局部正则化并不表征多类 PAC 可学习性
Local Regularization Does Not Characterize Multiclass PAC Learnability
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中文总结 AI 辅助
研究局部正则化是否表征多类 PAC 可学习性，通过构建最多二维的可数类，其 PAC 样本复杂度为\[O\!\left(\frac{1}{\varepsilon}\log\frac{1}{\delta}\right)\]，证明没有局部正则化器能学习，给出了否定答案。
中文摘要 AI 辅助
局部正则化为每个假设分配一个依赖于测试点的分数，并使用与样本一致的最低分数假设进行预测。阿西利斯等人询问这个原则是否表征多类 PAC 可学习性。我们给出了否定答案。存在一个最多二维的可数类，其具有可实现的 PAC 样本复杂度\[O\!\left(\frac{1}{\varepsilon}\log\frac{1}{\delta}\right)\]，但没有局部正则化器能学习。假设是完全图的边，实例是竞赛图。在测试竞赛图中，分数确定边的排名，而训练样本独立地排除竞争者。循环三角形迫使足够多的反转，使得存活的竞争者在任意大的样本量下产生恒定的总体误差。
英文摘要
Local regularization assigns each hypothesis a test-point-dependent score and predicts with a minimum-score hypothesis consistent with the sample. Asilis et al. asked whether this principle characterizes multiclass PAC learnability. We give a negative answer. There is a countable class of Daniely--Shalev-Shwartz dimension at most two with realizable PAC sample complexity \[ O\!\left(\frac{1}{\varepsilon}\log\frac{1}δ\right), \] that no local regularizer learns. Hypotheses are edges of complete graphs and instances are tournaments. At a test tournament, the scores fix an edge ranking while the training sample independently removes competitors. Cyclic triangles force enough inversions that surviving competitors produce constant population error at arbitrarily large sample sizes.
