x2x3 − x5x6 + x1(−x1 + x2 + x3 − x4 + x5 + x6) x ∈ [4.00, 6.36]6 , e ∈ [−ǫ, ǫ]15 , ǫ = 2−24 Dense SDP: (6+15+4

6+15 ) = 12650 variables ❀ Out of memory

Sparse SDP Real2Float tool: 15(6+1+4

6+1 ) = 4950 ❀ 759ǫ

Interval arithmetic: 922ǫ (10 × less CPU) Symbolic Taylor FPTaylor tool: 721ǫ (21 × more CPU) SMT-based rosa tool: 762ǫ (19 × more CPU)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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SLIDE 34

Preliminary Comparisons

R e a l 2 F l

  • a

t r

  • s

a F P T a y l

  • r

200 400 600 800 1,000 759ǫ 762ǫ 721ǫ CPU Time Error Bound (ǫ)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 18 / 28

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SLIDE 35

Extensions: Transcendental Programs

Reduce f ∗ := infx∈K f(x) to semialgebraic optimization

a y par+

a1

par+

a2

par−

a2

par−

a1

a2 a1 arctan m M

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 19 / 28

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SLIDE 36

Extensions: Programs with Conditionals

if (p(x) 0) f(x); else g(x); DIVERGENCE PATH ERROR: r⋆ := max{ max

p(x)0,p(x,e)0 | ˆ

f(x, e) − g(x) | max

p(x)0,p(x,e)0 | ˆ

g(x, e) − f(x) | max

p(x)0,p(x,e)0 | ˆ

f(x, e) − f(x) | max

p(x)0,p(x,e)0 | ˆ

g(x, e) − g(x) | }

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 20 / 28

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SLIDE 37

Comparison with rosa

Relative bound precision Relative execution time

a b c d e f g h i j k l m

  • p

q r t u v w x y z 10 100 −10 1 −1 0.5 −0.5

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 21 / 28

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SLIDE 38

Comparison with FPTaylor

Relative bound precision Relative execution time

a b c d e f g h i jk l m n

  • p

q r t u v w x α β γ δ 10 100 −10 1 −1 0.5 −0.5

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 22 / 28

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SLIDE 39

Introduction Semidefinite Programming for Polynomial Optimization Roundoff Error Bounds with Sparse SDP Formal Floating-Point Arithmetic Conclusion

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SLIDE 40

Interval Coefficient Polynomials

SOS certificate for 0 x1, x2 1 ∧ x2

1 x2 ⇒ x2 − 2x1 + 1 0:

x2 − 2x1 + 1 = (1 − x1)2 + x2 − x2

1

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 23 / 28

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SLIDE 41

Interval Coefficient Polynomials

SOS certificate for 0 x1, x2 1 ∧ x2

1 x2 ⇒ x2 − 2x1 + 1 0:

x2 − 2x1 + 1 = (1 − x1)2 + x2 − x2

1

SDP solvers only find approximate certificates: x2 − 2x1 + 1 ≃ 1.00007(0.99977 − 1.00022x1 − 0.00011x2)2 + 0.000332(−0.408035x1 + 0.816664x2 − 0.408126)2 + 0.000284x2 + 0.000116(1 − x2) + 1.00034(x2 − x2

1)

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 23 / 28

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SLIDE 42

Interval Coefficient Polynomials

SOS certificate for 0 x1, x2 1 ∧ x2

1 x2 ⇒ x2 − 2x1 + 1 0:

x2 − 2x1 + 1 = (1 − x1)2 + x2 − x2

1

SDP solvers only find approximate certificates: x2 − 2x1 + 1 ≃ 1.00007(0.99977 − 1.00022x1 − 0.00011x2)2 + 0.000332(−0.408035x1 + 0.816664x2 − 0.408126)2 + 0.000284x2 + 0.000116(1 − x2) + 1.00034(x2 − x2

1)

Exact error polynomial: ε(x) := 0.000232209x2

1 − 5.81334 × 10−7x1x2 − 0.0000297356x1

+ 0.000221436x2

2 + 0.0000621035x2 − 0.000201126

How to employ numerical certificates for formal verification?

