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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


  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

  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

  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

  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

  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

  6. Errors and Proofs G UARANTEED O PTIMIZATION 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

  7. Errors and Proofs G UARANTEED O PTIMIZATION Input : Linear problem (LP), geometric, semidefinite (SDP) Output : solution + certificate numeric-symbolic ❀ formal V ERIFICATION 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

  8. Errors and Proofs G UARANTEED O PTIMIZATION Input : Linear problem (LP), geometric, semidefinite (SDP) Output : solution + certificate numeric-symbolic ❀ formal V ERIFICATION 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

  9. Roundoff Error Bounds Real : p ( x ) : = x 1 × x 2 + x 3 p ( x , e ) : = [ x 1 x 2 ( 1 + e 1 ) + x 3 ]( 1 + e 2 ) Floating-point : ˆ Input variable constraints x ∈ S Finite precision ❀ bounds over e | e i | � 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

  10. Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 5 / 28

  11. Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Semialgebraic programs: | · | , √ , /, sup, inf 4 x x 1 + 1.11 Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 5 / 28

  12. Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Semialgebraic programs: | · | , √ , /, sup, inf 4 x x 1 + 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

  13. Existing Frameworks Classical methods : Abstract domains [Goubault-Putot 11] F LUCTUAT : intervals, octagons, zonotopes Interval arithmetic [Daumas-Melquiond 10] G APPA : interface with C OQ proof assistant Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 6 / 28

  14. Existing Frameworks Recent progress : Affine arithmetic + SMT [Darulova 14] rosa : sound compiler for reals (in S CALA ) Symbolic Taylor expansions [Solovyev 15] FPTaylor : certified optimization (in OC AML and H OL - LIGHT ) Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 6 / 28

  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

  16. 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 OC AML and C OQ ) Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 7 / 28

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

  18. What is Semidefinite Programming? Linear Programming (LP): ⊤ z min c z s.t. A z � d . Linear cost c Polyhedron Linear inequalities “ ∑ i A ij z j � d i ” Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 8 / 28

  19. What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ s.t. F i z i � F 0 . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 9 / 28

  20. What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ A z = d . s.t. F i z i � F 0 , i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 10 / 28

  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

  22. SDP for Polynomial Optimization Prove polynomial inequalities with SDP: p ( a , b ) : = a 2 − 2 ab + b 2 � 0 . � � � � � � z 1 z 2 a Find z s.t. p ( a , b ) = a b . z 2 z 3 b � �� � � 0 Find z s.t. a 2 − 2 ab + b 2 = z 1 a 2 + 2 z 2 ab + z 3 b 2 ( A z = d ) � z 1 � � 1 � � 0 � � 0 � � 0 � z 2 0 1 0 0 = z 1 + z 2 + z 3 � z 2 z 3 0 0 1 0 0 1 0 0 � �� � � �� � � �� � � �� � F 1 F 2 F 3 F 0 Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 12 / 28

  23. SDP for Polynomial Optimization Choose a cost c e.g. ( 1, 0, 1 ) and solve: ⊤ z min c z ∑ s.t. F i z i � F 0 , A z = d . i � 1 � z 1 � � z 2 − 1 Solution = � 0 (eigenvalues 0 and 2) − 1 z 2 z 3 1 � � 1 � � a � − 1 a 2 − 2 ab + b 2 = � = ( a − b ) 2 . a b − 1 1 b � �� � � 0 Solving SDP = ⇒ Finding S UMS OF S QUARES certificates Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 13 / 28

  24. SDP for Polynomial Optimization General case : Semialgebraic set S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } p ∗ : = min x ∈ S p ( x ) : NP hard Sums of squares (SOS) Σ [ x ] (e.g. ( x 1 − x 2 ) 2 ) � � σ 0 ( x ) + ∑ m Q ( S ) : = j = 1 σ j ( x ) g j ( x ) , with σ j ∈ Σ [ x ] Fix the degree 2 k of products: m � � ∑ Q k ( S ) : = σ 0 ( x ) + σ j ( x ) g j ( x ) , with deg σ j g j � 2 k j = 1 Victor Magron Certified Roundoff Error Bounds using SDP and Formal FP Arithmetic 14 / 28

  25. SDP for Polynomial Optimization Hierarchy of SDP relaxations : � � λ k : = sup λ : p − λ ∈ Q k ( S ) λ Convergence guarantees λ k ↑ p ∗ [Lasserre 01] Can be computed with SDP solvers ( CSDP , SDPA ) “No Free Lunch” Rule : ( n + 2 k 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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