Modern machine learning usually involves predictors in the overparametrised setting (number of trained parameters greater than dataset size), and their training yield not only good performances on training data, but also good generalisation capacity. This phenomenon challenges many theoretical results, and remains an open problem. To reach a better understanding, we provide novel generalisation bounds involving gradient terms. To do so, we combine the PAC-Bayes toolbox with Poincar\'e and Log-Sobolev inequalities, avoiding an explicit dependency on dimension of the predictor space. Our results highlight the positive influence of \emph{flat minima} (being minima with a neighbourhood nearly minimising the learning problem as well) on generalisation performances, involving directly the benefits of the optimisation phase.

现代机器学习通常涉及超参数设置（训练参数数量大于数据集大小），它们的训练结果不仅在训练数据上表现良好，而且具有良好的泛化能力。本文通过与PAC-Bayes工具箱相结合的Poincaré和Log-Sobolev不等式提供了新的包含梯度项的泛化界限，避免了对预测空间维度的明确依赖。我们的结果突出了“平坦极小值”对泛化性能的积极影响，直接涉及了优化阶段的益处。

泛化与平坦极小之间的PAC-Bayesian联系