Learning Conditional Invariances through Non-Commutativity
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Keywords: Invariance Learning, Domain Adaptation
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TL;DR: Non-commutatively mapping source domain samples to the representation space of the target domain can efficiently learn conditional invariances, satisfying the sample-complexity needs for generalization on the target with samples from the source.
Abstract: 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.
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Primary Area: learning theory
Submission Number: 1260
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