  Kernel conditional mean embeddings (CMEs) offer a powerful framework for
representing conditional distribution, but they often face scalability and
expressiveness challenges. In this work, we propose a new method that
effectively combines the strengths of deep learning with CMEs in order to
address these challenges. Specifically, our approach leverages the end-to-end
neural network (NN) optimization framework using a kernel-based objective. This
design circumvents the computationally expensive Gram matrix inversion required
by current CME methods. To further enhance performance, we provide efficient
strategies to optimize the remaining kernel hyperparameters. In conditional
density estimation tasks, our NN-CME hybrid achieves competitive performance
and often surpasses existing deep learning-based methods. Lastly, we showcase
its remarkable versatility by seamlessly integrating it into reinforcement
learning (RL) contexts. Building on Q-learning, our approach naturally leads to
a new variant of distributional RL methods, which demonstrates consistent
effectiveness across different environments.
