AutoGraph: Imperative-style Coding with Graph-based Performance

There is a perceived trade-off between machine learning code that is easy to write, and machine learning code that is scalable or fast to execute. In machine learning, imperative style libraries like Autograd and PyTorch are easy to write, but suffer from high interpretive overhead and are not easily deployable in production or mobile settings. Graph-based libraries like TensorFlow and Theano benefit from whole-program optimization and can be deployed broadly, but make expressing complex models more cumbersome. We describe how the use of staged programming in Python, via source code transformation, offers a midpoint between these two library design patterns, capturing the benefits of both. A key insight is to delay all type-dependent decisions until runtime, via dynamic dispatch. We instantiate these principles in AutoGraph, a software system that improves the programming experience of the TensorFlow library, and demonstrate usability improvements with no loss in performance compared to native TensorFlow graphs. We also show that our system is backend agnostic, and demonstrate targeting an alternate IR with characteristics not found in TensorFlow graphs

Efficient CHAD

We show how the basic Combinatory Homomorphic Automatic Differentiation (CHAD) algorithm can be optimised, using well-known methods, to yield a simple and generally applicable reverse-mode automatic differentiation (AD) technique that has the correct computational complexity that we would expect of a reverse AD algorithm. Specifically, we show that the standard optimisations of sparse vectors and state-passing style code (as well as defunctionalisation/closure conversion, for higher-order languages) give us a purely functional algorithm that is most of the way to the correct complexity, with (functional) mutable updates taking care of the final log-factors. We provide an Agda formalisation of our complexity proof. Finally, we discuss how the techniques apply to differentiating parallel functional programs: the key observations are 1) that all required mutability is (commutative, associative) accumulation, which lets us preserve task-parallelism and 2) that we can write down data-parallel derivatives for most data-parallel array primitives

Verifying an Effect-Handler-Based Define-By-Run Reverse-Mode AD Library

We apply program verification technology to the problem of specifying and verifying automatic differentiation (AD) algorithms. We focus on ``define-by-run'', a style of AD where the program that must be differentiated is executed and monitored by the automatic differentiation algorithm. We begin by asking, ``what is an implementation of AD?'' and ``what does it mean for an implementation of AD to be correct?'' We answer these questions both at an informal level, in precise English prose, and at a formal level, using types and logical assertions. After answering these broad questions, we focus on a specific implementation of AD, which involves a number of subtle programming language features, including dynamically allocated mutable state, first-class functions, and effect handlers. We present a machine-checked proof, expressed in a modern variant of Separation Logic, of its correctness. We view this result as an advanced exercise in program verification, with potential future applications to the verification of more realistic automatic differentiation systems and of other software components that exploit delimited control effects

Efficient Dual-Numbers Reverse AD via Well-Known Program Transformations

Where dual-numbers forward-mode automatic differentiation (AD) pairs each scalar value with its tangent value, dual-numbers \emph{reverse-mode} AD attempts to achieve reverse AD using a similarly simple idea: by pairing each scalar value with a backpropagator function. Its correctness and efficiency on higher-order input languages have been analysed by Brunel, Mazza and Pagani, but this analysis used a custom operational semantics for which it is unclear whether it can be implemented efficiently. We take inspiration from their use of \emph{linear factoring} to optimise dual-numbers reverse-mode AD to an algorithm that has the correct complexity and enjoys an efficient implementation in a standard functional language with support for mutable arrays, such as Haskell. Aside from the linear factoring ingredient, our optimisation steps consist of well-known ideas from the functional programming community. We demonstrate the practical use of our technique by providing a performant implementation that differentiates most of Haskell98

Dual-Numbers Reverse AD, Efficiently

Where dual-numbers forward-mode automatic differentiation (AD) pairs each scalar value with its tangent derivative, dual-numbers /reverse-mode/ AD attempts to achieve reverse AD using a similarly simple idea: by pairing each scalar value with a backpropagator function. Its correctness and efficiency on higher-order input languages have been analysed by Brunel, Mazza and Pagani, but this analysis was on a custom operational semantics for which it is unclear whether it can be implemented efficiently. We take inspiration from their use of /linear factoring/ to optimise dual-numbers reverse-mode AD to an algorithm that has the correct complexity and enjoys an efficient implementation in a standard functional language with resource-linear types, such as Haskell. Aside from the linear factoring ingredient, our optimisation steps consist of well-known ideas from the functional programming community. Furthermore, we observe a connection with classical imperative taping-based reverse AD, as well as Kmett's 'ad' Haskell library, recently analysed by Krawiec et al. We demonstrate the practical use of our technique by providing a performant implementation that differentiates most of Haskell98

Verifying an Effect-Handler-Based Define-By-Run Reverse-Mode AD Library

We apply program verification technology to the problem of specifying and verifying automatic differentiation (AD) algorithms. We focus on define-by-run, a style of AD where the program that must be differentiated is executed and monitored by the automatic differentiation algorithm. We begin by asking, "what is an implementation of AD?" and "what does it mean for an implementation of AD to be correct?" We answer these questions both at an informal level, in precise English prose, and at a formal level, using types and logical assertions. After answering these broad questions, we focus on a specific implementation of AD, which involves a number of subtle programming-language features, including dynamically allocated mutable state, first-class functions, and effect handlers. We present a machine-checked proof, expressed in a modern variant of Separation Logic, of its correctness. We view this result as an advanced exercise in program verification, with potential future applications to the verification of more realistic automatic differentiation systems and of other software components that exploit delimited-control effects