为带噪声输入的神经网络输出分布提供精确上下界，确保预测误差可保证。
Exact Upper and Lower Bounds for the Output Distribution of Neural Networks with Random Inputs
- 用ReLU神经网络逼近通用神经网络，构建输出分布上下界。
- 在任意分辨率下，上下界均收敛到真实累积分布函数。
- 适用于ReLU、tanh等激活函数，适合需要可靠推理的场景。
我们推导了在随机输入下神经网络输出在整个支持集上的累积分布函数（CDF）的精确上下界。随着分辨率提升，上下界逐步收敛至真实CDF。该方法适用于任意前馈神经网络及卷积神经网络，激活函数需为连续单调分段二阶连续可微（如ReLU、tanh、softmax）。其核心创新在于将一般神经网络用ReLU神经网络进行逼近，再基于此构建输出分布的上下界。实验表明，本方法能在支持集上提供预测输出分布的保证性边界，实现精确误差控制，优于现有方法。
原文摘要 · Abstract (English)
We derive exact upper and lower bounds for the cumulative distribution function (cdf) of the output of a neural network (NN) over its entire support subject to noisy (stochastic) inputs. The upper and lower bounds converge to the true cdf over its domain as the resolution increases. Our method applies to any feedforward NN using continuous monotonic piecewise twice continuously differentiable activation functions (e.g., ReLU, tanh and softmax) and convolutional NNs, which were beyond the scope of competing approaches. The novelty and instrumental tool of our approach is to bound general NNs with ReLU NNs. The ReLU NN-based bounds are then used to derive the upper and lower bounds of the cdf of the NN output. Experiments demonstrate that our method delivers guaranteed bounds of the predictive output distribution over its support, thus providing exact error guarantees, in contrast to competing approaches.
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