  We propose a conditional stochastic interpolation (CSI) approach to learning
conditional distributions. CSI learns probability flow equations or stochastic
differential equations that transport a reference distribution to the target
conditional distribution. This is achieved by first learning the drift function
and the conditional score function based on conditional stochastic
interpolation, which are then used to construct a deterministic process
governed by an ordinary differential equation or a diffusion process for
conditional sampling. In our proposed CSI model, we incorporate an adaptive
diffusion term to address the instability issues arising during the training
process. We provide explicit forms of the conditional score function and the
drift function in terms of conditional expectations under mild conditions,
which naturally lead to an nonparametric regression approach to estimating
these functions. Furthermore, we establish non-asymptotic error bounds for
learning the target conditional distribution via conditional stochastic
interpolation in terms of KL divergence, taking into account the neural network
approximation error. We illustrate the application of CSI on image generation
using a benchmark image dataset.
