On the complexity of nonsmooth automatic differentiation
This work addresses the efficiency of gradient computation in machine learning, particularly for nonsmooth functions common in neural networks, offering theoretical insights that could impact optimization algorithms, though it appears incremental by extending existing principles to nonsmooth cases.
The paper tackles the computational complexity of automatic differentiation for nonsmooth programs, showing that the backward mode's overhead is dimension-independent for certain function classes, extending the smooth cheap gradient principle. It establishes fast backpropagation for neural networks with standard activations and losses, contrasting with forward approaches that have dimension-dependent worst-case overhead.
Using the notion of conservative gradient, we provide a simple model to estimate the computational costs of the backward and forward modes of algorithmic differentiation for a wide class of nonsmooth programs. The overhead complexity of the backward mode turns out to be independent of the dimension when using programs with locally Lipschitz semi-algebraic or definable elementary functions. This considerably extends Baur-Strassen's smooth cheap gradient principle. We illustrate our results by establishing fast backpropagation results of conservative gradients through feedforward neural networks with standard activation and loss functions. Nonsmooth backpropagation's cheapness contrasts with concurrent forward approaches, which have, to this day, dimensional-dependent worst-case overhead estimates. We provide further results suggesting the superiority of backward propagation of conservative gradients. Indeed, we relate the complexity of computing a large number of directional derivatives to that of matrix multiplication, and we show that finding two subgradients in the Clarke subdifferential of a function is an NP-hard problem.