Diffeological Spaces and Denotational Semantics for Differential Programming

Transcription

1 Kammar, Staton, andvákár Diffeological Spaces and Denotational Semantics for Differential Programming Ohad Kammar, Sam Staton, and Matthijs Vákár MFPS 2018 Halifax 8 June 2018

2 What is differential programming? PL in which all (some) constructs are differentiable and derivatives can be computed mechanically. Compute derivatives compositionally, using chain rule, rather than symbolically or using finite differences. AKA Automatic Differentiation. Kammar, Staton, andvákár

3 Why study semantics of differential programming? Important: Efficient optimisation (in high dim); Efficient integration/sampling (in high dim); Often combined with concurrency and probability correctness non-trivial and good test-coverage hard to obtain; Kammar, Staton, andvákár

4 Why study semantics of differential programming? Non-Trivial: How to differentiate through traditional language constructs? conditionals iteration and term recursion higher order functions probability, state and other effects How to differentiate at structured types? inductive/recursive types refinement types function types quotient types So need to move beyond traditional calculus! Kammar, Staton, andvákár

5 Kammar, Staton, andvákár Concrete example of higher-order differential prog Finding Shortest Arc Length -- Integrate :: ([0,1] => real) => real -- Differentiate :: ([0,1] => real) => [0,1] => real ArcLength :: ([0,1] => real) => [0,infty) ArcLength(f) = Integrate(Sqrt(1 + Differentiate(f)^2)) Power :: (0,infty) => [0,1] => real Power(a)(t) = t^a -- Minimise :: ((0,infty) => real) => (0,infty) Minimal_a :: real Minimal_a = Minimise(\a -> ArcLength(Power(a))) -- Should be 1

6 Kammar, Staton, andvákár Is this not an artificial problem? Stan language: For specifying a smooth objective function and sampling from it / optimising it: conditionals (for custom combinations of objective functions), while loops, recursive functions (for iterative approximations); higher-order functions (ODE and algebraic solvers, used in pharmacokinetics); certain refinement types (constraints: intervals, unit vectors, simplices, symmetric positive definite matrices); fwd and rev mode AutoDiff through everything.

7 Kammar, Staton, andvákár Calculus 101 Refresher Traditional derivative (Jacobian) T x f : R n R m of f : U V at x, where U open in R n and V open in R m Best linear approximation: f(x + v) f(x) T x f(v) lim = 0 v 0 v

8 Categories for Smoothness? Category Open: objects - opens U in some R n ; morphisms f : U V - smooth (C ) functions from U to V ; R n f R m RU V Category Mfd of fin. dim. manifolds and smooth maps: Adds surfaces (certain refinements); Whitney Embedding: idempotent completion of Open; Kammar, Staton, andvákár

9 Kammar, Staton, andvákár Several Categories of Smooth Maps Many full subcategories: EuclSp Open Mfd FreSp FreMfd ConvSp FroSp Diffeo Sh(Open) [Open op, Set] Diffeo Grothendieck quasi-topos and well-pointed; Interprets tuples, function types, variant types, (co)inductive types, dependent types; Conservative extension: embeddings into Diffeo preserve all limits and coproducts, coequalizers of open covers;

10 Diffeological spaces [Souriau] Objects A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: Kammar, Staton, andvákár

11 Diffeological spaces [Souriau] Objects A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: constant functions c; α(r)=c c Kammar, Staton, andvákár

12 Diffeological spaces [Souriau] Objects A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: constant functions c; precomposition with a smooth φ : U V φ α Kammar, Staton, andvákár

13 Diffeological spaces [Souriau] Objects ( ) A diffeological space X = X, SX consists of: a carrier set X ; U X U for all U Open a set of plots SX such that the plots are closed under: constant functions c; precomposition with a smooth φ : U V gluing of compatible families along open covers U. case{u.αu U U } 7 Kammar, Staton, and Vákár

14 Diffeological spaces [Souriau] Objects A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that...3 axioms... Morphisms f : X Y Functions f : X Y such that: α S X = f α S Y Example: Traditional Spaces For manifold M, define (Yoneda) plots S U M := Mfd(U, M). Kammar, Staton, andvákár

15 Categorical structure Objects A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that...3 axioms... Products } S U X Y := {r ( α(r), β(r) ) α S U X, β S U Y Kammar, Staton, andvákár

16 Kammar, Staton, andvákár Categorical structure Morphisms f : X Y Functions f : X Y such that: α S X = f α S Y Function spaces Y X := Diffeo(X, Y ) S U Y X := { f : U Y X uncurry f Diffeo(U X, Y ) } NB: The exponential X U is the space of U-plots.

17 Kammar, Staton, andvákár Categorical structure Morphisms f : X Y Functions f : X Y such that: α S X = f α S Y Function spaces Y X := Diffeo(X, Y ) S U Y X := { f : U Y X uncurry f Diffeo(U X, Y ) } NB: The exponential X U is the space of U-plots. More structure Coproducts, limits, colimits,... (co)inductive types!

18 Kammar, Staton, andvákár Diffeo-morphisms are continuous! D-Topology Adjunction with topological spaces (M Top, X Diffeo): SDiffeoM U := Top(U, { M) } O TopX := B X U Open.α SX U, α 1 [X] O U Diffeo Top Diffeo = Top Generalises Euclidean topology. Set

19 Kammar, Staton, andvákár Derivatives? Covariant Derivative Fwd Mode R n f R m Write T U := U R n fst U T V := V R m fst V ; RU V And write T U T f T V x, v f(x), T x f(v) ; This defines a functor Open T Open Functoriality is chain rule! id cod Open.

