Invariance learning algorithms that conditionally filter out domain-specific random variables as distractors, do so based only on the data semantics, and not the target domain under evaluation. We show that a provably optimal and sample-efficient way of learning conditional invariances is by relaxing the invariance criterion to be non-commutatively directed towards the target domain. Under domain asymmetry, i.e., when the target domain contains semantically relevant information absent in the source, the risk of the encoder $\varphi^*$ that is optimal on average across domains is strictly lower-bounded by the risk of the target-specific optimal encoder $\Phi^*_\tau$. We prove that non-commutativity steers the optimization towards $\Phi^*_\tau$ instead of $\varphi^*$, bringing the $\mathcal{H}$-divergence between domains down to zero, leading to a stricter bound on the target risk. Both our theory and experiments demonstrate that non-commutative invariance (NCI) can leverage source domain samples to meet the sample complexity needs of learning $\Phi^*_\tau$, surpassing SOTA invariance learning algorithms for domain adaptation, at times by over $2\%$, approaching the performance of an oracle. Implementation is available at https://github.com/abhrac/nci.

翻译：条件不变性学习算法通过基于数据语义（而非目标评估域）有条件地过滤掉领域特定的随机变量作为干扰项。我们证明，学习条件不变性的一个可证明最优且样本高效的方式是，将不变性准则放宽为面向目标域的非交换方向性。在领域不对称情况下，即当目标域包含源域中缺失的语义相关信息时，在跨域平均意义下最优的编码器$\varphi^*$的风险严格受限于目标域特定最优编码器$\Phi^*_\tau$的风险。我们证明，非交换性将优化方向从$\varphi^*$引导至$\Phi^*_\tau$，使领域间的$\mathcal{H}$-散度降至零，从而得到更严格的目标风险上限。我们的理论和实验均表明，非交换不变性（NCI）能够利用源域样本满足学习$\Phi^*_\tau$的样本复杂度需求，在领域适应任务上超越现有最优的不变性学习算法，有时性能提升超过$2\%$，接近或acles性能。实现代码见https://github.com/abhrac/nci。