Abstract: 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 dataset consists of majorly unlabeled data and a small set of data with noisy reward labels. Our approach leverages a learned reward function on the smaller data set as a pseudolabeler to label the unlabelled data. After pseudo-labelling, a conditional diffusion model (CDM) is trained on the data and samples are generated by setting a target value $a$ as the condition in CDM. From a theoretical standpoint, we show that this directed generator can effectively learn and sample from the reward-conditioned data distribution: 1. our model is capable of recovering the data's latent subspace representation. 2. the model generates samples moving closer to the user-specified target. The improvement in rewards of samples is influenced by a 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.