  Adversarial Regression is a proposition to perform high dimensional
non-linear regression with uncertainty estimation. We used Conditional
Generative Adversarial Network to obtain an estimate of the full predictive
distribution for a new observation. Generative Adversarial Networks (GAN) are
implicit generative models which produce samples from a distribution
approximating the distribution of the data. The conditional version of it
(CGAN) takes the following expression: $\min\limits_G \max\limits_D V(D, G) =
\mathbb{E}_{x\sim p_{r}(x)} [log(D(x, y))] + \mathbb{E}_{z\sim p_{z}(z)} [log
(1-D(G(z, y)))]$. An approximate solution can be found by training
simultaneously two neural networks to model D and G and feeding G with a random
noise vector $z$. After training, we have that $G(z, y)\mathrel{\dot\sim}
p_{data}(x, y)$. By fixing $y$, we have $G(z|y) \mathrel{\dot\sim}
p{data}(x|y)$. By sampling $z$, we can therefore obtain samples following
approximately $p(x|y)$, which is the predictive distribution of $x$ for a new
$y$. We ran experiments to test various loss functions, data distributions,
sample size, size of the noise vector, etc. Even if we observed differences, no
experiment outperformed consistently the others. The quality of CGAN for
regression relies on fine-tuning a range of hyperparameters. In a broader view,
the results show that CGANs are very promising methods to perform uncertainty
estimation for high dimensional non-linear regression.
