Proof mining in convex optimization Andrei Sipos , (joint work with Laurent ¸iu Leus ¸tean and Adriana Nicolae) Institute of Mathematics of the Romanian Academy University of Bucharest May 2, 2017 PhDs in Logic IX Bochum, Deutschland
Proof mining Proof mining (introduced and developed by U. Kohlenbach) aims to obtain quantitative information from proofs of theorems (from various areas of mathematics) of a nature which is not (fully) constructive. A comprehensive reference is: U. Kohlenbach, Applied proof theory: Proof interpretations and their use in mathematics , Springer, Berlin/Heidelberg, 2008. An extensive survey detailing the intervening research can be found in: U. Kohlenbach, “Recent progress in proof mining in nonlinear analysis”, preprint 2016, to appear in forthcoming special issue of IFCoLog Journal of Logic and its Applications with invited articles by recipients of a G¨ odel Centenary Research Prize Fellowship. Andrei Sipos Proof mining in convex optimization ,
The general question Proof theory is one of the four main branches of logic and has as its scope of study proofs themselves (inside given logical systems), with a special aim upon consistency results, structural and substructural transformations, proof-theoretic ordinals et al. The driving question of proof mining / interpretative proof theory is the following: “What more do we know if we have proved a theorem by restricted means than if we merely know that it is true?” (posed by G. Kreisel in the 1950s) Andrei Sipos Proof mining in convex optimization ,
Kinds of information By analysing a specific proof of a mathematical theorem, one could obtain, in addition: Terms coding effective algorithms for witnesses and bounds for existentially quantified variables; Independence of certain parameters or at least continuity of the dependency; Weakening of premises. In order for this to work, we must impose well-behavedness conditions upon the logical system and upon the complexity of the statement of the theorem . Andrei Sipos Proof mining in convex optimization ,
The logical systems We generally use systems of arithmetic in all finite types, intuitionistic or classical, augmented by restricted non-constructive rules (such as choice principles) and by types referring to (metric/normed/Hilbert) spaces and functionals involving them. Two such systems are denoted by A ω i [ X , �· , ·� , C ] (intuitionistic) and A ω [ X , �· , ·� , C ] (classical). One typically uses proof interpretations to extract the necessary quantitative information. Metatheorems guaranteeing this fact were developed by Gerhardy and Kohlenbach in the 2000s. A sample metatheorem is the following, for classical logic, which uses G¨ odel’s functional interpretation, in its “monotone” variant introduced by Kohlenbach, combined with the negative translation. Andrei Sipos Proof mining in convex optimization ,
Proof interpretations We have mainly two proof interpretations at our disposal: monotone modified realizability, which: can extract bounds for all kinds of formulas; does not permit the use of excluded middle; monotone functional interpretation (combined with negative translation), which: can extract bounds only for Π 2 (that is, ∀∃ ) formulas; permits the use of excluded middle. These “interpretations” have corresponding metatheorems, which can be used to extract the required quantitative information. In some cases, where no set of restrictions is met, the two may be used in conjunction – see, e.g., Leus ¸tean (2014), A.S. (2016). Andrei Sipos Proof mining in convex optimization ,
What does “quantitative” mean? Let us see what kind of information we might hope to extract. An example from nonlinear analysis would be a limit statement of the form: ∀ ε > 0 ∃ N ε ∀ N ≥ N ε ( � x n − A n x n � < ε ) . What we want to get is a “formula” for N ε in terms of (obviously) ε and of some other arguments parametrizing our situation. Such a function is called a rate of convergence for the sequence. As the formula above is not in a Π 2 / ∀∃ form, and we generally work with the monotone functional interpretation, in some cases we are forced to only quantify its Herbrand normal form and obtain its so-called rate of metastability (in the sense of T. Tao). Andrei Sipos Proof mining in convex optimization ,
