Learning Is a Kan Extension
Authors: Matthew Pugh
Nick Harris
Corina Cirstea
Jo Grundy
Venue: Accepted by TMLR ()

Abstract: Previous work has demonstrated that efficient algorithms exist for computing Kan extensions and that some Kan extensions have interesting similarities to various machine learning algorithms. This paper closes the gap by proving that all error minimisation algorithms may be presented as a Kan extension. This result provides a foundation for future work to investigate the optimisation of machine learning algorithms through their presentation as Kan extensions. A corollary of this representation of error-minimising algorithms is a presentation of error from the perspective of lossy and lossless transformations of data.

Published in Transactions on Machine Learning Research (11/2025)
Learning Is a Kan Extension
Matthew Pugh
mp8g160soton.cco
School of Electronics and Computer Science University of Southampton University Road, S017 1BJ
Jo Grundy
j.grundy@sstonnacc.oo uk
School of Electronics and Computer Science University of Southampton University Road, S017 1BJ
Corina Cirstea
cc sol.csssoooo.cc.m uk
School of Electronics and Computer Science University of Southampton University Road, S017 1BJ
Nick Harris
nrh@ecs. uk
School of Electronics and Computer Science
Universily of Southampton
University Road, S017 1BJ
Reviewed on OpenReview: https: // openreview. net/ forum? id= dddeel.
Abstract
Previous work has demonstrated that efficient algorithms exist for computing Kan extensions and that some Kan extensions have interesting similarities to various machine learning algorithms. This paper closes the gap by proving that all error minimisation algorithms may be presented as a Kan extension. This result provides a foundation for future work to investigate the optimisation of machine learning algorithms through their presentation as Kan extensions. corollary of this representation of error-minimising algorithms a presentation of error from the perspective of lossy and lossless transformations of data.
1 Introduction
Recent work has indicated that Kan extensions have structural similarity to many machine learning algo- rithms Shiebler (2022); ; Pugh et al.
2024 There is tremendous amount of theory around the study of Kan extensions Perrone & Tholen (2022) Kelly (005) and even algorithms for computing left. Kan extensions efficiently Meyers et al. (2022). J. If the connection between Kan extensions and machine learning algorithms can be made more concrete, then would be possible to leverage this body of work in the study of machine learning algorithms.
This paper seeks to provide a concrete connection by proving that all error minimisation problems may be presented a left Kan extension (Thm 4.1
A definition of error minimisation using sets and functions is lifted into the category-theorctic domain by representing it with categories and functors (Def 3.7 It is shown that error may be represented by a lax 2-functor, which associates a form of information loss to transformations between datasets (morphisms in a category) (Def 3.3
The category-theoretie theoretic presentation of an error minimisation problem used show that left adjoint func- tors produce global error minimiser for any input dataset. (Thm 3.9h. Furthermore, the error minimiser

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is independent of the error, indicating that an appropriate choice of the category of datasets is sufficient to determine the global error minimisation solutions (Cor 3.10p. A consequence of this result is the connec- tion between adjoint functor theorems Porst (2024) and error minimisation problems, providing sufficient conditions to define when an optimal solution to an error minimisation problem must cxist (Cor 3.12
It is then shown that left Kan extensions are also error minimisers and that for any traditional or set- theoretic error minimisation problem, there is a 2-category whose left Kan extensions are precisely the global minimisers of the error minimisation problem (Thm 4.1
2
Background
2.1 Categories, Adjunctions, and Kan Extensions
The definitions of categories, functors, natural transforms, adjunctions, and Kan extensions are found in all of the following resources. (Riehl 2016 Fong & Spivak, 2018 Leinster 2016). The definition of a 2-category is adapted from its definition as an enriched category Kelly (2005).
A category is collection of objects and morphisms where every morphism has domain object and codomain object. Two morphisms may be composed if the domain of one equals the codomain of the other.
Definition 2.1 (Category). A category C consists of a class of objects Ob(C), and between any two objects v, E Ob(C) class of morphisms C(r,y) y) such that:
Any pair f E C( y) and g E C(y,=) =) can be composed to form gf E C(x,, z).
Composition is associative: (hg). h(gf).
Every object x E Ob(C) has an identity morphism Idg E C(x,x).
for any C(x,y) then fIdz = Idgf.
When clear from context, it is common to write x E Ob(C) as x E C and f E C(z,y) as f ; x y. One example of a category is Set whose objects are sets and whose morphisms are set functions. Morphisms are often considered to be structure preserving maps. As sets have no structure by design, their morphisms are just functions. An example of morphism between categories is a functor.
Definition 2.2 (Functor). A functor F' C' ( D, between categories C' and D sends every object x E Ob(C) to F(r) Ob(D), and every morphism f E C(x,)) y) to F(f) E D(F(z), F(y)) such that: F preserves composition: F(gs)= F())
F preserves identities: F(Idz) = Idr(s)
The product of two categories C and D may be written as C x D. Its objects are pairs of objects from C and D, and its morphisms are pairs of morphisms.
I E C(x,v) A D(w, 2) = (..9) E x D((2, w). (u,=))
(1)
The unit of the categorical product is the category 1 which has single object and a single morphism (which is the identity of its object). The categorical product of a category C with is isomorphic to C, meaning that there exists an invertible functor from the product into C. These invertible functors are referred to as the left and right unitors and
l:1C r:Cx1
(2)
The left and right unitors simply drop the single object from pairs of object in C'1 or 1xC. Le. 1(+,x)= and r(z, *) = x. The categorical product is also associative, as described by the existence of an invertible morphism a for triple of objects composed using the categorical product.
a: (C x D) x Cx
(4)
a simply rewrites nested tuples, a((2, v), = (x, 2)).
