How Futile are Mindless Assessments of Roundoff in Floating-Point
Certified Roundoff Error Bounds using Semidefinite Programming · 2017-11-26 · Nonlinear Programs...
Transcript of Certified Roundoff Error Bounds using Semidefinite Programming · 2017-11-26 · Nonlinear Programs...
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Certified Roundoff Error Bounds usingSemidefinite Programming
Victor Magron, CNRS VERIMAG
joint work with G. Constantinides and A. Donaldson
INRIA Mescal Team Seminar19 November 2015
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Errors and Proofs
Mathematicians and Computer Scientists want toeliminate all the uncertainties on their results. Why?
M. Lecat, Erreurs des Mathématiciens des origines à nosjours, 1935.
; 130 pages of errors! (Euler, Fermat, Sylvester, . . . )
Ariane 5 launch failure, Pentium FDIV bug
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Errors and Proofs
Mathematicians and Computer Scientists want toeliminate all the uncertainties on their results. Why?
M. Lecat, Erreurs des Mathématiciens des origines à nosjours, 1935.
; 130 pages of errors! (Euler, Fermat, Sylvester, . . . )
Ariane 5 launch failure, Pentium FDIV bug
Victor Magron Certified Roundoff Error Bounds using Semidefinite Programming 2 / 27
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Errors and Proofs
Mathematicians and Computer Scientists want toeliminate all the uncertainties on their results. Why?
M. Lecat, Erreurs des Mathématiciens des origines à nosjours, 1935.
; 130 pages of errors! (Euler, Fermat, Sylvester, . . . )
Ariane 5 launch failure, Pentium FDIV bug
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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 codesAerospace controlmolecular biology, robotics, code synthesis, . . .
Efficient Verification of Nonlinear Systems
Automated precision tuning of systems/programsanalysis/synthesis
Efficiency sparsity correlation patterns
Certified approximation algorithms
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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 codesAerospace controlmolecular biology, robotics, code synthesis, . . .
Efficient Verification of Nonlinear Systems
Automated precision tuning of systems/programsanalysis/synthesis
Efficiency sparsity correlation patterns
Certified approximation algorithms
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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 codesAerospace controlmolecular biology, robotics, code synthesis, . . .
Efficient Verification of Nonlinear Systems
Automated precision tuning of systems/programsanalysis/synthesis
Efficiency sparsity correlation patterns
Certified approximation algorithms
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Rounding Error Bounds
Real : p(x) := x1 × x2 + x3
Floating-point : p(x, e) := [x1x2(1 + e1) + x3](1 + e2)
Input variable uncertainties x ∈ SFinite precision ; bounds over e
| ei |6 2−m m = 24 (single) or 53 (double)
Guarantees on absolute round-off error | p− p | ?
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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))
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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))
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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))
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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
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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)
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Contributions
Maximal Rounding error of the program implementation of f :
r? := max |f (x, e)− f (x)|
Decomposition: linear term l w.r.t. e + nonlinear term h
r? 6 max |l(x, e)|+ max |h(x, e)|
Sparse SDP bounds for l
Coarse bound of h with interval arithmetic
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Contributions
1 Comparison with SMT and linear/affine arithmetic:; More 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)
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Introduction
Semidefinite Programming for Polynomial Optimization
Rounding Error Bounds with Sparse SDP
Conclusion
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What is Semidefinite Programming?
Linear Programming (LP):
minz
c>
z
s.t. A z > d .
Linear cost c
Linear inequalities “∑i Aij zj > di” Polyhedron
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What is Semidefinite Programming?
Semidefinite Programming (SDP):
minz
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
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What is Semidefinite Programming?
Semidefinite Programming (SDP):
minz
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
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Applications of SDP
Combinatorial optimization
Control theory
Matrix completion
Unique Games Conjecture (Khot ’02) :“A single concrete algorithm provides optimal guaranteesamong all efficient algorithms for a large class ofcomputational problems.”(Barak and Steurer survey at ICM’14)
Solving polynomial optimization (Lasserre ’01)
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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
)︸ ︷︷ ︸
<0
(ab
).
Find z s.t. a2 − 2ab + b2 = z1a2 + 2z2ab + z3b2 (A z = d)
(z1 z2z2 z3
)=
(1 00 0
)︸ ︷︷ ︸
F1
z1 +
(0 11 0
)︸ ︷︷ ︸
F2
z2 +
(0 00 1
)︸ ︷︷ ︸
F3
z3 <(
0 00 0
)︸ ︷︷ ︸
F0
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SDP for Polynomial Optimization
Choose a cost c e.g. (1, 0, 1) and solve:
minz
c>
z
s.t. ∑i
Fi zi < F0 , A z = d .
Solution(
z1 z2z2 z3
)=
(1 −1−1 1
)< 0 (eigenvalues 0 and 1)
a2 − 2ab + b2 =(a b
) ( 1 −1−1 1
)︸ ︷︷ ︸
<0
(ab
)= (a− b)2 .
