这篇论文从PAC-Bayes理论推导出一个基于Jacobian和Hessian的新正则化器，在CIFAR-10上验证有效，值得搞泛化理论的人看看。
本文研究利用平滑损失函数对PAC-Bayes界进行去随机化，以获得确定性预测器的高概率泛化界。通过后验均值从Gibbs预测器到确定性预测器的代价由Jensen gap类的泛化差距给出，并通过Rademacher复杂度控制。得到的界涉及参数Jacobian和得分映射Hessian表示的平坦度量，适用于有界和无界平滑损失，并特例化为线性预测器和平滑神经网络。理论中的Jacobian和Hessian量启发了一个实用的正则化器，对BatchNorm网络在CIFAR-10上进行了不同批量大小下的实验。
Smoothness-Based Derandomization of PAC-Bayes Bounds
We study PAC-Bayes derandomization for smooth loss functions. Our goal is to obtain generalization bounds that hold with high probability for deterministic predictors by exploiting smoothness properties of both the loss and the predictor class. We show that passing from the Gibbs predictor to the deterministic predictor at the posterior mean has a precise cost, given by the generalization gap of the Jensen gap class. We control this class through its Rademacher complexity, leading to bounds for deterministic predictors that involve flatness quantities expressed in terms of parameter Jacobians and Hessians of the score map. The framework applies to both bounded and unbounded smooth loss functions, and we specialize the results to linear predictors and smooth neural networks. Finally, the Jacobian and Hessian quantities appearing in the theory motivate a practical regularizer. For BatchNorm networks, we compute this regularizer with respect to effective BatchNorm weights obtained by folding the BatchNorm transformation into the adjacent affine weights. Experiments on CIFAR-10 illustrate the behavior of this regularizer under different batch sizes.