As well morphisms between categories it is also possible consider the existence of morphisms between functors, called natural transforms.
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Definition 2.3 (Natural Transform). Given functors F.G.C. between categories C and D, natural transformation 7) F F= G is family of morphisms 1/ F(x:) - G(z) in D for each object a E Ob(C), such that G(f) = (f) for any f D(r,)), ie. the following diagram commutes:
F(x)
9F
G(x)
F(f)
G())
F(y)
G(y)
By
A natural transform is a morphism between morphisms, referred to as a 2-morphism, whereas a morphism between objects a 1-morphism. When the definition category extended to include 2-morphisms it is referred as 2-category. An example of a 2-category Cat, whose objects are categories, 1-morphisms are functors, and 2- 2-morphisms are natural transforms. Given 1-morphisms f - y and g - y a two morphism from to g may be written 7) = g- Rather than hom classes 2-category has hom-categories. It is more concise to present the definition of a 2-calegory using composition functor and to present the identity morphisms with functor J/ 1 + C(I,T). The functor Ja selects on object of C(æ, x), were J=(()) = Idz. This also introduces an identity 2-morphism Id, for any 1-morphism f + y.
Definition 2.4 (2-category). A 2-category C consists of a class of objects Ob(C), and between any two objects y, E Ob(C)) 1-category of morphisms C(r, y) such that:
For any triple of objects x, v, E Ob(C) there a composition functor oz,y, C(y, 2) x C(.,v) + C(x, z).
Composition is associative: o (0,,,,0 x Ido(sa)) 22,2,0 (IdC(2,0) oz
Every object x E Ob(C') has an identity morphism J2 C(I,)). (J. C(x, v)) = and Ozz, (C x Jy) =r
The reason for writing the definition of a 2-category using functors rather than listing the axioms of its composition of 1-morphisms and 2-morphisms is because there is long list of axioms which are just consequence of its composition being functorial. For example, the horizontal and vertical composition of 2-morphisms. Because 2-morphisms are 1-morphisms of their hom categorics, two 2-morphisms 1/ : = g and 7 = h may be vertically composed to form 7M + h. Whereas, if the 2-morphisms are side by side they may be horizontally composed via the composition functor
10
A 1-morphism may be composed with a 2-morphism through the process of left or right whiskering. This is simply the horizontal composition of the 2-morphism with the identity of the 1-morphism.
yoldd z
f
x
y
x Idgon z
9'9
The results of this work concern the properties of adjunctions and left Kan extensions as error minimisers. Both of these constructions may be presented any 2-category, but the definition of an adjunction specific to adjoint functors will be of more use. A loose intuition of an adjunction between two functors is that each adjoint functor serves
approximate pseudo inverse for the other.
Definition 2.5 (Adjoint Functors (triangle)). Given two functors L : C D and R: D + C, is left adjoint to R, and is right adjoint to (written L 4 R) if and only if there exists natural transforms n Ide RL, called the adjunction unit, and natural transform LR = Idp, called the adjunction
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counit, which given any f: R(d) in C or g: L(c) + d in D there exists f: L(c) + Dr R(d) in C which are unique such that they satisfy the following commutative diagrams (triangle identities).
9c
RL(c)
R(d)
L(c)
LR(d)
R(F)
Lig)
Where is the adjunct of f constructed via f caL(f). and 9 the adjunction of g constructed via = R(g)l
Definition 2.6 (Left. Kan Extension (local)). Given 1-morphisms K C E, G C + D, a left Kan extension of K along G is 1-morphism Lang D + E' together with 2-morphism K = (LangK)G such that for any other such pair (H D + E, K + HG), There exists 2-morphism LongK + H. such that 7 = (a G)n.
D
Long K
C
E
2.2 Monoids, Preorders, and Lax 2-Functors
The categorical description of error presented in this paper is defined using a monoidal preorder. Though this structure can be described without the use category theory, is presented as a kind of 2-category so that may interact with other categorical components. Examples of the categorical definitions of otherwise describable objects are that of the Monoid and the Preorder.
Definition 2.7 (Monoid). A monoid C category with single object. Ob(C) {*}.
Definition 2.8 (Preorder). A proorder C' is a category with at most one morphism between any two objects.
Ya,y y C(f.g E C(æ,v) v) f f
Remark 2.9. The standard definition of proorder transitive and reflexive relation can be recovered by laking S if and only if there exists morphism f
Though the definition of error presented later does not necessarily use the real numbers, it does require that whatever order structure is used to compare errors has bottom least quantity of error.
Definition 2.10 (Bottom Element). Given a preorder P, an element 1 P bottom element of P if for all E P, 1 <
Understanding how monoid and preorder may be defined from categorical perspective makes the interpretation of the definition of monoidal preorder more apparent.
Definition 2.11 (Monoidal Preorder). A single object 2-category with at most one 2-morphism between any pair of 1-morphisms.
Johnson & Yau 2020
Definition 2.12 (Lax 2-Functor between 2-categories). A Lax 2-functor F - D, sends every object x E Ob(C) to F(x) E Ob(D), it has component functors Fzy C(r,y) + D(F(s), F()) and the following natural transforms:
orre) x Fz.9) + Foyy $ JF()) FJs
(5)
(6)
Which for all C(w, g C(z, v), and h Cly, 2) satisfy the following constraints.

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(IdF)( (9.1) (hg. Idr(s)) VId,.I (the Id)(()) = IdF(S)
IrIdd (Idp() o (w)
For the purposes of this paper, the relevant consequence of the definition of lax 2-functor is, due to the natural transforms & there cxists 2-morphism øg.5 F(f) = F(gf) in D for any composable morphisms f and g in C. When the codomain of the Lax 2-functor monoidal preorder, the existence of 2-morphism in the codomain
be reframed statement about the ordering of 1-morphisms.
