Cycle Consistency Loss is a type of loss used for generative adversarial networks that performs unpaired image-to-image translation. It was introduced with the CycleGAN architecture. For two domains $X$ and $Y$, we want to learn a mapping $G : X \rightarrow Y$ and $F: Y \rightarrow X$. We want to enforce the intuition that these mappings should be reverses of each other and that both mappings should be bijections. Cycle Consistency Loss encourages $F\left(G\left(x\right)\right) \approx x$ and $G\left(Y\left(y\right)\right) \approx y$. It reduces the space of possible mapping functions by enforcing forward and backwards consistency:

$$ \mathcal{L}_{cyc}\left(G, F\right) = \mathbb{E}_{x \sim p_{data}\left(x\right)}\left[||F\left(G\left(x\right)\right) - x||_{1}\right] + \mathbb{E}_{y \sim p_{data}\left(y\right)}\left[||G\left(F\left(y\right)\right) - y||_{1}\right] $$