We investigate the fundamental performance limitations of learning algorithms in several Domain Generalisation (DG) settings. Motivated by the difficulty with which previously proposed methods have in reliably outperforming Empirical Risk Minimisation (ERM), we derive upper bounds on the excess risk of ERM, and lower bounds on the minimax excess risk. Our findings show that in all the DG settings we consider, it is not possible to significantly outperform ERM. Our conclusions are limited not only to the standard covariate shift setting, but also two other settings with additional restrictions on how domains can differ. The first constrains all domains to have a non-trivial bound on pairwise distances, as measured by a broad class of integral probability metrics. The second alternate setting considers a restricted class of DG problems where all domains have the same underlying support. Our analysis also suggests how different strategies can be used to optimise the performance of ERM in each of these DG setting. We also experimentally explore hypotheses suggested by our theoretical analysis.

翻译：本文研究了多种领域泛化（DG）场景下学习算法的基本性能局限。鉴于先前提出的方法难以稳定超越经验风险最小化（ERM）的表现，我们推导了ERM超额风险的上界以及极小极大超额风险的下界。研究结果表明，在我们考察的所有DG场景中，算法性能均无法显著超越ERM。这一结论不仅适用于标准的协变量偏移场景，还适用于另外两种对领域差异施加额外约束的场景：第一种场景要求所有领域在积分概率度量定义的成对距离上存在非平凡上界；第二种受限场景考虑所有领域具有相同底层支撑集的DG问题。我们的分析同时揭示了如何针对不同DG场景采用相应策略优化ERM性能。最后通过实验验证了理论分析提出的假设。