Proposition 2.13. Given monoidal preorder S and lar functor F : : P S then for composable mor- phisms and in P.
F(g)F()) S F(g))
3 Error
Error minimisation atlempls achieve a particular output in one space E of given mapping In/ M + D by selecting an appropriate imput M. To compare different choices of there is bivariate function into the non negative real numbers Err : D D + R4 which allows some measurement of difference between the actual output [m) and the desired output d. Such a problem may be codified with sets and functions.
Definition 3.1 (Set Theoretic Error Minimisation Problem).
M, D E Set Inf E Set(M, D) Err E Set(D x D, R,) d= d' = Err(d, d') =0 de D minimise Err(d, Inf(m))
Given
To minimise Frr(d, Inf(m)) means to select a global error minimiser with respect to d. Definition 3.2 (Global Error Minimiser). Given an error minimisation problem (Def 3.1 x E M is a global error minimiser with respect to d E D if for any E M then Err(d, Inf(x)) S Err(d, Inf(m).
The choice to call the mapping between input and output Inf is in direct reference to the notion of model inference, where a machine learning model is used to predict an output given an input. Model inference is often referred to in the context of individual inputs vs outputs. The Inf function maps a particular parametrisation a machine learning model onto the dataset it produces when allowed to produce inference over the entire set of training imputs. From this perspective, M represents the set of all choices of parameters for the machine learning model and D is the set of all datasets that one may try to train against.
In order to apply the category theoretic constructions of adjunctions and Kan extensions to this definition, it needs to be lifted into a description using categories and functors. This is easily done with respect to In, by the statement that M and D should be categories and Inf: : M + D a functor. The next question how to categorify the notion of error. Morphisms are commonly thought of as structure preserving transformations. In the context of D this would suggest that the morphisms are transformations between datasets. Data transformations, functions, or programs can only lose information. They cannot add information that wasn't 't previously there. The better one dataset represents the information of another, the less information loss mapping between them may experience. This would indicate that error may be associated to category by assigning each morphism some quantity of error that represents its information loss. The values which represent error require some order structure so they can be compared and should reflect how error composes as morphisms compose. This suggests that the values of crror may be represented by a monoidal preorder (Del 2.11 Definition 3.3 (S Flavoured Error). Given a monoidal preorder S, where Id, is the bottom element, then S flavoured error D is lax 2-functor Err D S.
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To make use of the structure of D, the choice of error should respect the information that D contains. Namely, the composition of morphisms. A mapping of morphisms to morphisms which respects their composition would usually be indicative of a functor. However, though it would work, an error functor would be an excessively restrictive constraint. In practico, the information loss of the composite of two processes cannot usually be represented by the composition of the information loss of each of the processes individually. Two processes may lose the same portion of information, so their composite loss not much worse than their individual losses. In contrast, the lossy-ness of different pair of processes may affect entirely different portions of the information content, so their composite information loss would be much larger than their individual losses. It is much easier to represent the information loss of the composite of two morphisms via an inequality, which can be done using a lax 2-functor, encoding the relationship that the error of a composition of morphisms must be greater than or equal to some composition of their errors.
Frr(s) S Err(af)
(7)
One example of suitable monoidal preorder would be the single object category Rx, whose morphisms are the non-negative real numbers composed by taking the maximum and ordered by the standard ordering the reals. Imagine the case of the objects of D being data streams with morphisms being functions which map one data stream to another. In this case, the system is well described by an RA flavoured error on D. The functions between data streams have some associated information loss. If one is considering the error to be measured purely by the lost information and not just some invertible scrambling, then it wouldn't possible undo the error. So the error of those two functions composed together must always be greater than the maximum of the two errors.
Max Maz(Err(g) < Err(gf)
(8)
From a choice of S flavoured error, it is possible to recover an ordering of error associated with pairs of objects of D by looking at the best case scenario, the least errorful morphism.
Definition 3.4 (Error Comparison). Given S flavoured error on D. For objects v, E D then Err(r,y) y) Es Err(z, w) if and only if, for any f 2 + w there exists
+ and o Err(g) + Err(f).
Remark 3.5. The value Err(z, y) is notational convenience and is not an object in S. Instead, the important aspect of error in the traditional case is that it induces preorder on pairs of objects. Def 3.4 is a way of inducing a preorder on pairs of objects of D using S flavoured error.
Proposition 3.6. The error comparison of on S flavoured error on D defines preorder.
Proof. For any morphism f I >> there is an identity 2-morphism Id : Err(s) = Err(f) which implies that Err(n,y) Sg Err(x,8) y), demonstrating that the relation is reflexive.
If Err(z, v) S Err(w,2) 2) and Err(w, 2) < Err(a, b) then for any morphism f a->> b there is a morphism and 2-morphism h: y with associated 2-morphism &p Err(h) = Err(g), which by composition induces dyp Err(h) Err(f) for any f implying that Err(x, v) Is Err(a, b), demonstrating that the relation is transitive. As it is both reflexive and transitive the error comparison relation is a preorder.
g
Err(g) = Err(f). Given the existence of g, there must be morphism
By combining the requirement that Inf: : + D is a functor, with a choice of S flavoured error on D, one may produce a category-theoreti theoretic definition of an error minimisation problem. Definition 3.7 (Category Theoretic Error Minimisation Problem).
Given
M, D E Cat Inf E Cat(M, D) S flavoured error on D)
del DDD
return A global error minimiser with respect to d

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Remark 3.8. By fixing d, Def 3.7 may also serve definition for category theoretic loss minimisation.
