Provably Correct Automatic Sub-Differentiation for Qualified Programs

The Cheap Gradient Principle [Griewank and Walther, 2008] -- the computational cost of computing the gradient of a scalar-valued function is nearly the same (often within a factor of 5) as that of simply computing the function itself -- is of central importance in optimization; it allows us to quickly obtain (high dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing subderivatives: widely used ML libraries, including TensorFlow and PyTorch, do not correctly compute (generalized) subderivatives even on simple examples. This work considers the question: is there a Cheap Subgradient Principle? Our main result shows that, under certain restrictions on our library of nonsmooth functions (standard in nonlinear programming), provably correct generalized subderivatives can be computed at a computational cost that is within a (dimension-free) factor of 6 of the cost of computing the scalar function itself.