# What is the confusion loss for adversarial learning?

What is the confusion loss used in domain adaptation (DA) for adversarial learning/GANs? See this paper.

Two domains:

• $$s$$: source domain
• $$t$$: target domain

Generator/Discriminator setting:

• $$M_s:x_s\rightarrow f_t$$ generator of feature for the source domain; and $$M_t: x_t\rightarrow f_t$$ generator of feature for the target domain
• $$D:f\rightarrow [0,1]$$: discriminator, 1 if the features came from the source domain, 0 if from the target domain

The generator tries to minimize (equation (8) in the paper):

$$\mathcal{L}_{\mathrm{adv}_{M}}\left(\mathbf{X}_{s}, \mathbf{X}_{t}, D\right)=$$ $$-\sum_{d \in\{s, t\}} \mathbb{E}_{\mathbf{x}_{d} \sim \mathbf{X}_{d}}\left[\frac{1}{2} \log D\left(M_{d}\left(\mathbf{x}_{d}\right)\right)\right.$$ $$\left.+\frac{1}{2} \log \left(1-D\left(M_{d}\left(\mathbf{x}_{d}\right)\right)\right)\right]$$

If the samples come from $$s$$, then $$d=s$$; the generator tries to maximize the first part $$\log D(M_s(x_s))$$, which is doing the same thing as the discriminator? And tries to maximize $$\log(1-D(M_s(x_s))$$, which is the classical generator loss function (so the opposite of the first part, and I do not understand that). And the inverse for when $$d=t$$.