The value of translating the set-thcoretic definition into the category theoretic definition that the existence of morphisms between models and datasets provides more information about the structure of the problem. The first observation to those which respect the structure of the category. From practical perspective, this provides novel approach to the selection of an error function given a particular error minimisation problem, if one knows the morphisms bet ween datasets. The second observation is that because the choice of error is now dependent on the morphisms of D, the propertics of category thcorotic constructions which reference only morphisms may be translated into their consequences with respect to error. The first such consequence that adjunctions are error minimisers,
make about the additional information is that it constrains the choices of error
Theorem 3.9 (Adjunctions are Error Minimisers). Given a category-theores ervor ménimisation problem (Def where Inf: M D has left adjoint Alg: D M, then for all d E D, Alg(d) is global error minimiser with respect to d.
Proof. For any d E D show that Alg(d) is global error minimiser with respect to d by demonstrating that for any m E M, Err(d, Inf Ss Err(d, [mf(m)).
If there does not exist a morphism f d + Inf(m) then the error comparison requirement (Def 3.4 is trivially satisfied.
If there does exists a morphism f f: d (m) then by the definition of an adjunction (Def 2.5 for any morphism f - (m) there is a unique morphism f Alg(d) m such that Inf = f, where n Ido Alg is the adjunction unit.
By the definition of a lax 2-functor (Def 2.12 there exists 2-morphism.
a: Errr = Err(Inf(fmm) = Err(f)
Because the identity of S is the bottom element there is a 2-morphism &p Id. = Err(Tns(f)) which by right whiskering produces the 2-morphism
8 Err(na): Err(na) = (r((Ind(())
Compose this with a.
Err(na)) Err(na) Err(f)
As this is true for any choice of f this proves the error comparison.
Err(d, Alg(() Ss Err(d,In((m)
As the error comparison is true for any choice of m then Alg(d) global error minimiser with respect d.
Corollary 3.10. The left adjoint of Inf is an ervor minimiser for any choice of error on D. If one can compute the left adjoint of Inf than they may identify the global error minimiser, with respect to any d, without ever making particulor choice of ervor.
Remark 3.11. If true error minimising algorithun returns global error minimiser for any element d E D, then one could define ams as left adjoint to inference.
If an inverse to inference exists then it would make sense it would represent an error minimisation algorithm. Ideally, any algorithm would return model which produced exactly the intended dataset. In the case that one is not capable of reproducing exactly the desired dataset, then a pscudo inverse to inference would be an intuitive choice. In the context of category theory, the natural choice of pseudo inverse is an adjunction, but this does not have to be an adjunction functors. The definition of an adjunction can be generalised to any 2-category. If one believes that algorithms are left adjoint to inference, then whatever object one uses to represent the collection of models and datasets, if these objects exist in a suitable 2-category, then one can define what global error minimising amgorithm should be.
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Inf
M
D
Alg
Corollary 3.12. ervor manimising problem is presented in the categorical form, then Theorem 3.9 shows that may prove that global minimiser erists by proving that left adjoint to Inf erists. This stulement may be repackaged with the various adjuinl funclor theorems Porsl (2024) to produce sufficient conditions for the eristence of optimal solutions to error minimisation problems.
While the production of sufficient conditions for the existence of global error minimisers is an incredibly powerful result, is excessively strict to always require the existence of adjunction to discuss global error minimisers in category-theoretic context. Furthermore, there many practical problems where global error minimiser may exist for some datasets but not for others. Tt would be more helpful to provide a construction which exists as long as particular dataset has a global error minimiser. Such a construction would be the left Kan extension. The application of a left Kan extension to a category thcoretic error minimisation problem requires the problem to be represented as an extension triangle. This can be done with a suitable choice of 2-category.
Proposition 3.13 (Extension Error Minimisation). An exlension briangle (see below) in 2-culegory T with an S flavoured error on T(ô,T) defines a category theoretic error minimisation problem (Def 3.7
8
Proof. The 1-morphism + and the object t induce the functor T(GT): T(u, 7) + T(6,r). The functor T(4, r) is defined via precomposition, sending any object m T(p, to mi E T(ō, By renaming the functor and categories as Inf M D it is clear that they define category theoretic error minimisation problem.
Remark 3.14. It is also the case that any category-theoreti error minimisation problem may be presented as extension in 2-category T by directly defining the hom categories and composition functor of T to be the calegories and inference functor of the error minimisation problem. Proving this is also true for the set-theoretic error minimisation problem is slightly trickier (Thm 4.1
Theorem 3.15 (Kan Extensions are Error Minimisers). Given calegory theorelic ervor minimisation problem (Def 3.7) in the form of extension (Prop 8.13), the left Kan eatension Lan,d is, if it evists, global error minimiser with respect to d.
Proof. For any d D show that Lan,d d is a global error minimiser by demonstrating that for any m M, Err(d, (Lnn,dd))) Es Err(d, (nf)))).
If there does not. exist a morphism f : + Inf(m) then the error comparison requirement (Def 3.4 is trivially satisfied.
If there does exist morphism Inf(m) = m as described in Prop 3.13) in T. By the definition of a left Kan extension (Del 2.6 for any 2- 2-morphism f + mi there exists a 2- m-rophh in T correspond directly with 1-morphisms of D i.e. n d + Inf(Lan,d) and a- c: Inn d) + Inf(m) where (Lan.d) = dL..,d)..
+ Inf(m then this is also 2-morphism, f = (recalling that
Lan, d + m such that f= (a-1)) These 2-morphisms
By the definition of lax 2-functor (Def 2.12 there cxists 2-morphism.
Err(a Err(() Err((a t)m) Err(f)
Bocause the identity of is the bottom element there is a 2-morphism yp Id. = Err(a . i) which by right whiskering produces the 2-morphism
6 Err(n): Err(na) * Err(a () Err((a))

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Compose this with As this is true for any choice of f this proves the error comparison.
de o (er(())) Err(na) = Err(f)
Err(d, (Ln(La,,)) Es Err(d, Inf(m))
As the error comparison is true for any choice of them Lan,d is a global error minimiser.
