  We explore the methodology and theory of reward-directed generation via
conditional diffusion models. Directed generation aims to generate samples with
desired properties as measured by a reward function, which has broad
applications in generative AI, reinforcement learning, and computational
biology. We consider the common learning scenario where the data set consists
of unlabeled data along with a smaller set of data with noisy reward labels.
Our approach leverages a learned reward function on the smaller data set as a
pseudolabeler. From a theoretical standpoint, we show that this directed
generator can effectively learn and sample from the reward-conditioned data
distribution. Additionally, our model is capable of recovering the latent
subspace representation of data. Moreover, we establish that the model
generates a new population that moves closer to a user-specified target reward
value, where the optimality gap aligns with the off-policy bandit regret in the
feature subspace. The improvement in rewards obtained is influenced by the
interplay between the strength of the reward signal, the distribution shift,
and the cost of off-support extrapolation. We provide empirical results to
validate our theory and highlight the relationship between the strength of
extrapolation and the quality of generated samples.
