Distribution Theoretic Semantics for Non-Smooth Differentiable Programming (Chris Lam, UIUC CS)
With the rise in differentiable learning methods and differentiable ray tracing algorithms, computer scientists have begun turning their attention towards algorithms and semantics for differentiation operators. The major victory on this front has been the development of the theory of automatic differentiation, which efficiently computes derivatives of functions in very high dimensions via repeated applications of the chain rule along with some computation saving tricks. However, because this algorithm is inductively defined on the syntax of the language, it is vulnerable to two failure cases: functions that can be defined in multiple ways via conditional statements, and non-differentiable functions. In this talk, we will introduce some background on categorical programming language semantics, followed by a discussion of how to solve this problem by introducing Schwartz distributions to the semantics of a differentiable language via the category of diffeological spaces.