We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of polynomials.

中文翻译：

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PCF 中的自动微分

我们在正向和反向模式下，在高阶图灵完备语言（具有实数的 PCF）的上下文中研究自动微分 (AD) 的正确性。我们的主要结果是，在对语言中包含的原始函数的温和假设下，AD 几乎在所有地方都是正确的，也就是说，它计算所考虑程序的导数或梯度，除了一组 Lebesgue 测度为零。换句话说，存在 AD 不正确的输入，但随机选择一个这样的输入的概率为零。我们的结果实际上更精确，因为失败点集有更明确的描述：例如，如果原始函数只是常量、加法和乘法，则 AD 失败的点集包含在可数联合中多项式的零集。