Distribution Theoretic Semantics for Non-Smooth Differentiable Programming

Abstract: With the wide spread of deep learning and gradient descent inspired optimization algorithms, differentiable programming has gained traction. Nowadays it has found applications in many different areas as well, such as scientific computing, robotics, computer graphics and others. One of its notoriously difficult problems consists in interpreting programs that are not differentiable everywhere. In this work we define $\lambda_\delta$, a core calculus for non-smooth differentiable programs and define its semantics using concepts from distribution theory, a well-established area of functional analysis. We also show how $\lambda_\delta$ presents better equational properties than other existing semantics and use our semantics to reason about a simplified ray tracing algorithm. Further, we relate our semantics to existing differentiable languages by providing translations to and from other existing differentiable semantic models. Finally, we provide a proof-of-concept implementation in PyTorch of the novel constructions in this paper.