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Forward- or reverse-mode automatic differentiation: What's the difference?

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Title: Forward- or reverse-mode automatic differentiation: What's the difference?
Authors: van den Berg, Birthe1 (AUTHOR) birthe.vandenberg@kuleuven.be, Schrijvers, Tom1 (AUTHOR) tom.schrijvers@kuleuven.be, McKinna, James2 (AUTHOR) j.mckinna@hw.ac.uk, Vandenbroucke, Alexander3 (AUTHOR) alexander.vandenbroucke@sc.com
Source: Science of Computer Programming. Jan2024, Vol. 231, pN.PAG-N.PAG. 1p.
Subjects: Automatic differentiation, Kronecker delta
Abstract: Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although many researchers appreciate and know how to apply AD, it remains a challenge to truly understand the underlying processes. From an algebraic point of view, however, AD appears surprisingly natural: it originates from the differentiation laws. In this work we use Algebra of Programming techniques to reason about different AD variants, leveraging Haskell to illustrate our observations. Our findings stem from three fundamental algebraic abstractions : (1) the notion of semimodule, (2) Nagata's construction of the 'idealization of a module', and (3) Kronecker's delta function, that together allow us to write a single-line abstract definition of AD. From this single-line definition, and by instantiating our algebraic structures in various ways, we derive different AD variants, that have the same extensional behaviour, but different intensional properties, mainly in terms of (asymptotic) computational complexity. We show the different variants equivalent by means of Kronecker isomorphisms , a further elaboration of our Haskell infrastructure which guarantees correctness by construction. With this framework in place, this paper seeks to make AD variants more comprehensible, taking an algebraic perspective on the matter. • Basic forward mode AD is the fusion of two semiring homomorphisms: symbolic differentiation and evaluation. • Three fundamental algebraic abstractions lay the foundations of a single-line definition of AD algorithms. • Different AD algorithms can be obtained using isomorphisms. • Our approach supports common extensions: more primitives, overloading, let-sharing, higher derivatives, non-commutativity. [ABSTRACT FROM AUTHOR]
Copyright of Science of Computer Programming is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
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  Data: Forward- or reverse-mode automatic differentiation: What's the difference?
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  Data: <searchLink fieldCode="AR" term="%22van+den+Berg%2C+Birthe%22">van den Berg, Birthe</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> birthe.vandenberg@kuleuven.be</i><br /><searchLink fieldCode="AR" term="%22Schrijvers%2C+Tom%22">Schrijvers, Tom</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> tom.schrijvers@kuleuven.be</i><br /><searchLink fieldCode="AR" term="%22McKinna%2C+James%22">McKinna, James</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> j.mckinna@hw.ac.uk</i><br /><searchLink fieldCode="AR" term="%22Vandenbroucke%2C+Alexander%22">Vandenbroucke, Alexander</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> alexander.vandenbroucke@sc.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Science+of+Computer+Programming%22">Science of Computer Programming</searchLink>. Jan2024, Vol. 231, pN.PAG-N.PAG. 1p.
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  Data: Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although many researchers appreciate and know how to apply AD, it remains a challenge to truly understand the underlying processes. From an algebraic point of view, however, AD appears surprisingly natural: it originates from the differentiation laws. In this work we use Algebra of Programming techniques to reason about different AD variants, leveraging Haskell to illustrate our observations. Our findings stem from three fundamental algebraic abstractions : (1) the notion of semimodule, (2) Nagata's construction of the 'idealization of a module', and (3) Kronecker's delta function, that together allow us to write a single-line abstract definition of AD. From this single-line definition, and by instantiating our algebraic structures in various ways, we derive different AD variants, that have the same extensional behaviour, but different intensional properties, mainly in terms of (asymptotic) computational complexity. We show the different variants equivalent by means of Kronecker isomorphisms , a further elaboration of our Haskell infrastructure which guarantees correctness by construction. With this framework in place, this paper seeks to make AD variants more comprehensible, taking an algebraic perspective on the matter. • Basic forward mode AD is the fusion of two semiring homomorphisms: symbolic differentiation and evaluation. • Three fundamental algebraic abstractions lay the foundations of a single-line definition of AD algorithms. • Different AD algorithms can be obtained using isomorphisms. • Our approach supports common extensions: more primitives, overloading, let-sharing, higher derivatives, non-commutativity. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Science of Computer Programming is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.scico.2023.103010
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      – Code: eng
        Text: English
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        Type: general
      – SubjectFull: Kronecker delta
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      – TitleFull: Forward- or reverse-mode automatic differentiation: What's the difference?
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              Text: Jan2024
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              Y: 2024
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