Current PAC-Bayes generalisation bounds are restricted to scalar metrics of performance, such as the loss or error rate. However, one ideally wants more information-rich certificates that control the entire distribution of possible outcomes, such as the distribution of the test loss in regression, or the probabilities of different mis classifications. We provide the first PAC-Bayes bound capable of providing such rich information by bounding the Kullback-Leibler divergence between the empirical and true probabilities of a set of M error types, which can either be discretized loss values for regression, or the elements of the confusion matrix (or a partition thereof) for classification. We transform our bound into a differentiable training objective. Our bound is especially useful in cases where the severity of different mis-classifications may change over time; existing PAC-Bayes bounds can only bound a particular pre-decided weighting of the error types. In contrast our bound implicitly controls all uncountably many weightings simultaneously.

翻译：当前PAC-Bayes泛化界仅限于标量性能指标（如损失或错误率）。然而，理想情况下需要信息更丰富的认证，能够控制整个可能结果的分布，例如回归中测试损失的分布或不同分类错误的概率。我们首次提出一种PAC-Bayes界，该界通过约束M类误差类型的经验概率与真实概率之间的Kullback-Leibler散度，提供此类丰富信息——其中误差类型可以是回归中离散化的损失值，也可以是分类中混淆矩阵（或其分区）的元素。我们将该界转化为可微的训练目标。该界在以下场景中尤为有用：当不同分类错误的严重性随时间变化时，现有PAC-Bayes界仅能约束预先确定的误差类型加权组合；而我们的界能同时隐式控制所有不可数无穷多种加权组合。