4 Universal representation
It has been shown that Kan extensions represent global error minimisers for category-theoretic error min- imisation problems, but it may not be clear that this also applies to the set-theoretic crror minimisation problem. It is actually possible to convert any sel-theoretic error minimisation problem into category- theoretic error minimisation problem, namely as an extension problem in a 2-category, such that the left Kan extensions of the extension problem are exactly the global crror minimisers of the set-thcoretic error minimisation problem.
Theorem 4.1 (Machine Learning representation). Given sel theorelic error minimisation problem (Def 8. 3.1 there ezists 2-category T such that M = =((,,T), D = T(,,),, Inf = T(, and an object m E M is a global error minimiser with respect to if and only if m Lan,d
Proof. Construct T have three objects, 14 6, and 7. The hom objects will be selected such that Inf becomes composition morphism, and the 2-morphisms (morphisms of the hom category) constructed to artificially select minimising element if it exists.
Define the following singleton categories
(1) (Tds) = (Tdz)
Define M such that Obj(M) = M and that for any m, m' M there is unique morphism if and only if (m) = [m'). Define D to be the category whose objects are the elements of D. Let U C D be the subset of datasets for which an error minimising model exists, and let Alg : U - M be a function which selects an error minimising model for each d D under the constraint that if there exists an m M such that Inf(m d, then Alg(d) = m.
m'
Define the hom sets of D with the following piecewise function.
{Ida} d=d
D(d, =
(+) (nNd'=In((AA((()) A A Ad40 d'
0 clse
Composition is defined in the obvious way. For objects d, d' d" D consider the form of the composition morphism.
a D(d,d) x D(d' + D(d,
Whenever (dd,'') or D(d', d') is empty, then the product is empty, making the composition morphism the unique map from the emply set. When both D(d,d') or D(d', d') are non emply, they must. both be singleton. Therefore the following must be true.
d= d' v (d' Inf(Alg(d)) Ad+1) d' = d' v (d" = Inf(Alg( ^ dd' +d))

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Which may be simplified to form the following.
d = d' v d= = (Id=II((A
d' = d" V ct =In[(Al(((d))
Combining these statements produces the following deduction.
D(d, x d(d',"") =
dd' Inf(Ag((d))
A
(d=''Ad'=d")
(d= A = -In((Alg(d))
v v Inf(Alg(d)) Ad' d')
V v Inf(Alg(d) A d" Inf(Alg(d')
(d d")
Vd = (Alg(())
V (d' = Ins((AAg(d))) = IIn((Ag((d)))
When d' = (Alg(()) then for m Alg(d)
(rr(In((m), d') Err( Err(In((Agg(()) Err(d'd') d')= =0
By the definition of Alg this forces Alg(d') = Alg(d", which by the construction of M means that Inf (Alg(dd)) = Inf((lg(()) d'. This allows the deduction to be simplified to the following implication.
D(d,d) x
(d= dd) = Inf(Alg(d)
V (d' = Inf(AA(()) A d" = Inf((Ag(dd))
(d = Inf[vl=
V (d' Inn((Ag((() dd d'))
(d II
d(d,d)) 1
Making the composition morphism in this case, the unique morphism between singleton sets. Using the above-defined categories, define the hom-categories of T as follows.
T(-.-)
8
14
0
M
8
{
{Id
}
D
0
0
{Id,}
The only composition morphism which is not fixed by identity laws or the empty categories is the following
0g,.r T(5, x T(G,)) + T( r)
Substituting the known hom objects, this is rewritten as.
08, -
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Because {c} M = M, the composition morphism can be defined by the inference function which maps all morphisms of M to the relevant identity morphisms
O = Inf
Finally, consider the following Kan extension problem in T.
P
m
8
d
If an error minimising m does not exist for the given d then no Kan extension can exist. as there is no morphism from d into the image of Inf : {(} x M However, if an error minimising m does exist then by construction there is a morphism in and consequently 2-morphism in T of the form d = Alg(d) (Alg(d))) For any m for which there also exists 2-morphism d m then as such 2-morphism from d into the image of Inf is unique, then Alg(d) = m. This makes Alg(d), when it exists, a left Kan extension in T
[n which by the construction of M means that
The strategy for constructing I' amounts to appending morphisms to M and D such that the Kan extension selects the minimising elements of the error functions, if they exist. The morphisms in D effectively present a relation which points from any element. E D to its respective error minimiser, when it exists. The appended morphisms and sets are the minimal structure required to admit a Kan extension. Consequently, the construction of T acts as a minimal or free construction regarding the relevant set theoretic error min- imisation problem. is not necessarlly unique as a 2-category which admits a Kan extension for a given error minimisation problem, but it will be a subcategory of any other 2-category which does admit such an extension. The intuition of this construction can be seen by considering the cases of convex optimisation and linear regression.
Example 4.2 (Convex optimisation). Let M R = be a closed convex proper subset of the 2D planc, D. The standard euclidean distance Err || yll R2 I R can be used as an error function to determine the distance between two points in D. Because M is subset, its inclusion in D may be presented as the injective function Inf M - D. These definitions of Inf and Err satisfy the requirements of a set theoretic error minimisation problem. Given some point d E D, the error minimiser will be the unique point in M which is closest to d. Consequently, by Thm 4.1 there exists a two category T such that these nearest points are Kan extensions. Figure shows an example morphism in when * Alg(d) = d'. In this instance, as d is outside of M it cannot be equal to its minimiser. Therefore D(d, d') = (+). Because D is a hom-category of T, the 1-morphism * : d d' in D is the 2-morphism in T required to make Alg(d) a Kan extension of d along 1. If d was selected such that d M, then d = Inf (nlg(A)()) d' and D(d, (Tda). This identity morphism would then form the 2-morphism required in T to form the requisite Kan extension.
