$\lambda_S$: Computable Semantics for Differentiable Programming with Higher-Order Functions and Datatypes

TL;DR

The paper introduces $ ext{\lambda}_S$, a new differentiable programming language with a formal semantics for higher-order functions and datatypes, enabling automatic derivatives for complex objectives like integration and root-finding.

Contribution

It provides the first computable semantics for higher-order functions and datatypes in differentiable programming, implemented as an embedded Haskell language, facilitating advanced differentiable algorithms.

Findings

Developed $ ext{\lambda}_S$ with formal semantics for higher-order functions.

Constructed differentiable libraries for probability distributions and surfaces.

Demonstrated novel differentiable algorithms like Hausdorff distance computation.

Abstract

Deep learning is moving towards increasingly sophisticated optimization objectives that employ higher-order functions, such as integration, continuous optimization, and root-finding. Since differentiable programming frameworks such as PyTorch and TensorFlow do not have first-class representations of these functions, developers must reason about the semantics of such objectives and manually translate them to differentiable code. We present a differentiable programming language, $\lambda_S$, that is the first to deliver a semantics for higher-order functions, higher-order derivatives, and Lipschitz but nondifferentiable functions. Together, these features enable $\lambda_S$ to expose differentiable, higher-order functions for integration, optimization, and root-finding as first-class functions with automatically computed derivatives. $\lambda_S$'s semantics is computable, meaning that values can be computed to arbitrary precision, and we implement $\lambda_S$ as an embedded language in Haskell. We use $\lambda_S$ to construct novel differentiable libraries for representing probability distributions, implicit surfaces, and generalized parametric surfaces -- all as instances of higher-order datatypes -- and present case studies that rely on computing the derivatives of these higher-order functions and datatypes. In addition to modeling existing differentiable algorithms, such as a differentiable ray tracer for implicit surfaces, without requiring any user-level differentiation code, we demonstrate new differentiable algorithms, such as the Hausdorff distance of generalized parametric surfaces.