  Dual learning has been successfully applied in many machine learning
applications including machine translation, image-to-image transformation, etc.
The high-level idea of dual learning is very intuitive: if we map an $x$ from
one domain to another and then map it back, we should recover the original $x$.
Although its effectiveness has been empirically verified, theoretical
understanding of dual learning is still very limited. In this paper, we aim at
understanding why and when dual learning works. Based on our theoretical
analysis, we further extend dual learning by introducing more related mappings
and propose multi-step dual learning, in which we leverage feedback signals
from additional domains to improve the qualities of the mappings. We prove that
multi-step dual learn-ing can boost the performance of standard dual learning
under mild conditions. Experiments on WMT 14 English$\leftrightarrow$German and
MultiUNEnglish$\leftrightarrow$French translations verify our theoretical
findings on dual learning, and the results on the translations among English,
French, and Spanish of MultiUN demonstrate the effectiveness of multi-step dual
learning.
