Automatic Differentiation in Functional Programming via First-Class Operators
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Solution Overview
Problem
Existing functional-programming languages like SCHEME, ML, and HASKELL face difficulties in implementing automatic differentiation (AD) effectively, as traditional AD systems are not modular and struggle with handling higher-order derivatives and gradients through processes involving AD-based optimization or PDE solvers.
Innovation Solution
A novel framework is introduced that treats AD operators as first-class higher-order functions mapping first-class function objects to first-class function objects, allowing for compositional construction of derivatives and gradients, and is implemented in a new language called VLAD, which supports AD better than existing functional programming languages.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If traditional AD systems are used in functional programming languages, then basic differentiation functionality is provided, but modularity is poor and handling higher-order derivatives is difficult
Solution Approach 1:
The patent segments the AD system into independent first-class function components that can be composed modularly. Each AD operator is a separate first-class function that can be independently defined, passed as arguments, and combined to build complex derivative computations, thereby improving modularity while managing complexity through functional composition.
Solution Approach 2:
The patent introduces dual numbers as an intermediary data structure that enables automatic differentiation through algebraic operations. Dual numbers carry both primal values and derivative information, allowing derivatives to be computed through standard arithmetic operations while maintaining modularity and supporting higher-order derivatives through nested dual number constructions.
2Adaptability or versatility
If AD operators are implemented as first-class higher-order functions, then modularity and composability improve, but implementation complexity increases
Solution Approach 1:
The patent implements self-service through automatic code generation that transforms primal function definitions into their derivative counterparts. The system automatically generates the adjoint code from the original function, reducing manual implementation effort while maintaining the benefits of first-class higher-order function composability.
Solution Approach 2:
The patent performs preliminary actions by pre-defining a set of primitive AD operators with known derivative rules. These primitive operators serve as building blocks that can be composed to handle complex functions, reducing implementation complexity by leveraging pre-established derivative relationships rather than computing everything from scratch.
3Reliability
If AD-based optimization or PDE solvers are used, then solution capability improves, but taking gradients through these processes becomes difficult
Solution Approach 1:
The patent implements feedback mechanisms through adjoint computations that propagate derivative information backward through the computational graph. This feedback loop allows gradients to be computed through complex processes like AD-based optimization and PDE solvers by systematically tracking and propagating sensitivity information through each computational step.
Solution Approach 2:
The patent creates a universal AD framework that works across multiple computational domains including optimization, PDE solving, and general function differentiation. The first-class higher-order function approach provides a unified mechanism for computing gradients through any differentiable process, reducing complexity by applying the same fundamental principles across diverse applications.
Data Source
AI summary
The disclosed system provides a transformation-based implementation of forward-mode and reverse-mode automatic differentiation as a built-in, first-class function in a functional programming language. Each of these constructs imposes only a small constant factor of the computational burden (time) of the function itself, and the forward construct has the same properties in terms of space. The functions can be applied to any function, including those involving derivatives and nested closures.