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 23 / 28

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SLIDE 43

How to use numerical certificates in COQ?

tactic strategy ε(x) := 0.000232209x2

1 − 5.81334 × 10−7x1x2 − 0.0000297356x1

+ 0.000221436x2

2 + 0.0000621035x2 − 0.000201126

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 24 / 28

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SLIDE 44

How to use numerical certificates in COQ?

tactic strategy MICROMEGA uses heuristics to get an exact representation ε′(x) =

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 24 / 28

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SLIDE 45

How to use numerical certificates in COQ?

tactic strategy MICROMEGA uses heuristics to get an exact representation NLCERTIFY gives lower bound on ε by exact computations ε∗ := 0.000232209x2

1−5.81334 × 10−7x1x2−0.0000297356x1

+ 0.000221436x2

2 + 0.0000621035x2−0.000201126

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 24 / 28

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SLIDE 46

How to use numerical certificates in COQ?

tactic strategy MICROMEGA uses heuristics to get an exact representation NLCERTIFY gives lower bound on ε by exact computations NLVERIFY use interval arithmetics to bound ε ε∗ := interval enclosure of ε

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 24 / 28

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SLIDE 47

Interval Coefficient Polynomials

Floating point numbers F(p) := Fr,p with radix r and precision p Fast, certified inside COQ (FLOCQ, Boldo/Melquiond). In this talk r = 10, in the implementation r = 2. Intervals Ip := Ir,p with floating point bounds Fp Keep track of roundoff errors. Box constraints: x ∈ B Coefficient enclosure [·]p and variable enclosure |·|B

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 25 / 28

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SLIDE 48

Interval Coefficient Polynomials

Replace coefficients by intervals to speed up computation: f := 1 3x − 1 3x = 0 [f]2 = [0.33, 0.34]x − [0.33, 0.34]x [0]2 = [0.00, 0.00]

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 25 / 28

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SLIDE 49

Interval Coefficient Polynomials

Replace variables by intervals to obtain bounds on the function: With B = [−1, 1] × [0, 1] × [0, 1], x1(x2 − x3) = x1x2 − x2x3 |x1(x2 − x3)|B = [−1, 1][−1, 1] = [−1, 1] |x1x2 − x1x3|B = [−1, 1] − [−1, 1] = [−2, 2]

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 25 / 28

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SLIDE 50

Coq Implementation

Theorem:

  • [f]p
  • B ⊆ [ℓ, ∞) ⇒ f ℓ on B.

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 26 / 28

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SLIDE 51

Coq Implementation

Theorem:

  • [f]p
  • B ⊆ [ℓ, ∞) ⇒ f ℓ on B.

COQVersion:

Lemma toPolI_ok p box x : x ∈ box → eval x p ∈ Vencl box (toPolI p).

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 26 / 28

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SLIDE 52

Comparison with FPTaylor

Relative informal execution time Relative formal execution time

a b c d e f g h i 10 100 10 100

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 27 / 28

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SLIDE 53

Introduction Semidefinite Programming for Polynomial Optimization Roundoff Error Bounds with Sparse SDP Formal Floating-Point Arithmetic Conclusion

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SLIDE 54

Conclusion

Sparse SDP relaxations analyze NONLINEAR PROGRAMS: Polynomial and transcendental programs Handles conditionals, input uncertainties, . . . Certified ❀ Formal roundoff error bounds Real2Float open source tool:

http://nl-certify.forge.ocamlcore.org/real2float.html

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 28 / 28

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SLIDE 55

Conclusion

Further research: Improve formal polynomial checker Roundoff error analysis with while/for loops Automatic FPGA code generation

Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 28 / 28

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SLIDE 56

End

Thank you for your attention! http://www-verimag.imag.fr/~magron

  • V. Magron, G. Constantinides, A. Donaldson. Certified

Roundoff Error Bounds Using Semidefinite Programming, arxiv.org/abs/1507.03331