  Human can extrapolate well, generalize daily knowledge into unseen scenarios,
raise and answer counterfactual questions. To imitate this ability via
generative models, previous works have extensively studied explicitly encoding
Structural Causal Models (SCMs) into architectures of generator networks. This
methodology, however, limits the flexibility of the generator as they must be
carefully crafted to follow the causal graph, and demands a ground truth SCM
with strong ignorability assumption as prior, which is a nontrivial assumption
in many real scenarios. Thus, many current causal GAN methods fail to generate
high fidelity counterfactual results as they cannot easily leverage
state-of-the-art generative models. In this paper, we propose to study
counterfactual synthesis from a new perspective of knowledge extrapolation,
where a given knowledge dimension of the data distribution is extrapolated, but
the remaining knowledge is kept indistinguishable from the original
distribution. We show that an adversarial game with a closed-form discriminator
can be used to address the knowledge extrapolation problem, and a novel
principal knowledge descent method can efficiently estimate the extrapolated
distribution through the adversarial game. Our method enjoys both elegant
theoretical guarantees and superior performance in many scenarios.