20 Kammar, Staton, andvákár C made difficult objects: morphisms a : A A 0, b : B B 0 in C; morphisms f 0, f 1 : C (a, b) := f 0 C(A 0,B 0 ) C/A 0(a, f 0 b) using pullback (suppose exist); projection functor fst : C C; A f 1 f a f 0 B B f 0 b b f0 A 0 B 0

21 Kammar, Staton, andvákár C made difficult objects: morphisms a : A A 0, b : B B 0 in C; morphisms f 0, f 1 : C (a, b) := f 0 C(A 0,B 0 ) C/A 0(a, f 0 b) using pullback (suppose exist); projection functor fst : C C; A a f 1 f 0 B B f 0 b b f0 A 0 B 0

22 Kammar, Staton, andvákár Dual Fibration: Polynomials/Containers Poly(C) objects: morphisms a : A A 0, b : B B 0 in C; morphisms f 0, f 1 : Poly(C)(a, b) := f 0 C(A 0,B 0 ) C/A 0(f0 b, a) using pullback; projection functor fst : Poly(C) C; A f 1 a f 0 B B f 0 b b f0 A 0 B 0

23 Kammar, Staton, andvákár Contravariant Derivative Rev Mode Write T x f : R m = R m R Txf R R n R = R n ; Define polynomials/containers T U := U R n fst U T V := V R m fst V and f T V T f T U x, ξ This defines a functor x, T x f(ξ) ; Open T Poly (Open) Functoriality is chain rule! id cod Open. (Poly : slight cheating, objects product projections, so pb)

24 Kammar, Staton, andvákár Derivatives on Diffeo Observe that Diffeo and Poly(Diffeo) are cocomplete (and complete); Define through left Kan extension Poly (Open) T Open T Open Poly(y) y y Poly(Diffeo) Diffeo Lan y (T ; Poly(y)) Lan y (T ; y ) Diffeo Functor by construction chain rule! y is full and faithful extends usual definition (on Mfd); But many other ways to define in literature! Do they coincide..?

25 Kammar, Staton, andvákár Derivatives are dependently typed! (Co)tangent bundles not all trivial (product projections): take S 2. dependent types! Type Formers: Given Γ X type, we get Γ, x : X T x X type and Γ, x : X T x X type. T X is syntactic sugar for Σ x:x T x X and T X for Σ x:x T x X. Term Formers: Given Γ, x : X f(x) : Y, we get Γ, x : X, v : T x X T x f(v) : T f(x) Y and Γ, x : X, ξ : T f(x) Y T x f(ξ) : T x X.

26 Our example revisited Finding Shortest Arc Length -- Integrate :: ([0,1] => real) => real -- Differentiate :: ([0,1] => real) => [0,1] => real ArcLength :: ([0,1] => real) => [0,infty) ArcLength(f) = Integrate(Sqrt(1 + Differentiate(f)^2)) Power :: (0,infty) => [0,1] => real Power(a)(t) = t^a -- Minimise :: ((0,infty) => real) => (0,infty) Minimal_a :: real Minimal_a = Minimise(\a -> ArcLength(Power(a))) -- Should be 1 Uncurry to see Power is smooth; Integrate sends smooth functions of two arguments to smooth functions, so it is smooth. Kammar, Staton, andvákár

27 Diffeological Domains [Kammar, Staton, Vákár] Category ωdiffeo Objects: diffeological spaces X with an ω-cpo structure X, s.t. S X is closed under lubs of ω-chains; Morphisms: Scott continuous Diffeo-morphisms; Gives a (co)complete ccc ωdiffeo (locally presentable); Recursive types through Fiore s axiomatic domain theory! Define T, T as enriched Kan extensions, but working with ωdiffeo fib and its dual fibration? 3 More Equivalent Characterisations ωcpo-enriched separated sheaves on Open; Models of an essentially algebraic theory; Internal language ω-cpos in Diffeo. Kammar, Staton, andvákár

28 Conclusion Summary Diffeo: very simple, well-behaved and rich setting for semantics of differential languages Rich type theory allows for syntactic proofs Some Questions Derivative of a recursive function and T, T on ωdiffeo? An adequate semantics for differential programming? Prove corrects NUTS? Analyse the (co)tangent bundle of function types? C.f (co)tangent bundles in literature? Smooth probability monad? Relationship cotangent bundle and CPS? Boman s theorem: why not restrict to S R X? Kammar, Staton, andvákár

29 Thanks! Recall - definition A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: Kammar, Staton, andvákár

30 Thanks! Recall - definition A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: constant functions c; α(r)=c c Kammar, Staton, andvákár

31 Thanks! Recall - definition A diffeological space X = ( X, S X ) consists of: a carrier set X ; a set of plots S U X X U for all U Open such that the plots are closed under: constant functions c; precomposition with a smooth φ : U V φ α Kammar, Staton, andvákár

32 Thanks! Recall - definition ( ) A diffeological space X = X, SX consists of: a carrier set X ; U X U for all U Open a set of plots SX such that the plots are closed under: constant functions c; precomposition with a smooth φ : U V gluing of compatible families along open covers U. case{u.αu U U } 7 Kammar, Staton, and Vákár