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On Correctness of Automatic Differentiation for Non-Differentiable Functions

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Abstract

Differentiation lies at the core of many machine-learning algorithms, and is wellsupported by popular autodiff systems, such as TensorFlow and PyTorch. Originally, these systems have been developed to compute derivatives of differentiable functions, but in practice, they are commonly applied to functions with non-differentiabilities. For instance, neural networks using ReLU define nondifferentiable functions in general, but the gradients of losses involving those functions are computed using autodiff systems in practice. This status quo raises a natural question: are autodiff systems correct in any formal sense when they are applied to such non-differentiable functions? In this paper, we provide a positive answer to this question. Using counterexamples, we first point out flaws in oftenused informal arguments, such as: non-differentiabilities arising in deep learning do not cause any issues because they form a measure-zero set. We then investigate a class of functions, called PAP functions, that includes nearly all (possibly nondifferentiable) functions in deep learning nowadays. For these PAP functions, we propose a new type of derivatives, called intensional derivatives, and prove that these derivatives always exist and coincide with standard derivatives for almost all inputs. We also show that these intensional derivatives are what most autodiff systems compute or try to compute essentially. In this way, we formally establish the correctness of autodiff systems applied to non-differentiable functions.
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Dates and versions

hal-03081582 , version 1 (18-12-2020)

Identifiers

  • HAL Id : hal-03081582 , version 1

Cite

Wonyeol Lee, Hangyeol Yu, Xavier Rival, Hongseok Yang. On Correctness of Automatic Differentiation for Non-Differentiable Functions. NeurIPS 2020 - 34th Conference on Neural Information Processing Systems, Dec 2020, Vancouver / Virtual, Canada. ⟨hal-03081582⟩
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