Having demonstrated how the nearest point within a convex set may be presented as a Kan extension, is only a small modification to present this result in the context of linear regression.
Example 4.3 (Lincar regression). Consider a dataset of n ordered pairs (r,, d)) E RR x R. The line of best fit is the affine function fzz = ax + which minimises the root mean squared error.
Err(d, = ((( - d)2
One can consider the datasct of values to be the n-tuple of values, d E R The set of models M is the set of affine functions ʃ IR + R which can be parametrised as a pair of real numbers (a, b) E =² = M. The inference function, Inf M + D, maps M into D by evaluating a given affine function f at every poInt 2,, Inf(f), = f(x)). Inf a linear map from R² R" which, in the context of linear regression, is commonly referred as the design matrix. Using this construction of Inf the error function becomes the standard euclidean distance on D. Analogous to the previously presented convex optimisation problem,
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d
(ng(d))
M
Figure 1: M is presented convex subset set of the plane. A morphism d (Alg(g)) is appended between and the closest point to d in M such that Kan extension of d identifies Inf(Alg(d) as the only possible extension.
the selection of a global minima given a particular amounts to selecting the point in the convex subset M which is closest to d. As before, so that Kan extension exists in whenever an error minimiser exists, a morphism - Inf (Alg(()) is added to D whenever minimiser exists for every choice of d. This means that not only
exists and d + Inf (Alg())) For the
particular of linear regression, can the error be presented as Kan extension in T, but that they also form an adjunction Inf | Alg. As the inference mapping in this case is linear and the error function is euclidean distance, the mapping which identifies the error minimiser, Alg, is the Moore-Penrose pseudo inverse of Inf. This connection re-enforces the relationship between adjunctions, optimisation algorithms, and pseudo inverses.
The special case of linear regression presenting Alg as the Moore-Penrose pseudo inverse indicates a more general aspect of the T construction. one has an algorithm for Computing Inf M + D and Alg U + M then it is possible to query information about the 2-morphisms Tf with a worst case time complexity the sum of the time complexities of Inf and Alg. This comes directly from determining the hom-set D(d, d'), which when d ش d' and d E U is equal the set { (Alg(() d'). This can be used to show that the time complexity of computing T is entirely dependent algorithm which computes the function f X + Y there is known time complexity lower bound of Of. This function may be presented as the solution to an error minimisation problem.
the problem domain. Presume that for any
Proposition 4.4 (Error Minimisation Function Representation). For any set function f X - there exists a set theorefic error minimisation problem E, (Def 3.1) with Inf ; Y + X such that for any point d X Y the point [dl {dd is its global error minimiser.
Proof. Construct the Inf function as the canonical inclusion Inf : X X+Y, and define Err (X ()+ ()+ R the following piecewise function.
Err(d,d) { d= f(d) d'
else
This defines set theoretic error minimisation function according to Def 3.1
The function {f. Idy]
Y selects an error minimiser. This function may be defined explicitly as follows.
(f, f(d) xex dEY
d
In the first case assume d Y.
Ery(d,)(1,I&y)())) Err(d, =0
In the second case assume d E X
Err(d, idy = Err(d, f(d))
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In both cases, as 0 is the smallest possible value of the given error function, then {f. Idy Md) is the global error minimiser with respect to d.
By using Thm to construct T with respect to Ey, as shown in Prop 4.4 one can see that the worst case time complexity for computing D(d, d") when d + d' is the time complexity of iff, Idy]: which in the worst case has lower bound O,. This demonstrates that one can construct error minimisation problems such that querying the construction of their respective 2-category T must have a worst-case time complexity lower bound of O₃- As the function i{f, Idy) also serves to compute Kan extensions, by identifying error minimisers, the computational complexity of computing Kan extensions would also be Oj. By selecting arbitrarily hard problems, one can see that the possible complexity of constructing T is unbounded. One can infer from this that any 2-category which solves Es via Kan extensions would have a worst case time complexity lower bound of y for the computation of these Kan extensions. This can either be seen directly as these categories compute f for which the time complexity is known, or as these categories contain as a subeategory.
While an efficient algorithm is known, it would be unnecessary to compute T directly; however, the opposite problem may be susceptible to Kan extension approach. If f X + Y is function which is desired to be computed but has no known efficient algorithm of computation, then by presenting this function as the solution to an error minimisation problem, it may be possible to develop methodologies for its computation via the known processes of computing Kan extensions. Particularly when these Kan extensions exist between categorics with well known structures. While should again be noted that the time complexities of these algorithms are highly dependent on the problem domain, it provides another avenue of attack by which researchers may approach the problem of algorithm design.
5 Conclusion
This paper has introduced an important bridge in the field of category theory for machine learning, providing a connection that allows the application of powerful category theoretic tools to old problems.
In addition to the theorems relating to the category-theoretic constructions, this methodology has also introduced insights that may impact how views machine learning problems.
Firstly, the introduction of S flavoured error supplements rigorous notions of how one might select an error function. Suggesting that error should be associated with the transformations between datasets gives an indication of how one may appropriately choose an error for a given problem. From this perspective, traditional notions such as distance, accuracy, and information loss reveal themselves as measurements of the minimal transformation necessary convert one dataset into another.