Our setting Let us describe our first tentative object of study. Set H to be a Hilbert space. We say that a multi-valued operator A : H → 2 H is monotone if for all x , y , u , v ∈ H with u ∈ A ( x ) and v ∈ A ( y ) we have that � x − y , u − v � ≥ 0 . We call it maximally monotone if it is maximal among monotone operators considered as subsets of H × H . The proximal point algorithm’s goal is to find zeroes of A , i.e. points x ∈ H s.t. 0 ∈ A ( x ). Andrei Sipos Proof mining in convex optimization ,
Resolvents The main tool to use is the resolvent of A – that is, the mapping defined by: J A := ( id + A ) − 1 . We have the following classical results: if A is maximally monotone, then J A is single-valued; for all x ∈ H , x is a zero of A iff x is a fixed point of J A . Now we can state the PPA theorem. Andrei Sipos Proof mining in convex optimization ,
The PPA theorem Theorem (Proximal Point Algorithm) Let A : H → 2 H be a maximally monotone operator that has at least one zero and let ( γ n ) n ∈ N ⊆ (0 , ∞ ) be such that � ∞ n =0 γ 2 n = ∞ . Let x ∈ H. Set x 0 := x and for all n ∈ N , x n +1 := J γ n A x n . Then the sequence ( x n ) n ∈ N converges weakly to a zero of A. A particular case of it, which we shall analyse first is: if the set of zeroes of A has nonempty interior, then the convergence is strong. Andrei Sipos Proof mining in convex optimization ,
Metastability This is where we are going to use the idea of metastability announced before. Metastability can be formulated as: 1 ∀ k ∈ N ∀ g : N → N ∃ n ∀ i , j ∈ [ n , n + g ( n )] d ( x i , x j ) ≤ k + 1 , which can be seen to be a Π 2 statement in the extended system. A rate of metastability will be a bound Ψ( k , g ) on n . Andrei Sipos Proof mining in convex optimization ,
The “nonempty interior” case – quantitatively Theorem (L.L., A.N., A.S.) For any g : N → N , define χ g : N → N recursively, as follows: χ g (0) := 0 χ g ( n + 1) := χ g ( n ) + g ( χ g ( n )) Set now, for any k ∈ N and g : N → N , �� b 2 ( k + 1) �� Φ b , r ( k , g ) := χ g . 2 r Then, in this case, Φ b , r is a rate of metastability for ( x n ) n ∈ N . This is a simple application of Tao’s “finite convergence principle”. Andrei Sipos Proof mining in convex optimization ,
The uniform case The most interesting case for bound extraction will be when the operator is “uniformly monotone”, i.e. satisfies the stronger inequality: � x − y , u − v � ≥ φ ( � x − y � ) with respect to an increasing function φ : [0 , ∞ ) → [0 , ∞ ) which vanishes only at 0. In this case, it is known that the zero is unique and the convergence is necessarily strong. Andrei Sipos Proof mining in convex optimization ,
Briseid’s work Following Briseid’s work in his PhD thesis, one might try to quantify the following uniqueness statement for a fixed point: ∀ x ∀ y ( Tx = x ∧ Ty = y → x = y ) . For that, we exploit the implementation of the equality sign in our system, i.e. we write the above as: ∀ x ∀ y ( ∀ δ ( d ( Tx , x ) ≤ δ ∧ d ( Ty , y ) ≤ δ ) → ∀ ε ( d ( x , y ) ≤ ε ) . Andrei Sipos Proof mining in convex optimization ,
The modulus of uniqueness The quantified version is: ∀ x ∀ y ∀ ε (( d ( Tx , x ) ≤ δ ( ε ) ∧ d ( Ty , y ) ≤ δ ( ε )) → ( d ( x , y ) ≤ ε ) , where δ ( ε ) is the quantity extracted by proof mining, called the “modulus of uniqueness”. Eyvind Briseid has observed in his PhD thesis that this modulus together with an associated asymptotic regularity lemma might help in obtaining the rate of convergence directly, regardless of the principles used in the proof (except in said lemma). Andrei Sipos Proof mining in convex optimization ,
Our case In our case, given that we use a whole family of mappings for which the zero sought is a fixed point, we had to modify the idea, so that we use the modulus of uniqueness only implicitly. The relevant lemma is the following: Lemma In this framework, if x ∈ H and z is a zero of A, we have that: γφ ( d ( J γ A x , z )) ≤ d ( x , J γ A x ) d ( J γ A x , z ) . The corresponding asymptotic regularity result will be: Lemma Let Σ b : N → N be defined, for all k, by Σ b ,θ ( k ) := θ ( b 2 ( k + 1) 2 ) . For all k ∈ N and all n ≥ Σ b ,θ ( k ) , we have that d ( x n , x n +1 ) 1 ≤ k + 1 . γ n Andrei Sipos Proof mining in convex optimization ,
The rate of convergence Putting all this together, we obtain our main quantitative result, which is: Theorem (L.L., A.N., A.S.) Set, for any k ∈ N , 2 b + 1 , Ψ b ,θ,φ ( k ) := Σ b ,θ � � 1 φ k +1 where Σ b ,θ is the one from the lemma before. Then Ψ b ,θ,φ is a rate of convergence for ( x n ) n ∈ N . Andrei Sipos Proof mining in convex optimization ,
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