Solving SDP =⇒ Finding SUMS OF SQUARES certificates
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SDP for Polynomial Optimization
General case:
Semialgebraic set S := {x ∈ Rn : g1(x) > 0, . . . , gm(x) > 0}
p∗ := minx∈S
p(x): NP hard
Sums of squares (SOS) Σ[x] (e.g. (x1 − x2)2)
Q(S) :={
σ0(x) + ∑mj=1 σj(x)gj(x), with σj ∈ Σ[x]
}Fix the degree 2k of sums of squaresQk(S) := Q(S) ∩R2k[x]
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SDP for Polynomial Optimization
Hierarchy of SDP relaxations:
λk := supλ
{λ : p− λ ∈ Qk(S)
}Convergence guarantees λk ↑ p∗ [Lasserre 01]
Can be computed with SDP solvers (CSDP, SDPA)
“No Free Lunch” Rule: (n+2kn ) SDP variables
Extension to semialgebraic functions r(x) = p(x)/√
q(x)[Lasserre-Putinar 10]
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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+2kκ )
variables
C1 := {1, 4}C2 := {1, 2, 3, 5}C3 := {1, 3, 5, 6}Dense SDP: 210 variablesSparse SDP: 115 variables
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Introduction
Semidefinite Programming for Polynomial Optimization
Rounding Error Bounds with Sparse SDP
Conclusion
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Polynomial Programs
Input: exact f (x), floating-point f (x, e), x ∈ S, | ei |6 2−m
Output: Bound for f − f1: Error r(x, e) := f (x)− 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′+2kn+n′ ) SDP variables
Sparse relaxation: n′(n+1+2kn+1 ) SDP variables
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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 ∈ [−ε, ε]15 , ε = 2−24
Dense SDP: (6+15+46+15 ) = 12650 variables ; Out of memory
Sparse SDP Real2Float tool: 15(6+1+46+1 ) = 4950 ; 789ε
Interval arithmetic: 2023ε (17 × less CPU)
Symbolic Taylor FPTaylor tool: 936ε (16.3 ×more CPU)
SMT-based rosa tool: 789ε (4.6 ×more CPU)Victor Magron Certified Roundoff Error Bounds using Semidefinite Programming 18 / 27
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Preliminary Comparisons
Real2Float
rosa
FPTaylor
0
200
400
600
800
1,000
789ε 789ε
936ε
CPU Time
Erro
rBo
und
(ε)
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Extensions: Transcendental Programs
Given K a compact set and f a transcendental function, boundf ∗ = inf
x∈Kf (x) and prove f ∗ > 0
f is under-approximated by a semialgebraic function fsa
Reduce the problem f ∗sa := infx∈K fsa(x) to a polynomialoptimization problem (POP)
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Extensions: Transcendental Programs
Approximation theory: Chebyshev/Taylor models
mandatory for non-polynomial problems
hard to combine with Sum-of-Squares techniques (degreeof approximation)
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Maxplus Approximations
Initially introduced to solve Optimal Control Problems[Fleming-McEneaney 00]
Curse of dimensionality reduction [McEaneney Kluberg,Gaubert-McEneaney-Qu 11, Qu 13].Allowed to solve instances of dim up to 15 (inaccessible bygrid methods)
In our context: approximate transcendental functions
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Maxplus Approximations
DefinitionLet γ > 0. A function φ : Rn → R is said to be γ-semiconvex ifthe function x 7→ φ(x) + γ
2 ‖x‖22 is convex.
a
y
par+a1
par+a2
par−a2
par−a1
a2a1
arctan
m M
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Maxplus Approximations
Exact parsimonious maxplus representations
a
y
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Maxplus Approximations
Exact parsimonious maxplus representations
a
y
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Extensions: Programs with Conditionals
if (p(x) 6 0) f (x); else g(x);
DIVERGENCE PATH ERROR:
r? := max{max
p(x)60,p(x,e)>0| f (x, e)− g(x) |
maxp(x)>0,p(x,e)60
| g(x, e)− f (x) |
maxp(x)>0,p(x,e)>0
| f (x, e)− f (x) |
maxp(x)60,p(x,e)60
| g(x, e)− g(x) |
}
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Benchmarks
Doppler
u ∈ [−100, 100]v ∈ [20, 20000]T ∈ [−30, 50]
let t1 = 331.4 + 0.6T in−t1v
(t1 + u)2
FPTaylor
Real2Float
rosa
0
1,000
2,000
3,000
4,000
5,000
1,415ε
2,567ε
4,477ε
CPU Time
Erro
rBo
und
(ε)
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Benchmarks
Kepler2
x ∈ [4, 6.36]6
x1x4(−x1 + x2 + x3 − x4+x5 + x6) + x2x5(x1 − x2+x3 + x4 − x5 + x6)
+x3x6(x1 + x2 − x3+x4 + x5 − x6)
−x2x3x4 − x1x3x5−x1x2x6 − x4x5x6
Real2Float
FPTaylor
rosa
0
0.5
1
1.5
2
·104
17,114ε
23,509ε
19,456ε
CPU Time
Erro
rBo
und
(ε)
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Benchmarks
verhulst
x ∈ [0.1, 0.3]
4x
1 +x
1.11
Real2Float
FPTaylor
rosa
0
2
4
6
3.07ε 3.16ε
6.15ε
CPU Time
Erro
rBo
und
(ε)
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Benchmarks
logexp
x ∈ [−8, 8]
log(1 + exp(x))
Real2Float
FPTaylor
0
5
10
15
20
13.5ε
15.4ε
CPU Time
Erro
rBo
und
(ε)
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Introduction
Semidefinite Programming for Polynomial Optimization
Rounding Error Bounds with Sparse SDP
Conclusion
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Conclusion
Sparse SDP relaxations analyze NONLINEAR PROGRAMS:
Polynomial and transcendental programs
Handles conditionals, input uncertainties, . . .
Certified ; Formal rounding error bounds
Real2Float open source tool:
https://forge.ocamlcore.org/projects/nl-certify
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Conclusion
Further research:
Improve formal polynomial checker
Alternative polynomial bounds using geometricprogramming (T. de Wolff, S. Iliman)
Mixed linear/SDP certificates (trade-off CPU/precision)
More program verification: while loops
Automatic FPGA code generation
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End
Thank you for your attention!
http://www-verimag.imag.fr/~magron
V. Magron, G. Constantinides, A. Donaldson. CertifiedRoundoff Error Bounds Using Semidefinite Programming,arxiv.org/abs/1507.03331