Secondly, the independence of the left adjoint to choices of error may indicate that there are more fundamental ways of selecting a globally optimal model with respect to dataset without referring to error. Re-framing model inference as way of producing a dataset from model, mapping from a space of models to a space of datasets, allows one to think of algorithms as pscudo inverses to inference. In category theory, the natural choice of pseudo inverse is the adjunction, which is not constrained to an adjunction of functors. The definition of an adjunction can be generalised to any 2-category. Suppose it is more appropriate to represent the spaces of models and datasets as some other object, such as manifolds or mcasure spaces. In that case, finding an appropriate choice of 2-morphisms will indicate how to construct an algorithm as left adjoint to inference.
Inf T Alg
M
D
Finally, the demonstration that left. Kan extensions may represent any error minimisation problem provides an interesting connection between the purpose of machine learning and the presentation. It is often intuitive to think that machine learning models find patterns within a datasct, allowing it to extend the alrcady present data. However, this is only possible if one introduces additional assumptions about the data. Assumptions such linearity, distance, smoothness, and maximum likelihood are all examples of assumptions machine
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learning models utilise to extend datasets. The nature of taking some aspect of data and extending it to different context is precisely the structure encoded by Kan extension, connecting intuitions about machine learning to a rigorous algebraic representation.
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Joshua Meyers, David 1. Spivak, and Ryan Wisnesky. Fast Left Kan Extensions Using The Chase, May 2022.
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0455/ arXiv:2101.04531 [math].
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abs/2310. 19528 arXiv:2310.19525 [math].
Matthew Pugh, Jo Grundy, Corina Cirstea, and Nick Harris. Using Kan Extensions to Motivate the Design a Surprisingly Effective Unsupervised Lincar SVM on the Occupancy Datasct. Mathematical and Compulational Applications, 29(5):74, October 2024. ISSN 2297-8747. doi: 10.3390/mc2290500077. URL https://wwis https::/www.mmpp..or htwp..ppp...o//27777//////////// Number: Publisher: Multidisciplinary Digital Publishing Institute.
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14


Reviews:
[{"review_id": "OJqWp0ZMCS", "reviewer": "Reviewer_KEQY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- This work builds a bridge between machine learning (ML) and category theory by showing that any error minimization problem in ML can be reformulated as a Kan extension problem.\n- the authors prove that given an arbitrary learning task (with a pre-defined space of models, that of data/outcomes, a prediction map, and a loss function), one can construct a categorical framework in which finding an optimal model is equivalent to computing a left Kan extension.\n- In doing so, it opens the door to applying powerful categorical tools and theorems to machine learning problems.\n\nStrength\n\n- The paper’s main theorem is quite general – it does not just draw an analogy between one algorithm and a Kan extension, but shows every error minimization problem fits into this paradigm. This unification is a significant strength, as it consolidates many specific observations into one framework.\n- By making the connection to Kan extensions precise, the work allows researchers to apply known category-theoretic results to ML. For example, a strong insight is treating loss functions as morphism-induced measures of information loss. Another insight is that a learning algorithm can be seen as a left adjoint (a kind of generalized inverse) to the data-generating process.\n- The reduction of learning to adjunctions and Kan extensions is carried out in a mathematically rigorous way. For instance, Theorem 3.9 is a direct consequence of the adjunction properties and is proved by showing the required error inequality holds universally. Theorem 4.1 is constructive – the authors explicitly build the required 2-category and verify that a left Kan extension in it matches the known minimizer.\n\n\nWeakness\n- A notable weakness is the high level of abstraction. The paper operates in the realm of category theory (2-categories, lax 2-functors, Kan extensions), which is far from the usual toolkit of most machine learning researchers. This means the work might be challenging to digest for the typical reader in ML or even theoretical CS who isn’t already versed in category theory.\n- The paper is almost entirely theoretical. While it references prior examples (e.g. anomaly detection algorithm derived via Kan extensions in separate work, and a classification algorithm in Shiebler’s work), it does not demonstrate the framework on a specific well-known learning problem within the paper.\n- The main Theorem 4.1, though powerful, is somewhat non-constructive. In its proof, the authors assume the existence of a minimizer in order to construct the category $T$ – for example, they define a function $Alg: U \\to M$ that picks an error-minimizing model for each dataset $d$ (in the subset $U \\subseteq D$ where a minimizer exists). This means the framework encodes a solution that is already known or assumed to exist, rather than providing a method to find a solution when one is not known.\n- The flip side of the paper’s generality is that the constructed 2-category in Theorem 4.1 can feel tailored or artificial to each problem. The authors essentially build $T$ in a custom way for a given learning task, encoding the error criterion into the morphisms of $T$. Some may question whether this is just a tautological embedding – after all, if you already have an optimal model, you can always embed it into a category somehow.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper’s core results – notably Theorem 3.9 and Theorem 4.1 – are stated with clear hypotheses and conclusions, and their proofs leverage standard categorical reasoning.", "audience_yn": "Yes", "audience_explanation": "I guess only those people working on KAN will be interested.", "requested_changes": "- Include at least one worked-out example of a standard learning problem (e.g., linear regression, classification with cross-entropy loss) reformulated as a Kan extension. This would demonstrate practical relevance and clarify the constructions.\n- The construction of the 2-category T is valid but quite dense and hard to follow. Currently, the argument is heavy with logical case-splitting.\n- The paper frames all learning problems as Kan extensions, but does not sufficiently acknowledge the limitations (e.g., non-existence of minimizers, the somewhat ad hoc construction of T).", "broader_impact_concerns": "Not many people will read this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "6NoEa7mKNZ", "reviewer": "Reviewer_SpYQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper recasts supervised error minimization using category theory. It defines an $S$ flavoured error as a lax 2-functor into a monoidal preorder so that information loss composes via an inequality. It then shows that left adjoints to inference yield global minimizers, independent of the chosen error, with existence ensured by adjoint functor theorems. Finally, it proves that left Kan extensions are error minimizers and that any set-theoretic error minimization problem admits a 2-categorical representation where global minimizers coincide with left Kan extensions (Theorem 4.1). This reframes algorithm design as constructing left adjoints to inference and choosing error structures from dataset morphisms.\n\nStrengths:\n\nS1. Establishes that all set-theoretic error minimization problems admit a 2-categorical encoding where global minimizers coincide with left Kan extensions (Theorem 4.1)—a clean bridge from ML objectives to categorical machinery.\n\nS2. The $S$ flavoured error (lax 2-functor into a monoidal preorder) gives a principled way to model information loss and its composition via $Err(g)Err(f) \\leq Err(gf)$. This both respects structure and guides error selection based on dataset morphisms.\n\nS3. Shows adjunctions are error minimizers and connects to adjoint functor theorems to obtain sufficient conditions for optimal solutions without fixing a loss function.\n\nWeaknesses:\n\nW1. The paper gives representation theorem but does not demonstrate concrete instantiations on standard ML problems (e.g., explicit choices of $S,D,M,$ and computed Kan extensions), leaving practical impact to future work.\n\nW2. The proof of Theorem 4.1 relies on building a 2-category and hom-sets engineered to select minimizers, which may feel mathematically tautological and offers limited guidance for natural problem categories.\n\nW3. While $S$ flavoured error is elegant, how to pick $S$ and the lax 2-functor for real tasks is mostly left abstract; the requirement of a bottom element and inequality-based composition may not align cleanly with common ML losses without possible additional design choices.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The core category-theoretic claims are accurately and clearly supported by formal proofs within the paper’s framework. Though, the practical significance is more speculative, due to the engineered nature of the universal construction and the absence of applications to particular error minimization approaches.", "audience_yn": "Yes", "audience_explanation": "Yes it will interest certain researchers in theoretical machine learning who value new abstractions that unify learning objectives via adjoints and Kan extensions.", "requested_changes": "1. Please refine the exposition, especially the introduction, and make the connections among definitions, theorems, and implications explicit. Add motivation and high-level intuition so that readers without a background in category theory can follow the results and understand their significance. The overall writing needs improvement.\n\n2. Please add a concrete application on a standard ML task that explicitly instantiates $S,D,$ and $M$, defines dataset morphisms, and computes the left Kan extension. A brief, step by step walk-through interpreting the resulting predictor would improve accessibility and demonstrate practical utility.\n\n3. The conclusion notes potential applications; however, developing at least one concrete instantiation would substantially enhance the paper’s appeal. I acknowledge this may be beyond scope, but in its current form the manuscript reads largely as a collection of definitions and proofs with limited guidance on practical impact.\n\n4. Please ensure all figures are clearly labeled and explicitly referenced in the main text, with concise captions that explain their relevance to the narrative.", "broader_impact_concerns": "Not very applicable here.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gFIPzpRCZN", "reviewer": "Reviewer_Jc1D", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors prove a general result showing that any error minimization problem can be represented as a left Kan extension in a suitably defined 2-category. This builds on prior work noting that certain ML algorithms resemble Kan extensions, but goes further by showing that all error minimization algorithms admit such a representation. The paper develops the notion of “S-flavored error” via lax 2-functors, proves that adjunctions are global error minimizers (Theorem 3.9), generalizes this to left Kan extensions (Theorem 3.15), and finally shows that any set-theoretic error minimization problem can be embedded into a 2-category whose left Kan extensions are exactly the global minimizers (Theorem 4.1). This provides a clean categorical formalism that connects machine learning optimization to a large body of existing results in category theory.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper makes several claims and proves them directly in the paper. I highlight the claims made and how well they are supported below, noting particularly strong aspects of the paper:\n\n1. Theorem 4.1 is a very general and elegant unification, subsuming prior special-case results and putting all error minimisation problems into the same categorical framework.\n\n2. The “S-flavoured error” abstraction offers a principled way to relate error to morphisms between datasets and is a useful conceptual contribution.\n\n3. The discussion of adjunctions as pseudo-inverses to inference gives a clear link between abstract category theory and the more intuitive notion of an “ideal” algorithm.\n\n4. The results open the door to applying existing categorical tools such as adjoint functor theorems and colimit constructions directly to machine learning optimisation.", "audience_yn": "Yes", "audience_explanation": "This paper presents a substantial theoretical contribution, establishing a broad categorical unification of error minimization problems via left Kan extensions. The work will be of interest to the theory-oriented portion of TMLR’s readership, particularly those at the intersection of category theory and machine learning. That said, the paper’s impact and accessibility would be improved by adding at least one worked-through example, some discussion of computational aspects, and lighter notation where possible.", "requested_changes": "I give some explicit requested changes below, together with whether or not each change is critical for securing acceptance:\n\n1. Please provide at least one fully worked example showing how a standard ML problem (e.g., linear or logistic regression) can be represented in this framework, including how the Kan extension arises (critical for securing acceptance).\n\n2. Please add some discussion of computational feasibility. In the constructed 2-categories of Theorem 4.1, when is computing a left Kan extension tractable? Are there practical cases where this could be done efficiently? (critical for securing acceptance)\n\n3. Add diagrams and informal explanations alongside key definitions to help readers build intuition. (recommended but not critical for acceptance)\n\n4. Minor corrections for clarity and correctness (recommended but not critical):\n\n- Break up long formal definitions with brief plain-language restatements\n\n- Fix small typos such as “them Alg(d) is a global error minimizer” -> “then Alg(d) is a global error minimizer” (on page 7)", "broader_impact_concerns": "This is primarily a theoretical paper with no immediate societal risks beyond those of ML in general. As such, I do not believe a broader impact statement is required.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)